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/@rhi{10 div/rhi X/rhiSeen true N}B/@llx{/llx X}B/@lly{/lly X}B/@urx{
/urx X}B/@ury{/ury X}B/magscale true def end/@MacSetUp{userdict/md known
{userdict/md get type/dicttype eq{userdict begin md length 10 add md
maxlength ge{/md md dup length 20 add dict copy def}if end md begin
/letter{}N/note{}N/legal{}N/od{txpose 1 0 mtx defaultmatrix dtransform S
atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{
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transform{itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll
curveto}}{{closepath}}pathforall newpath counttomark array astore/gc xdf
pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}
if}N/txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1
-1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3
get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip
yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub
neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{
noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop
90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get
neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr
1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr
2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4
-1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S
TR}if}N/cp{pop pop showpage pm restore}N end}if}if}N/normalscale{
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currentpoint TR/psf$ury psfts/psf$urx psfts/psf$lly psfts/psf$llx psfts
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psf$urx psf$llx sub div N/psf$sy psf$y psf$ury psf$lly sub div N psf$sx
psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub
TR/showpage{}N/erasepage{}N/setpagedevice{pop}N/copypage{}N/p 3 def
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newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto
closepath clip newpath moveto}N/endTexFig{end psf$SavedState restore}N
/@beginspecial{SDict begin/SpecialSave save N gsave normalscale
currentpoint TR @SpecialDefaults count/ocount X/dcount countdictstack N}
N/@setspecial{CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs
neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate
rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse
scale llx neg lly neg TR}{rhiSeen{rhi ury lly sub div dup scale llx neg
lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx
ury lineto llx ury lineto closepath clip}if/showpage{}N/erasepage{}N
/setpagedevice{pop}N/copypage{}N newpath}N/@endspecial{count ocount sub{
pop}repeat countdictstack dcount sub{end}repeat grestore SpecialSave
restore end}N/@defspecial{SDict begin}N/@fedspecial{end}B/li{lineto}B
/rl{rlineto}B/rc{rcurveto}B/np{/SaveX currentpoint/SaveY X N 1
setlinecap newpath}N/st{stroke SaveX SaveY moveto}N/fil{fill SaveX SaveY
moveto}N/ellipse{/endangle X/startangle X/yrad X/xrad X/savematrix
matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc
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%%EndProcSet
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueID 5000791 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR12
%!PS-AdobeFont-1.1: CMR12 1.0
%%CreationDate: 1991 Aug 20 16:38:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-34 -251 988 750}readonly def
/UniqueID 5000794 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueID 5000820 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTI12
%!PS-AdobeFont-1.1: CMTI12 1.0
%%CreationDate: 1991 Aug 18 21:06:53
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-36 -251 1103 750}readonly def
/UniqueID 5000829 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTT9
%!PS-AdobeFont-1.1: CMTT9 1.0
%%CreationDate: 1991 Aug 20 16:46:24
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTT9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch true def
end readonly def
/FontName /CMTT9 def
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/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-6 -233 542 698}readonly def
/UniqueID 5000831 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI6
%!PS-AdobeFont-1.1: CMMI6 1.100
%%CreationDate: 1996 Jul 23 07:53:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{11 -250 1241 750}readonly def
/UniqueID 5087381 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTI9
%!PS-AdobeFont-1.1: CMTI9 1.0
%%CreationDate: 1991 Aug 18 21:08:07
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI9 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-35 -250 1148 750}readonly def
/UniqueID 5000827 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI12
%!PS-AdobeFont-1.1: CMMI12 1.100
%%CreationDate: 1996 Jul 27 08:57:55
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
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end readonly def
/FontName /CMMI12 def
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/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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/FontBBox{-30 -250 1026 750}readonly def
/UniqueID 5087386 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX7
%!PS-AdobeFont-1.1: CMBX7 1.0
%%CreationDate: 1991 Aug 20 16:35:49
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-55 -250 1289 751}readonly def
/UniqueID 5000765 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: LINEW10
%!PS-AdobeFont-1.1: LINEW10 1.001
%%CreationDate: 1992 Oct 23 20:22:06
%%RevisionDate: 2001 Jun 05 20:22:06
% Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.001) readonly def
/Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def
/FullName (LINEW10) readonly def
/FamilyName (LaTeX) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /LINEW10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /a0 put
dup 64 /a64 put
readonly def
/FontBBox{-220 -220 1040 1040}readonly def
/UniqueID 5011941 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTT10
%!PS-AdobeFont-1.1: CMTT10 1.00B
%%CreationDate: 1992 Apr 26 10:42:42
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTT10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch true def
end readonly def
/FontName /CMTT10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-4 -235 731 800}readonly def
/UniqueID 5000832 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR6
%!PS-AdobeFont-1.1: CMR6 1.0
%%CreationDate: 1991 Aug 20 16:39:02
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-20 -250 1193 750}readonly def
/UniqueID 5000789 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMEX10
%!PS-AdobeFont-1.1: CMEX10 1.00
%%CreationDate: 1992 Jul 23 21:22:48
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMEX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMEX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 26 /braceleftbigg put
dup 80 /summationtext put
dup 81 /producttext put
dup 88 /summationdisplay put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueID 5000774 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI9
%!PS-AdobeFont-1.1: CMMI9 1.100
%%CreationDate: 1996 Jul 23 07:53:55
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI9 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-29 -250 1075 750}readonly def
/UniqueID 5087384 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY9
%!PS-AdobeFont-1.1: CMSY9 1.0
%%CreationDate: 1991 Aug 15 07:22:27
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY9 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-30 -958 1146 777}readonly def
/UniqueID 5000819 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: LINE10
%!PS-AdobeFont-1.1: LINE10 1.001
%%CreationDate: 1992 Oct 23 20:22:05
%%RevisionDate: 2001 Jun 05 20:22:05
% Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.001) readonly def
/Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def
/FullName (LINE10) readonly def
/FamilyName (LaTeX) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /LINE10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 27 /a27 put
dup 45 /a45 put
dup 63 /a63 put
readonly def
/FontBBox{-150 -150 1020 1020}readonly def
/UniqueID 5011940 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY5
%!PS-AdobeFont-1.1: CMSY5 1.0
%%CreationDate: 1991 Aug 15 07:21:16
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY5 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{21 -944 1448 791}readonly def
/UniqueID 5000815 def
currentdict end
currentfile eexec
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29969FA66B8D10FF51F5F9C785876AF93182741AAB34573F1F5F6C6471223443
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cleartomark
%%EndFont 
%%BeginFont: LASY10
%!PS-AdobeFont-1.1: LASY10 1.001
%%CreationDate: 1992 Oct 23 20:19:17
%%RevisionDate: 2001 Jun 05 20:19:17
% Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.001) readonly def
/Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def
/FullName (LASY10) readonly def
/FamilyName (LaTeX) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /LASY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 50 /a50 put
readonly def
/FontBBox{-19 -192 944 683}readonly def
/UniqueID 5011949 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMCSC10
%!PS-AdobeFont-1.1: CMCSC10 1.0
%%CreationDate: 1991 Aug 18 17:46:49
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMCSC10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMCSC10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{14 -250 1077 750}readonly def
/UniqueID 5000772 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY7
%!PS-AdobeFont-1.1: CMSY7 1.0
%%CreationDate: 1991 Aug 15 07:21:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMSY7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-15 -951 1252 782}readonly def
/UniqueID 5000817 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI5
%!PS-AdobeFont-1.1: CMMI5 1.100
%%CreationDate: 1996 Aug 02 08:21:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI5 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{37 -250 1349 750}readonly def
/UniqueID 5087380 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR5
%!PS-AdobeFont-1.1: CMR5 1.00B
%%CreationDate: 1992 Feb 19 19:55:02
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR5) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR5 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-341 -250 1304 965}readonly def
/UniqueID 5000788 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR10
%!PS-AdobeFont-1.1: CMR10 1.00B
%%CreationDate: 1992 Feb 19 19:54:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
/UniqueID 5000793 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI7
%!PS-AdobeFont-1.1: CMMI7 1.100
%%CreationDate: 1996 Jul 23 07:53:53
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{0 -250 1171 750}readonly def
/UniqueID 5087382 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR7
%!PS-AdobeFont-1.1: CMR7 1.0
%%CreationDate: 1991 Aug 20 16:39:21
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-27 -250 1122 750}readonly def
/UniqueID 5000790 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI10
%!PS-AdobeFont-1.1: CMMI10 1.100
%%CreationDate: 1996 Jul 23 07:53:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMMI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
/UniqueID 5087385 def
currentdict end
currentfile eexec
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%%BeginFont: LCIRCLE10
%!PS-AdobeFont-1.1: LCIRCLE10 1.001
%%CreationDate: 1992 Oct 23 20:21:59
%%RevisionDate: 2001 Jun 05 20:21:59
% Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.001) readonly def
/Notice (Copyright (C) 1997, 2001 American Mathematical Society. All Rights Reserved) readonly def
/FullName (LCIRCLE10) readonly def
/FamilyName (LaTeX) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /LCIRCLE10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 24 /a24 put
dup 25 /a25 put
dup 26 /a26 put
dup 27 /a27 put
dup 32 /a32 put
dup 33 /a33 put
dup 34 /a34 put
dup 35 /a35 put
dup 107 /a107 put
readonly def
/FontBBox{-2000 -2000 2040 2040}readonly def
/UniqueID 5011942 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR9
%!PS-AdobeFont-1.1: CMR9 1.0
%%CreationDate: 1991 Aug 20 16:39:59
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR9 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-39 -250 1036 750}readonly def
/UniqueID 5000792 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTI10
%!PS-AdobeFont-1.1: CMTI10 1.00B
%%CreationDate: 1992 Feb 19 19:56:16
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMTI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-163 -250 1146 969}readonly def
/UniqueID 5000828 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX10
%!PS-AdobeFont-1.1: CMBX10 1.00B
%%CreationDate: 1992 Feb 19 19:54:06
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMBX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-301 -250 1164 946}readonly def
/UniqueID 5000768 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR17
%!PS-AdobeFont-1.1: CMR17 1.0
%%CreationDate: 1991 Aug 20 16:38:24
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def
/FullName (CMR17) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR17 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /.notdef put
readonly def
/FontBBox{-33 -250 945 749}readonly def
/UniqueID 5000795 def
currentdict end
currentfile eexec
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(J.)f(T)-7 b(romp,)28 b(M.)g(Y)-7 b(annak)i(akis,)26
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(ann)n(ual)h(A)n(CM)h(Sym-)731 2897 y(p)r(osiom)i(on)g(Theory)f(of)h
(Computing)h(\(STOC91\),)f(New)g(Orleans,)f(Ma)n(y)h(1991,)e(pp.)731
2997 y(328{336.)k FG(Journal)30 b(of)g(the)g(A)n(CM)p
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b FK(G.)21 b(Kissin,)g(J.)f(T)-7 b(romp,)20 b FG(The)25
b(ener)l(gy)f(c)l(omplexity)g(of)g(thr)l(eshold)h(and)f(other)731
3263 y(functions)p FK(,)j(CWI)h(T)-7 b(ec)n(hnical)27
b(Rep)r(ort)h(CS-R9101,)d(Jan)n(uary)h(1991.)523 3429
y FH(Chapter)33 b(5:)40 b FK(J.)e(T)-7 b(romp,)37 b(P)-7
b(.)37 b(v)-5 b(an)37 b(Emde-Boas,)f FG(Asso)l(ciative)k(Stor)l(age)f
(Mo)l(di\014c)l(ation)731 3528 y(Machines)p FK(,)26 b(in:)f
(\\Complexit)n(y)e(Theory",)g(Am)n(b)r(os-Spies,)h(Homer,)g(and)g(Sc)
3115 3507 y(\177)3113 3528 y(honing)731 3628 y(\(editors\),)j(Cam)n
(bridge)f(Univ)n(ersit)n(y)h(Press.)523 3794 y FH(Chapter)33
b(6:)40 b FK(J.)e(T)-7 b(romp,)37 b FG(How)i(to)g(Construct)f(an)h(A)n
(tomic)g(V)-6 b(ariable)p FK(,)39 b(Pro)r(c.)e(of)g(the)731
3894 y(3rd)24 b(In)n(ternational)g(W)-7 b(orkshop)24
b(on)h(Distributed)h(Algorithms,)f FG(L)l(e)l(ctur)l(e)h(Notes)i(in)731
3993 y(Computer)i(Scienc)l(e)g(392)p FK(,)f(Springer-V)-7
b(erlag,)25 b(pp.)j(292{302,)c(1989.)523 4159 y FH(Chapter)33
b(7:)40 b FK(M.)22 b(Li,)f(J.)g(T)-7 b(romp,)21 b(P)-7
b(.M.B.)21 b(Vit\023)-42 b(an)n(yi,)21 b FG(How)j(to)g(Shar)l(e)g
(Concurr)l(ent)g(Wait-)731 4259 y(F)-6 b(r)l(e)l(e)29
b(V)-6 b(ariables)p FK(,)29 b(\(under)f(revision)e(for)h
FG(Journal)j(of)g(the)g(A)n(CM)p FK(\).)523 4425 y FH(Chapter)j(8:)40
b FK(J.)26 b(T)-7 b(romp,)25 b(J-H.)h(Ho)r(epman,)g FG(Binary)j
(Snapshots)p FK(,)d(Pro)r(c.)f(of)h(the)g(5th)g(In-)731
4525 y(ternational)d(W)-7 b(orkshop)23 b(on)h(Distributed)h
(Algorithms,)e FG(L)l(e)l(ctur)l(e)j(Notes)h(in)f(Com-)731
4624 y(puter)j(Scienc)l(e)h(725)p FK(,)f(Springer-V)-7
b(erlag,)25 b(pp.)j(18{25)d(1993.)523 4790 y FH(Chapter)33
b(9:)40 b FK(J.)32 b(T)-7 b(romp,)32 b FG(On)h(Up)l(date-L)l(ast)h
(Schemes)p FK(,)f(P)n(arallel)d(Pro)r(cessing)g(Letters)731
4890 y(3\(1\),)d(pp.)h(25{28,)d(1993.)606 5073 y(The)i(co-authors)e(of)
i(the)g(ab)r(o)n(v)n(e)e(pap)r(ers)h(ha)n(v)n(e)g(greatly)f(con)n
(tributed)i(to)g(the)g(thesis.)g(I)523 5172 y(w)n(ould)d(lik)n(e)g(to)h
(thank)f(all)h(of)f(them)i(for)e(their)g(inspiring)g(co)r(op)r
(eration,)g(and)g(the)h(referees)523 5272 y(of)j(the)g(journal)e
(publications)i(for)f(their)g(helpful)i(commen)n(ts.)p
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1802 y(ideas)25 b(with)h(me:)f(Victor)g(Allis,)h(Krzysztof)e(Apt,)j
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(that)h(is)f(roughly)f(prop)r(ortional)g(to)h(the)h(length)523
3595 y(of)27 b(the)h(input)g(in)g(some)f(standard)f(notation.)h(When)h
(inputs)g(are)e(w)n(ords)g(in)i(some)f(input)523 3694
y(language,)21 b(then)j(their)e(size)h(is)g(simply)f(de\014ned)i(as)e
(their)g(length.)h(If)h(an)e(input)i(is)f(a)f(graph)523
3794 y(for)g(instance,)g(then)i(the)f(n)n(um)n(b)r(er)f(of)g(no)r(des)h
(plus)f(the)h(n)n(um)n(b)r(er)g(of)f(edges)g(is)h(a)f(reasonable)523
3894 y(de\014nition)k(of)g(size.)f(Usually)-7 b(,)26
b(the)g(bigger)e(the)i(input,)g(the)g(more)f(resources)f(are)h(needed.)
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(quic)n(kly)f(the)h(use)g(of)g(resources)d(gro)n(ws)h(with)523
4093 y(input)37 b(size:)f(it)g(is)g(a)g(function)g(that)h(giv)n(es)d
(for)i(eac)n(h)f(size)h(the)g(maxim)n(um)g(amoun)n(t)g(of)523
4193 y(resource)24 b(used,)i(o)n(v)n(er)d(all)j(inputs)g(of)g(that)g
(size.)f(Apart)g(from)h(this)g FG(worst-c)l(ase)g FK(measure,)523
4292 y(one)c(can)h(also)e(consider)h(an)g FG(aver)l(age)27
b(c)l(ase)c FK(complexit)n(y)-7 b(,)22 b(where)g(the)h(a)n(v)n(erage)d
(of)j(resource)523 4392 y(used)28 b(is)f(tak)n(en)g(o)n(v)n(er)f(all)h
(inputs.)606 4491 y(Chapter)34 b(4)f(considers)g(an)h(practically)f
(signi\014can)n(t)g(resource)g(for)g(circuits,)h(namely)523
4591 y(energy)39 b(consumption.)h(In)h(con)n(v)n(en)n(tional)d(tec)n
(hnologies,)h(whenev)n(er)g(the)i(input)g(to)f(a)523
4691 y(circuit)28 b(c)n(hanges,)e(some)i(subset)f(of)h(the)h(gates)d
(and)i(wires)f(will)h(switc)n(h)g(and)g(the)g(energy)523
4790 y(dissipated)36 b(hereb)n(y)f(in)i(the)f(from)g(of)g(heat)g(is)g
(prop)r(ortional)e(to)j(the)f(sum)g(area)f(of)h(all)523
4890 y(switc)n(hing)27 b(elemen)n(ts.)606 4990 y(In)39
b FG(r)l(andomize)l(d)h FK(algorithms,)e(c)n(hoices)f(can)i(b)r(e)g
(made)g(based)f(on)g(the)i(outcome)e(of)523 5089 y(random)32
b(coin)g(\015ips.)h(Randomness)f(is)g(used)h(either)f(to)h(ensure)f(a)g
(high)g(probabilit)n(y)g(of)523 5189 y(giving)g(the)h(correct)e(answ)n
(er,)g(or)h(to)h(limit)g(the)g(\(exp)r(ected\))g(time)h(to)e(\014nd)h
(the)g(correct)523 5289 y(answ)n(er.)26 b(In)i(either)f(case,)g(n)n(um)
n(b)r(er)h(of)f(coin)g(\015ips)h(is)g(a)f(useful)h(resource.)p
eop end
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(resource)e(of)h FG(queries)p FK(.)i(A)f(query)e(is)i(a)f(question)523
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(allo)n(w)n(ed)e(questions,)g(to)i(alw)n(a)n(ys)d(b)r(e)523
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f(coin)h(\015ip,)h(this)g(is)f(one)g(external)g(bit)h(of)523
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(w)n(a)n(y)-7 b(.)606 905 y(The)32 b(last)f(`resource')f(w)n(e'll)h
(discuss)g(is)h(the)f(size)h(of)f(the)h(output.)g(This)g(b)r(ecomes)f
(in-)523 1005 y(teresting)d(when)h(for)g(eac)n(h)f(input,)i(more)e
(than)h(one)f(output)i(solution)e(is)h(p)r(ossible.)g(The)523
1104 y(closer)36 b(in)h(size)g(a)g(solution)f(is)h(to)g(the)h(minimal)f
(one,)g(the)g(b)r(etter.)h(Chapter)f(2)f(is)h(ex-)523
1204 y(clusiv)n(ely)c(concerned)g(with)h(this)g(output)h(appro)n
(ximation)d(complexit)n(y)h(for)g(a)h(sp)r(eci\014c)523
1303 y(problem.)523 1611 y FL(1.3)116 b(T)-10 b(rade-o\013s)523
1810 y FK(Complexit)n(y)24 b(measures)f(can)h(rarely)f(b)r(e)i
(considered)f(in)h(isolation.)e(This)i(is)f(b)r(ecause)g(one)523
1910 y(resource)29 b(can)i(often)g(b)r(e)g(minimized)h(at)e(the)i(cost)
e(of)h(sev)n(eral)e(others.)h(Most)h(resources)523 2009
y(can)19 b(b)r(e)h(minimized)g(at)g(the)g(cost)f(of)g(making)g(the)h
(time)g(exp)r(onen)n(tial)f(in)h(the)g(size)f(of)h(input.)523
2109 y(Since)26 b(exp)r(onen)n(tial)g(time)g(\(or)g(for)f(that)i
(matter)f(exp)r(onen)n(tial)f(an)n(ything\))h(is)g(considered)523
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g(to)h(ha)n(v)n(e,)e(this)i(trade-o\013)f(is)h(hardly)523
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b(are)f(in)n(terested)g(in)h(trade-o\013s)f(b)r(et)n(w)n(een)g
(resources)f(that)i(all)523 2408 y(remain)i(p)r(olynomial.)606
2507 y(In)34 b(Chapter)g(3,)f(a)h(trade-o\013)f(is)h(established)f(b)r
(et)n(w)n(een)h(time)h(and)f(space)f(for)g(a)h(par-)523
2607 y(ticular)27 b(exploration-t)n(yp)r(e)f(problem.)i(It)g(is)g(sho)n
(wn)f(that)h(v)-5 b(arious)26 b(di\013eren)n(t)i(algorithms)523
2707 y(ha)n(v)n(e)33 b(the)i(same)f(space-time-pro)r(duct)f(complexit)n
(y)-7 b(,)34 b(up)g(to)g(constan)n(t)g(factors.)f(Some-)523
2806 y(times)h(algorithms)e(are)h(designed)g(with)i(a)e(single)g
(parameter)f(that)i(can)g(b)r(e)g(v)-5 b(aried)33 b(to)523
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(domain)e(of)h(circuits,)g(the)h(familiar)e(time-space)g(trade-o\013)h
(translates)e(in)n(to)i(a)523 3105 y(depth-area)i(tradeo\013.)g(It)i
(is)f(kno)n(wn)f(for)h(instance)f(that)i(a)e(logarithmic)g(depth)i
(circuit)523 3205 y(on)e FC(n)h FK(inputs)g(needs)f(on)g(the)h(order)e
(of)h FC(n)14 b FK(log\()p FC(n)p FK(\))32 b(area)e(to)i(b)r(e)f(em)n
(b)r(edded)h(in)g(the)g(plane)523 3304 y(\(assuming)38
b(the)h FC(n)f FK(inputs)h(lie)g(on)f(a)g(con)n(v)n(ex)f(b)r
(oundary\).)h(If)h(the)g(depth)g(is)g(relaxed,)523 3404
y(then)f(often)f(a)g(linear)f(em)n(b)r(edding)h(is)g(p)r(ossible.)g(F)
-7 b(or)36 b(circuits,)h(logarithmic)f(depth)i(is)523
3504 y(considered)18 b(just)i(as)f(desirable)g(as)f(p)r(olynomial)h
(time)h(is)g(for)e(sequen)n(tial)h(mac)n(hines.)g(Th)n(us,)523
3603 y(in)37 b(Chapter)e(4,)h(the)h(resource)d(of)j(energy)e(is)h
(minimized)h(under)f(the)g(requiremen)n(t)g(of)523 3703
y(logarithmic)26 b(depth.)523 4010 y FL(1.4)116 b(Ov)m(erview)523
4209 y FK(In)31 b(Chapter)g(2,)f(w)n(e)h(consider)e(the)j(follo)n(wing)
e(problem:)g(giv)n(en)g(a)h(collection)f(of)h(strings)523
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4321 y FA(m)886 4309 y FK(,)30 b(\014nd)h(the)g(shortest)e(string)h
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4408 y(\(a)i(consecutiv)n(e)f(blo)r(c)n(k\))g(of)h FC(s)p
FK(.)g(Although)g(this)g(problem)f(is)h(kno)n(wn)f(to)g(b)r(e)i
(NP-hard,)d(a)523 4508 y(simple)24 b(greedy)f(pro)r(cedure)f(app)r
(ears)h(to)h(do)f(quite)h(w)n(ell)g(and)f(is)h(routinely)f(used)h(in)g
(DNA)523 4608 y(sequencing)k(and)h(data)g(compression)f(practice,)g
(namely:)h(rep)r(eatedly)g(merge)f(the)h(pair)523 4707
y(of)h(\(distinct\))g(strings)f(with)h(maxim)n(um)f(o)n(v)n(erlap)f(un)
n(til)i(only)f(one)g(string)g(remains.)g(Let)523 4807
y FC(n)23 b FK(denote)g(the)g(length)h(of)f(an)f(optimal)h
(\(shortest\))g(sup)r(erstring.)f(A)i(common)e(conjecture)523
4907 y(states)28 b(that)h(the)g(ab)r(o)n(v)n(e)e(greedy)g(pro)r(cedure)
h(pro)r(duces)g(a)g(sup)r(erstring)g(of)g(length)h FC(O)r
FK(\()p FC(n)p FK(\))523 5006 y(\(in)23 b(fact,)g(2)p
FC(n)p FK(\),)f(y)n(et)h(the)g(only)f(previous)f(non)n(trivial)h(b)r
(ound)h(kno)n(wn)f(for)g FG(any)h FK(p)r(olynomial-)523
5106 y(time)28 b(algorithm)f(is)g(a)g(recen)n(t)g FC(O)r
FK(\()p FC(n)14 b FK(log)h FC(n)p FK(\))28 b(result.)606
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5305 y(pro)n(ximation,)j(pro)n(ving)f(an)i(upp)r(er)g(b)r(ound)h(of)f
(4)p FC(n)p FK(.)g(F)-7 b(urthermore,)25 b(w)n(e)g(presen)n(t)h(a)g
(simple)p eop end
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FG(MAX)h(SNP-har)l(d)p FK(,)523 706 y(whic)n(h)30 b(implies)h(that)g(a)
f(p)r(olynomial-time)g(appro)n(ximation)f(sc)n(heme)h(for)f(this)i
(problem)523 805 y(is)d(unlik)n(ely)-7 b(.)606 905 y(In)19
b(Chapter)g(3,)g(w)n(e)g(examine)g(the)g(space)g(complexit)n(y)f(of)i
(\015o)r(o)r(d-\014lling.)e(Fill)i(algorithms)523 1005
y(are)28 b(commonly)f(used)i(for)f(c)n(hanging)f(the)i(color)e(of)h(a)g
(region)f(of)i(pixels.)f(A)h(\015o)r(o)r(d-\014ll)f(al-)523
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FG(se)l(e)l(d)39 b FK(pixel)31 b(from)g(whic)n(h)f(it)i(starts)d
(exploring)h(the)h(region)523 1204 y(delimited)f(b)n(y)f(a)g
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(ev)n(en)h(more)f(space)g(than)h(is)g(dev)n(oted)523
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n(v)n(el)g(construction)523 2300 y(is)32 b(describ)r(ed)g(that)h
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(recognize)g(in)h(p)r(olynomial)f(time)h(exactly)523
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(thesis.)h(The)f(Asso)r(ciativ)n(e)f(Storage)523 3296
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b(e)30 b(sho)n(w)g(ho)n(w)f(to)h(v)n(erify)g(these)g(constructions)f(b)
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(b)r(er)i(of)f(accesses)f(p)r(er)i(Read/W)-7 b(rite)29
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706 y(where)27 b(eac)n(h)f(cell)i(con)n(tains)e(a)h(single)g(bit.)h(W)
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(of)f(storing)f(an)g(index,)i(and)f(deriv)n(e)f(exact)h(b)r(ounds)523
1503 y(on)e(their)h(space)f(complexit)n(y)-7 b(.)p eop
end
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(of)h(at)f(most)h(500)e(bases)h(can)g(b)r(e)h(sequenced)523
4043 y(at)32 b(a)h(time.)g(Then)f(from)g(h)n(undreds,)h(thousands,)f
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(Therefore)g(\014nding)i(a)e(go)r(o)r(d)h(appro)n(ximation)523
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(implies)i(e\016cien)n(t)f(learnabilit)n(y)g(or)f(infer-)523
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(that)i(if)f(the)h(optimal)f(solution)f(sa)n(v)n(es)f
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b(w)n(e)g(giv)n(e)g(a)g(simple)h(mo)r(di\014ed)g(greedy)523
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h(b)r(ound)h(of)f(4)p FC(n)p FK(,)g(and)g(then)h(presen)n(t)523
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(organized)f(as)h(follo)n(ws:)f(Section)i(2.2)f(con)n(tains)f
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(\\substring-free")e(in)i(that)g(no)g(string)f FC(s)3173
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FA(i)2606 3504 y FK(,)g(the)h(string)e FC(s)i FK(can)e(b)r(e)523
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eop end
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FC(s;)14 b(t)p FK(\))23 b(is)f(the)i(shortest)523 1005
y(sup)r(erstring)k(of)g FC(s)h FK(and)f FC(t)h FK(in)g(whic)n(h)g
FC(s)f FK(app)r(ears)g(\(strictly\))h(b)r(efore)f FC(t)p
FK(,)h(and)f(is)h(also)e(called)523 1104 y(the)k FG(mer)l(ge)38
b FK(of)31 b FC(s)g FK(and)g FC(t)p FK(.)g(F)-7 b(or)30
b FC(s)1528 1116 y FA(i)1556 1104 y FC(;)14 b(s)1632
1116 y FA(j)1696 1104 y FD(2)29 b FC(S)5 b FK(,)31 b(w)n(e)f(will)h
(abbreviate)f FG(pr)l(ef)19 b FK(\()p FC(s)2813 1116
y FA(i)2841 1104 y FC(;)14 b(s)2917 1116 y FA(j)2951
1104 y FK(\))32 b(to)f(simply)523 1204 y FG(pr)l(ef)18
b FK(\()p FC(i;)c(j)5 b FK(\),)33 b(and)f FG(d)9 b FK(\()p
FC(s)1190 1216 y FA(i)1218 1204 y FC(;)14 b(s)1294 1216
y FA(j)1329 1204 y FK(\))33 b(and)f FG(ov)10 b FK(\()p
FC(s)1721 1216 y FA(i)1749 1204 y FC(;)k(s)1825 1216
y FA(j)1860 1204 y FK(\))32 b(to)h FG(d)9 b FK(\()p FC(i;)14
b(j)5 b FK(\))32 b(and)h FG(ov)9 b FK(\()p FC(i;)14 b(j)5
b FK(\))33 b(resp)r(ectiv)n(ely)-7 b(.)32 b(The)523 1303
y(o)n(v)n(erlap)f(b)r(et)n(w)n(een)j(a)f(string)f(and)i(itself)g(is)f
(called)g(a)g FG(self-overlap)5 b FK(.)36 b(As)e(an)f(example)g(of)523
1403 y(self-o)n(v)n(erlap,)18 b(w)n(e)j(ha)n(v)n(e)e(for)h(the)h
(string)f FC(s)j FK(=)g(undergrounder)18 b(an)j(o)n(v)n(erlap)d(of)j
FG(ov)9 b FK(\()p FC(s;)14 b(s)p FK(\))24 b(=)e(5)523
1503 y(Also,)31 b FG(pr)l(ef)19 b FK(\()p FC(s;)14 b(s)p
FK(\))29 b(=)g(undergro)h(and)h FG(d)9 b FK(\()p FC(s;)14
b(s)p FK(\))30 b(=)f(8.)i(The)h(string)f FC(s)e FK(=)g(alfalfa)o(,)j
(for)f(whic)n(h)523 1602 y FG(ov)10 b FK(\()p FC(s;)k(s)p
FK(\))23 b(=)g(4,)d(sho)n(ws)f(that)i(the)g(o)n(v)n(erlap)e(is)h(not)h
(limited)g(to)g(half)f(the)h(total)g(string)f(length.)606
1702 y(Giv)n(en)e(a)g(list)h(of)f(strings)g FC(s)1411
1714 y FA(i)1434 1722 y Fy(1)1471 1702 y FC(;)c(s)1547
1714 y FA(i)1570 1722 y Fy(2)1606 1702 y FC(;)g(:)g(:)g(:)g(;)g(s)1830
1714 y FA(i)1853 1722 y Fx(r)1891 1702 y FK(,)k(w)n(e)g(de\014ne)h(the)
g(sup)r(erstring)e FC(s)23 b FK(=)g FD(h)p FC(s)3054
1714 y FA(i)3077 1722 y Fy(1)3114 1702 y FC(;)14 b(:)g(:)g(:)g(;)g(s)
3338 1714 y FA(i)3361 1722 y Fx(r)3399 1702 y FD(i)523
1802 y FK(to)43 b(b)r(e)h(the)f(string)f FG(pr)l(ef)19
b FK(\()p FC(i)1394 1814 y FB(1)1431 1802 y FC(;)14 b(i)1497
1814 y FB(2)1534 1802 y FK(\))p FG(pr)l(ef)19 b FK(\()p
FC(i)1782 1814 y FB(2)1819 1802 y FC(;)14 b(i)1885 1814
y FB(3)1921 1802 y FK(\))g FD(\001)g(\001)g(\001)g FG(pr)l(ef)19
b FK(\()p FC(i)2294 1814 y FA(r)r Fw(\000)p FB(1)2415
1802 y FC(;)14 b(i)2481 1814 y FA(r)2518 1802 y FK(\))p
FC(s)2589 1814 y FA(i)2612 1822 y Fx(r)2650 1802 y FK(.)43
b(That)g(is,)g FC(s)g FK(is)g(the)523 1901 y(shortest)37
b(string)g(suc)n(h)g(that)h FC(s)1520 1913 y FA(i)1543
1921 y Fy(1)1580 1901 y FC(;)14 b(s)1656 1913 y FA(i)1679
1921 y Fy(2)1716 1901 y FC(;)g(:)g(:)g(:)g(;)g(s)1940
1913 y FA(i)1963 1921 y Fx(r)2039 1901 y FK(app)r(ear)36
b FG(in)k(or)l(der)47 b FK(in)38 b(that)g(string.)f(F)-7
b(or)523 2001 y(a)26 b(sup)r(erstring)f(of)i(a)f(substring-free)f(set,)
h(this)h(order)e(is)h(w)n(ell-de\014ned,)g(since)h(substrings)523
2100 y(cannot)32 b(`start')h(or)f(`end')h(at)f(the)i(same)e(p)r
(osition,)g(and)h(if)g(substring)f FC(s)2831 2112 y FA(j)2899
2100 y FK(starts)g(b)r(efore)523 2200 y FC(s)562 2212
y FA(k)603 2200 y FK(,)d(then)g FC(s)884 2212 y FA(j)947
2200 y FK(m)n(ust)g(also)f(end)h(b)r(efore)f FC(s)1769
2212 y FA(k)1810 2200 y FK(.)h(De\014ne)g FG(\014rst)7
b FK(\()p FC(s)p FK(\))25 b(=)g FC(s)2528 2212 y FA(i)2551
2220 y Fy(1)2616 2200 y FK(and)k FG(last)8 b FK(\()p
FC(s)p FK(\))25 b(=)g FC(s)3169 2212 y FA(i)3192 2220
y Fx(r)3230 2200 y FK(.)k(In)523 2300 y(eac)n(h)e(iteration)g(of)g
(GREED)n(Y)h(the)g(follo)n(wing)e(in)n(v)-5 b(arian)n(t)27
b(holds:)523 2466 y Fv(Claim)32 b FH(2.1)40 b FK(F)-7
b(or)21 b(t)n(w)n(o)f(distinct)h(strings)f FC(s)h FK(and)g
FC(t)g FK(in)g(GREED)n(Y's)g(set)g(of)g(strings,)f(neither)523
2565 y FG(\014rst)7 b FK(\()p FC(s)p FK(\))28 b(nor)f
FG(last)8 b FK(\()p FC(s)p FK(\))28 b(is)g(a)f(substring)g(of)g
FC(t)p FK(.)523 2731 y Fv(Pr)n(oof.)d FK(Initially)-7
b(,)25 b FG(\014rst)7 b FK(\()p FC(s)p FK(\))23 b(=)g
FG(last)8 b FK(\()p FC(s)p FK(\))24 b(=)e FC(s)j FK(for)f(all)g
(strings,)g(so)g(the)h(claim)f(follo)n(ws)g(from)523
2831 y(the)32 b(fact)f(that)h FC(S)k FK(is)31 b(substring-free.)g(Supp)
r(ose)g(that)h(the)f(in)n(v)-5 b(arian)n(t)31 b(is)g(in)n(v)-5
b(alidated)31 b(b)n(y)523 2931 y(a)d(merge)f(of)h(t)n(w)n(o)f(strings)g
FC(t)1389 2943 y FB(1)1454 2931 y FK(and)h FC(t)1646
2943 y FB(2)1711 2931 y FK(in)n(to)g(a)f(string)h FC(t)23
b FK(=)g FD(h)p FC(t)2389 2943 y FB(1)2427 2931 y FC(;)14
b(t)2494 2943 y FB(2)2531 2931 y FD(i)28 b FK(that)h(has,)e(sa)n(y)-7
b(,)27 b FG(\014rst)7 b FK(\()p FC(s)p FK(\))523 3030
y(as)31 b(a)h(substring.)f(Let)h FC(t)f FK(=)f FC(u)h
FG(\014rst)7 b FK(\()p FC(s)p FK(\))33 b FC(v)s FK(.)f(Since)h
FG(\014rst)7 b FK(\()p FC(s)p FK(\))32 b(is)g(not)g(a)g(substring)f(of)
h(either)523 3130 y FC(t)553 3142 y FB(1)629 3130 y FK(or)38
b FC(t)772 3142 y FB(2)810 3130 y FK(,)h(it)g(m)n(ust)g(prop)r(erly)f
(`con)n(tain')g(the)i(piece)f(of)g(o)n(v)n(erlap)d(b)r(et)n(w)n(een)j
FC(t)3018 3142 y FB(1)3095 3130 y FK(and)f FC(t)3297
3142 y FB(2)3335 3130 y FK(,)523 3230 y FG(i.e.)p FK(,)e
FD(j)p FG(\014rst)7 b FK(\()p FC(s)p FK(\))p FD(j)35
b FC(>)e FG(ov)10 b FK(\()p FC(t)1280 3242 y FB(1)1317
3230 y FC(;)k(t)1384 3242 y FB(2)1422 3230 y FK(\))34
b(and)g FD(j)p FC(u)p FD(j)g FC(<)g FG(d)9 b FK(\()p
FC(t)1996 3242 y FB(1)2033 3230 y FC(;)14 b(t)2100 3242
y FB(2)2137 3230 y FK(\).)35 b(Hence,)f FG(ov)10 b FK(\()p
FC(t)2655 3242 y FB(1)2692 3230 y FC(;)k(s)p FK(\))35
b FC(>)e FG(ov)10 b FK(\()p FC(t)3085 3242 y FB(1)3122
3230 y FC(;)k(t)3189 3242 y FB(2)3227 3230 y FK(\);)34
b(a)523 3329 y(con)n(tradiction.)2271 b Fu(2)606 3479
y FK(So)37 b(when)g(GREED)n(Y)g(\(or)f(its)i(v)-5 b(ariation)36
b(MGREED)n(Y)g(that)i(w)n(e)e(in)n(tro)r(duce)h(later\))523
3578 y(c)n(ho)r(oses)h FC(s)h FK(and)h FC(t)f FK(as)g(ha)n(ving)f(the)i
(maxim)n(um)f(o)n(v)n(erlap,)f(then)i(this)f(o)n(v)n(erlap)f
FG(ov)9 b FK(\()p FC(s;)14 b(t)p FK(\))523 3678 y(in)41
b(fact)g(equals)g FG(ov)9 b FK(\()p FG(last)g FK(\()p
FC(s)p FK(\))p FC(;)14 b FG(\014rst)7 b FK(\()p FC(t)p
FK(\)\),)42 b(and)f(as)f(a)h(result,)g(the)g(merge)f(of)h
FC(s)g FK(and)g FC(t)g FK(is)523 3778 y FD(h)p FG(\014rst)7
b FK(\()p FC(s)p FK(\))p FC(;)14 b(:)g(:)g(:)g(;)g FG(last)8
b FK(\()p FC(s)p FK(\))p FC(;)14 b FG(\014rst)8 b FK(\()p
FC(t)p FK(\))p FC(;)14 b(:)g(:)g(:)g(;)g FG(last)8 b
FK(\()p FC(t)p FK(\))p FD(i)p FK(.)32 b(W)-7 b(e)31 b(can)f(therefore)g
(sa)n(y)g(that)h(GREED)n(Y)523 3877 y FG(or)l(ders)45
b FK(the)37 b(substrings,)g(b)n(y)g(\014nding)g(the)g(shortest)g(sup)r
(erstring)f(in)h(whic)n(h)g(the)h(sub-)523 3977 y(strings)27
b(app)r(ear)f(in)i(that)g(order.)606 4076 y(W)-7 b(e)32
b(can)f(rephrase)f(the)i(ab)r(o)n(v)n(e)e(in)i(terms)f(of)h(p)r(erm)n
(utations.)f(F)-7 b(or)31 b(a)g(p)r(erm)n(utation)g FC(\031)523
4176 y FK(on)d(the)g(set)g FD(f)p FK(1)p FC(;)14 b(:)g(:)g(:)e(;)i(m)p
FD(g)p FK(,)28 b(let)g FC(S)1516 4188 y FA(\031)1585
4176 y FK(=)23 b FD(h)p FC(s)1744 4191 y FA(\031)r FB(\(1\))1874
4176 y FC(;)14 b(:)g(:)g(:)g(;)g(s)2098 4191 y FA(\031)r
FB(\()p FA(m)p FB(\))2253 4176 y FD(i)p FK(.)29 b(In)f(a)f(shortest)g
(sup)r(erstring)g(for)523 4276 y FC(S)5 b FK(,)22 b(the)h(substrings)f
(app)r(ear)g(in)g(some)g(total)h(order,)e(sa)n(y)g FC(s)2321
4291 y FA(\031)r FB(\(1\))2451 4276 y FC(;)14 b(:)g(:)g(:)g(;)g(s)2675
4291 y FA(\031)r FB(\()p FA(m)p FB(\))2830 4276 y FK(,)23
b(hence)g(it)g(m)n(ust)523 4375 y(equal)k FC(S)793 4387
y FA(\031)838 4375 y FK(.)606 4475 y(W)-7 b(e)30 b(will)f(consider)f(a)
h(tra)n(v)n(eling)e(salesman)h(problem)h(on)f(a)h(w)n(eigh)n(ted)f
(directed)h(com-)523 4575 y(plete)35 b(graph)e FC(G)1040
4587 y FA(S)1123 4575 y FK(deriv)n(ed)h(from)g FC(S)39
b FK(and)34 b(sho)n(w)f(that)i(one)f(can)g(ac)n(hiev)n(e)f(a)h(factor)g
(of)g(4)523 4674 y(appro)n(ximation)24 b(for)h(TSP)g(on)g(that)h
(graph,)e(yielding)h(a)g(factor)g(of)h(4)f(appro)n(ximation)e(for)523
4774 y(the)35 b(shortest-common-sup)r(erstring)c(problem.)i(Graph)h
FC(G)2442 4786 y FA(S)2524 4774 y FK(=)g(\()p FC(V)5
b(;)14 b(E)5 b(;)14 b(d)p FK(\))34 b(has)g FC(m)g FK(v)n(er-)523
4873 y(tices)c FC(V)45 b FK(=)27 b FD(f)p FK(1)p FC(;)14
b(:)g(:)g(:)e(;)i(m)p FD(g)p FK(,)30 b(and)f FC(m)1571
4843 y FB(2)1638 4873 y FK(edges)g FC(E)j FK(=)26 b FD(f)p
FK(\()p FC(i;)14 b(j)5 b FK(\))27 b(:)f(1)h FD(\024)f
FC(i;)14 b(j)31 b FD(\024)26 b FC(m)p FD(g)p FK(.)k(Here)f(w)n(e)g(tak)
n(e)523 4973 y(as)g(w)n(eigh)n(t)h(function)g(the)h(distance)e
FG(d)9 b FK(\()p FC(;)14 b FK(\):)31 b(edge)e(\()p FC(i;)14
b(j)5 b FK(\))30 b(has)g(w)n(eigh)n(t)f FG(d)9 b FK(\()p
FC(i;)14 b(j)5 b FK(\))27 b(=)g FG(d)9 b FK(\()p FC(s)3164
4985 y FA(i)3192 4973 y FC(;)14 b(s)3268 4985 y FA(j)3302
4973 y FK(\),)523 5073 y(to)24 b(obtain)h(the)g FG(distanc)l(e)i(gr)l
(aph)p FK(.)f(This)f(graph)e(is)h(similar)g(to)h(one)f(considered)f(b)n
(y)h(T)-7 b(urner)523 5172 y(in)27 b(the)g(end)h(of)e(his)h(pap)r(er)g
([16)o(].)g(Later)f(w)n(e)h(will)g(tak)n(e)f(the)h(o)n(v)n(erlap)e
FG(ov)10 b FK(\()p FC(;)k FK(\))27 b(as)f(the)h(w)n(eigh)n(t)523
5272 y(function)h(to)f(obtain)g(the)h FG(overlap)k(gr)l(aph)p
FK(.)c(W)-7 b(e)28 b(will)g(call)f FC(s)2377 5284 y FA(i)2432
5272 y FK(the)h(string)e FG(asso)l(ciate)l(d)j FK(with)p
eop end
%%Page: 11 21
TeXDict begin 11 20 bop 2515 299 a FF(2.2.)53 b(Preliminaries)156
b(11)523 506 y FK(v)n(ertex)36 b FC(i)p FK(,)h(and)g(let)g
FG(pr)l(ef)19 b FK(\()p FC(i;)14 b(j)5 b FK(\))39 b(=)f
FG(pr)l(ef)18 b FK(\()p FC(s)1864 518 y FA(i)1892 506
y FC(;)c(s)1968 518 y FA(j)2003 506 y FK(\))38 b(b)r(e)f(the)g(string)g
(asso)r(ciated)f(with)h(edge)523 606 y(\()p FC(i;)14
b(j)5 b FK(\).)606 706 y(As)24 b(examples)f(w)n(e)g(dra)n(w)f(in)i
(Figure)f(2.1)g(the)h(o)n(v)n(erlap)e(graph)g(and)i(the)g(distance)f
(graph)523 805 y(for)h(our)h(previous)f(example)g FC(S)1494
817 y FB(0)1554 805 y FK(=)f FD(f)i FK(ate,)f(half,)i(lethal,)f(alpha,)
f(alfalfa)g FD(g)p FK(.)h(All)h(edges)e(not)523 905 y(sho)n(wn)29
b(ha)n(v)n(e)g(o)n(v)n(erlap)f(0.)i(Note)g(that)h(the)f(sum)g(of)g(the)
h(distance)f(and)g(o)n(v)n(erlap)e(w)n(eigh)n(ts)523
1005 y(on)f(an)h(edge)f(\()p FC(i;)14 b(j)5 b FK(\))27
b(is)h(the)g(length)g(of)f(the)h(string)f FC(s)2132 1017
y FA(i)2160 1005 y FK(.)606 1104 y(Notice)43 b(no)n(w)f(that)h(TSP\()p
FC(G)1525 1116 y FA(S)1573 1104 y FK(\))49 b FD(\024)f
FK(OPT)o(\()p FC(S)5 b FK(\))29 b FD(\000)f FG(ov)9 b
FK(\()p FG(last)g FK(\()p FC(s)p FK(\))p FC(;)14 b FG(\014rst)7
b FK(\()p FC(s)p FK(\)\))49 b FD(\024)f FK(OPT)o(\()p
FC(S)5 b FK(\),)523 1204 y(where)29 b(TSP\()p FC(G)1025
1216 y FA(S)1073 1204 y FK(\))h(is)f(the)h(cost)f(of)h(the)g(minim)n
(um)g(w)n(eigh)n(t)f(Hamiltonian)g(cycle)g(on)g FC(G)3286
1216 y FA(S)3335 1204 y FK(.)523 1303 y(The)i(reason)f(is)h(that)h
(turning)f(an)n(y)f(sup)r(erstring)g(in)n(to)h(a)g(Hamiltonian)g(cycle)
g(b)n(y)g(o)n(v)n(er-)523 1403 y(lapping)i(its)h(last)g(and)f(\014rst)h
(substring)f(sa)n(v)n(es)f(on)h(cost)g(b)n(y)h(c)n(harging)e
FG(last)8 b FK(\()p FC(s)p FK(\))34 b(for)f(only)523
1503 y FG(d)9 b FK(\()p FG(last)f FK(\()p FC(s)p FK(\))p
FC(;)14 b FG(\014rst)8 b FK(\()p FC(s)p FK(\)\))28 b(instead)g(of)f
(its)h(full)g(length.)606 1602 y(W)-7 b(e)23 b(no)n(w)f(de\014ne)g
(some)g(notation)g(for)g(dealing)f(with)i(directed)f(cycles)g(in)h
FC(G)2965 1614 y FA(S)3013 1602 y FK(.)g(Call)f(t)n(w)n(o)523
1702 y(strings)k FC(s;)14 b(t)26 b FK(equiv)-5 b(alen)n(t,)27
b FC(s)c FD(\021)g FC(t)p FK(,)j(if)i(they)e(are)g(cyclic)h(shifts)g
(of)f(eac)n(h)g(other,)g FG(i.e.)p FK(,)j(if)e(there)523
1802 y(are)e(strings)g FC(u;)14 b(v)28 b FK(suc)n(h)e(that)g
FC(s)d FK(=)f FC(uv)29 b FK(and)d FC(t)d FK(=)f FC(v)s(u)p
FK(.)k(If)g FC(c)g FK(is)g(a)f(directed)h(cycle)f(in)h
FC(G)3122 1814 y FA(S)3196 1802 y FK(with)523 1901 y(v)n(ertices)32
b FC(i)858 1913 y FB(0)895 1901 y FC(;)14 b(:)g(:)g(:)g(;)g(i)1109
1913 y FA(r)r Fw(\000)p FB(1)1264 1901 y FK(in)33 b(order)f(around)h
FC(c)p FK(,)g(w)n(e)g(de\014ne)h FG(strings)6 b FK(\()p
FC(c)p FK(\))34 b(to)g(b)r(e)g(the)f(equiv)-5 b(a-)523
2001 y(lence)29 b(class)f([)p FG(pr)l(ef)19 b FK(\()p
FC(i)1167 2013 y FB(0)1204 2001 y FC(;)14 b(i)1270 2013
y FB(1)1307 2001 y FK(\))p FG(pr)l(ef)k FK(\()p FC(i)1554
2013 y FB(1)1592 2001 y FC(;)c(i)1658 2013 y FB(2)1694
2001 y FK(\))g FD(\001)g(\001)g(\001)30 b FG(pr)l(ef)18
b FK(\()p FC(i)2082 2013 y FA(r)r Fw(\000)p FB(1)2204
2001 y FC(;)c(i)2270 2013 y FB(0)2306 2001 y FK(\)])30
b(and)f FG(strings)7 b FK(\()p FC(c;)14 b(i)2936 2013
y FA(k)2976 2001 y FK(\))30 b(the)f(rota-)523 2100 y(tion)d(starting)e
(with)i FG(pr)l(ef)19 b FK(\()p FC(i)1402 2112 y FA(k)1442
2100 y FC(;)14 b(i)1508 2112 y FA(k)q FB(+1)1633 2100
y FK(\),)26 b FG(i.e.)p FK(,)h(the)f(string)e FG(pr)l(ef)19
b FK(\()p FC(i)2467 2112 y FA(k)2508 2100 y FC(;)14 b(i)2574
2112 y FA(k)q FB(+1)2698 2100 y FK(\))g FD(\001)g(\001)g(\001)g
FG(pr)l(ef)k FK(\()p FC(i)3070 2112 y FA(k)q Fw(\000)p
FB(1)3196 2100 y FC(;)c(i)3262 2112 y FA(k)3302 2100
y FK(\),)523 2200 y(where)33 b(subscript)f(arithmetic)h(is)g(mo)r(dulo)
g FC(r)r FK(.)i(Let)e(us)g(sa)n(y)f(that)h(an)g(equiv)-5
b(alence)33 b(class)523 2300 y([)p FC(s)p FK(])c(has)g
FG(p)l(erio)l(dicity)38 b FC(k)32 b FK(\()p FC(k)d(>)24
b FK(0\),)29 b(if)h FC(s)f FK(is)g(in)n(v)-5 b(arian)n(t)27
b(under)i(a)g(rotation)f(b)n(y)g FC(k)k FK(c)n(haracters)523
2399 y(\()p FC(s)23 b FK(=)g FC(uv)j FK(=)c FC(v)s(u;)14
b FD(j)p FC(u)p FD(j)23 b FK(=)f FC(k)s FK(\).)j(Ob)n(viously)-7
b(,)23 b([)p FC(s)p FK(])h(has)g(p)r(erio)r(dicit)n(y)g
FD(j)p FC(s)p FD(j)p FK(.)h(A)f(momen)n(t's)g(re\015ection)523
2499 y(sho)n(ws)30 b(that)h(the)h(minim)n(um)g(p)r(erio)r(dicit)n(y)f
(of)g([)p FC(s)p FK(])g(m)n(ust)g(equal)g(the)g(n)n(um)n(b)r(er)g(of)g
(distinct)523 2599 y(rotations)22 b(of)g FC(s)p FK(.)h(This)g(is)g(the)
g(size)g(of)g(the)g(equiv)-5 b(alence)23 b(class)f(and)g(denoted)h(b)n
(y)g FG(c)l(ar)l(d)9 b FK(\([)p FC(s)p FK(]\).)523 2698
y(F)-7 b(urthermore,)25 b(it)h(is)f(easily)g(pro)n(v)n(en)f(that)i(if)g
([)p FC(s)p FK(])f(has)g(p)r(erio)r(dicities)h FC(a)f
FK(and)h FC(b)p FK(,)f(then)h(it)g(has)523 2798 y(p)r(erio)r(dicit)n(y)
i(gcd)o(\()p FC(a;)14 b(b)p FK(\))29 b(as)e(w)n(ell.)h(\(See,)g
FG(e.g.)p FK(,)i([4)o(].\))f(It)f(follo)n(ws)g(that)g(all)g(p)r(erio)r
(dicities)g(are)523 2897 y(a)f(m)n(ultiple)h(of)f(the)h(minim)n(um)g
(one.)f(In)g(particular,)f(w)n(e)h(ha)n(v)n(e)f(that)i
FD(j)p FC(s)p FD(j)f FK(is)g(a)g(m)n(ultiple)h(of)523
2997 y FG(c)l(ar)l(d)9 b FK(\([)p FC(s)p FK(]\).)606
3097 y(In)33 b(general,)f(w)n(e)h(will)g(denote)g(a)g(cycle)f
FC(c)h FK(with)h(v)n(ertices)e FC(i)2477 3109 y FB(1)2514
3097 y FC(;)14 b(:)g(:)g(:)f(;)h(i)2727 3109 y FA(r)2797
3097 y FK(in)33 b(the)g(order)f(b)n(y)523 3196 y(\\)p
FC(i)594 3208 y FB(1)671 3196 y FD(!)40 b(\001)14 b(\001)g(\001)41
b(!)f FC(i)1084 3208 y FA(r)1161 3196 y FD(!)g FC(i)1313
3208 y FB(1)1350 3196 y FK(.")e(Also,)g(let)g FC(w)r
FK(\()p FC(c)p FK(\),)h(the)g FG(weight)47 b FK(of)38
b(cycle)f FC(c)p FK(,)i(equal)e FD(j)p FC(s)p FD(j)p
FC(;)14 b(s)40 b FD(2)523 3296 y FG(strings)7 b FK(\()p
FC(c)p FK(\).)36 b(F)-7 b(or)34 b(con)n(v)n(enience,)g(w)n(e)h(will)g
(sa)n(y)f(that)i FC(s)2243 3308 y FA(j)2313 3296 y FK(is)f(in)h
FC(c)p FK(,)f(or)f(\\)p FC(s)2793 3308 y FA(j)2863 3296
y FD(2)i FC(c)p FK(")f(if)h FC(j)k FK(is)35 b(a)523 3396
y(v)n(ertex)27 b(of)g(the)h(cycle)f FC(c)p FK(.)606 3495
y(No)n(w,)d(a)g(few)h(preliminary)e(facts)i(ab)r(out)f(cycles)g(in)h
FC(G)2292 3507 y FA(S)2340 3495 y FK(.)g(Let)g FC(c)e
FK(=)f FC(i)2709 3507 y FB(0)2769 3495 y FD(!)h(\001)14
b(\001)g(\001)23 b(!)g FC(i)3130 3507 y FA(r)r Fw(\000)p
FB(1)3275 3495 y FD(!)523 3595 y FC(i)552 3607 y FB(0)617
3595 y FK(and)29 b FC(c)816 3565 y Fw(0)867 3595 y FK(b)r(e)g(cycles)f
(in)g FC(G)1382 3607 y FA(S)1431 3595 y FK(.)g(F)-7 b(or)28
b(an)n(y)g(string)f FC(s)p FK(,)i FC(s)2156 3565 y FA(k)2225
3595 y FK(denotes)f(the)h(string)e(consisting)h(of)523
3694 y FC(k)j FK(copies)26 b(of)i FC(s)g FK(concatenated)e(together.)
523 3861 y Fv(Claim)32 b FH(2.2)40 b FK(Eac)n(h)30 b(string)h
FC(s)1453 3873 y FA(i)1476 3881 y Fx(j)1542 3861 y FK(in)h
FC(c)f FK(is)g(a)g(substring)f(of)h FC(s)2371 3830 y
FA(k)2443 3861 y FK(for)g(all)f FC(s)f FD(2)h FG(strings)6
b FK(\()p FC(c)p FK(\))32 b(and)523 3960 y(su\016cien)n(tly)c(large)e
FC(k)s FK(.)523 4066 y Fv(Pr)n(oof.)j FK(By)g(induction,)h
FC(s)1404 4078 y FA(i)1427 4086 y Fx(j)1492 4066 y FK(is)f(a)g
(pre\014x)g(of)g FG(pr)l(ef)19 b FK(\()p FC(i)2195 4078
y FA(j)2230 4066 y FC(;)14 b(i)2296 4078 y FA(j)s FB(+1)2414
4066 y FK(\))g FD(\001)g(\001)g(\001)g FG(pr)l(ef)k FK(\()p
FC(i)2786 4078 y FA(j)s FB(+)p FA(l)p Fw(\000)p FB(1)2979
4066 y FC(;)c(i)3045 4078 y FA(j)s FB(+)p FA(l)3151 4066
y FK(\))g FC(s)3236 4078 y FA(i)3259 4087 y Fx(j)r Fy(+)p
Fx(l)523 4166 y FK(for)36 b(an)n(y)g FC(l)j FD(\025)e
FK(0)f(\(addition)g(mo)r(dulo)h FC(r)r FK(\).)g(T)-7
b(aking)36 b FC(k)k FK(=)e FD(dj)p FC(s)2446 4178 y FA(i)2469
4186 y Fx(j)2504 4166 y FD(j)p FC(=w)r FK(\()p FC(c)p
FK(\))p FD(e)f FK(and)g FC(l)i FK(=)e FC(k)s(r)i FK(w)n(e)523
4265 y(get)26 b(that)g FC(s)877 4277 y FA(i)900 4285
y Fx(j)961 4265 y FK(is)g(a)f(pre\014x)g(of)h FG(pr)l(ef)19
b FK(\()p FC(i)1650 4277 y FA(j)1684 4265 y FC(;)14 b(i)1750
4277 y FA(j)s FB(+1)1869 4265 y FK(\))g FD(\001)g(\001)g(\001)g
FG(pr)l(ef)k FK(\()p FC(i)2241 4277 y FA(j)s FB(+)p FA(k)q(r)r
Fw(\000)p FB(1)2481 4265 y FC(;)c(i)2547 4277 y FA(j)s
FB(+)p FA(k)q(r)2702 4265 y FK(\))24 b(=)e FG(strings)7
b FK(\()p FC(c;)14 b(i)3227 4277 y FA(j)3261 4265 y FK(\))3293
4235 y FA(k)3335 4265 y FK(,)523 4365 y(whic)n(h)28 b(itself)g(is)f(a)g
(substring)g(of)h FC(s)1609 4335 y FA(k)q FB(+1)1761
4365 y FK(for)f(an)n(y)g FC(s)c FD(2)h FG(strings)6 b
FK(\()p FC(c)p FK(\).)740 b Fu(2)523 4581 y Fv(Claim)32
b FH(2.3)40 b FK(If)d(eac)n(h)f(of)h FD(f)p FC(s)1437
4593 y FA(j)1464 4601 y Fy(1)1500 4581 y FC(;)14 b(:)g(:)g(:)g(;)g(s)
1724 4593 y FA(j)1751 4601 y Fx(r)1789 4581 y FD(g)36
b FK(is)g(a)g(substring)g(of)h FC(s)2550 4551 y FA(k)2627
4581 y FK(for)f(some)g(string)g FC(s)i FD(2)523 4681
y FG(strings)7 b FK(\()p FC(c)p FK(\))25 b(and)f(su\016cien)n(tly)g
(large)f FC(k)s FK(,)i(then)f(there)h(exists)f(a)g(cycle)g(of)g(w)n
(eigh)n(t)g FD(j)p FC(s)p FD(j)f FK(=)g FC(w)r FK(\()p
FC(c)p FK(\))523 4780 y(con)n(taining)k(all)g(these)h(strings.)523
4886 y Fv(Pr)n(oof.)33 b FK(In)g(a)g(\(in\014nite\))i(rep)r(etition)f
(of)f FC(s)p FK(,)h(ev)n(ery)e(string)g FC(s)2467 4898
y FA(i)2528 4886 y FK(app)r(ears)h(as)f(a)h(substring)523
4986 y(at)e(ev)n(ery)e(other)i FD(j)p FC(s)p FD(j)f FK(c)n(haracters.)f
(This)i(naturally)f(de\014nes)h(a)f(circular)f(ordering)h(of)g(the)523
5085 y(strings)23 b FD(f)p FC(s)869 5097 y FA(j)896 5105
y Fy(1)933 5085 y FC(;)14 b(:)g(:)g(:)f(;)h(s)1156 5097
y FA(j)1183 5105 y Fx(r)1221 5085 y FD(g)24 b FK(and)g(the)h(strings)e
(in)i FC(c)f FK(whose)f(successiv)n(e)g(distances)h(sum)g(to)h
FD(j)p FC(s)p FD(j)p FK(.)523 5185 y Fu(2)523 5324 y
Fv(Claim)32 b FH(2.4)40 b FK(The)28 b(sup)r(erstring)f
FD(h)p FC(s)1639 5336 y FA(i)1662 5344 y Fy(0)1699 5324
y FC(;)14 b FD(\001)g(\001)g(\001)f FC(;)h(s)1922 5336
y FA(i)1945 5344 y Fx(r)q Ft(\000)p Fy(1)2057 5324 y
FD(i)27 b FK(is)h(a)f(substring)g(of)h FG(strings)6 b
FK(\()p FC(c;)14 b(i)3106 5336 y FB(0)3143 5324 y FK(\))p
FC(s)3214 5336 y FA(i)3237 5344 y Fy(0)3274 5324 y FK(.)p
eop end
%%Page: 12 22
TeXDict begin 12 21 bop 2515 299 a FF(2.2.)53 b(Preliminaries)156
b(12)647 2805 y @beginspecial 0 @llx 0 @lly 310.500000
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%%BeginDocument: dist.ps
%!
%%Creator: tromp@wolf.cwi.nl (John Tromp,M231,4078,02207-16322)
%%Title: dist.x (xpic)
%%CreationDate: Thu Sep  9 12:20:24 1993
%%Pages: 1
%%BoundingBox: 0 0 310.5 203.4
%	(in inches) at 0 0, width 4.3125, height 2.825
%%EndComments
% Prolog for xpic to PostScript converter
% Author: Mark Moraes
% $Header: x2ps.pro,v 1.2 88/03/19 16:50:09 moraes Exp 
%	%d D - change style SOLID, DOTTED, SHORT-DASH, LONG-DASH, DOT-DASH
%	%s F - change font to fontname
%	%d S - change size (font size in points)
%	(%s) rj %d t - text right just. (%d is TOPLINE, MIDLINE, BOTLINE)
%	(%s) lj %d t - text left just. (%d is TOPLINE, MIDLINE, BOTLINE)
%	(%s) ce %d t - text centered (%d is TOPLINE, MIDLINE, BOTLINE)
%	%d %d l - lineto
%	%d %d m - moveto
%	%d %d s - spline segment
%	x - flush line, spline
%	<wid> <ht> <x> <y> b - box
%	<wid> <ht> <x> <y> e - ellipse
%	%d ss - setscale
%	%d W - change linewidth
%	getpagesize - gets the values of PAGEHEIGHT and PAGEWIDTH
%	%d %d flip - translate by %d, PAGEHEIGHT - %d (this 
%		transforms to X windows coordinates)
save 50 dict begin /xpic exch def
/StartXpic {newpath 0 0 moveto [] 0 setdash 0 setgray 1 setlinecap} def
% Set defaults
/fontname /Times-Roman def
/ptsize 12 def
% halign has the values for MIDLINE, TOPLINE, BOTLINE
/halign 3 array def
/s {rcurveto} def
/x {stroke} def
/l {lineto} def
/m {moveto} def
/b {
	/ury exch def /urx exch def /lly exch def /llx exch def 
	llx lly moveto urx lly lineto urx ury lineto 
	llx ury lineto llx lly lineto stroke
} def
/mtrx matrix def
/e {
	/yc exch def /xc exch def /yrad exch def /xrad exch def
	xc xrad add yc moveto
	/savematrix mtrx currentmatrix def
	xc yc translate
	xrad yrad scale
	0 0 1 0 360 arc
	savematrix setmatrix stroke
} def
% The next three take the text string, and moveto the right horiz. position
% leaving the string on the stack.
/lj {} def
/rj {dup stringwidth pop neg 0 rmoveto} def
/ce {dup stringwidth pop 2 div neg 0 rmoveto} def
% And this is invoked after one of the three above, and 
% computes the vert. pos, and then displays the string.
/t {halign exch get 0 exch rmoveto show newpath} def
% Store an array of patterns in /styles - a pattern is an array consisting 
% of an array and an offset. Corresp to xpic patterns
% solid, dotted, short-dashed, long-dashed, dot-dashed
/styles [ [] 0 ] [ [1 3] 0 ] [ [4 4] 0 ] [ [8 4] 0 ] [ [1 4 4 4] 0 ]
	5 array astore def
% change style to arg.
/D {stroke styles exch get aload pop setdash newpath} def
/W {stroke 0.5 mul setlinewidth newpath} def
% fontbox takes a fontname off the stack, and returns an array
% containing the values of the bottom line of the bounding box, the
% mid line of the bounding box, and the top line of the bounding box
% of that font, taken from the baseline, scaled to a font of size 1
/fontbox {
	findfont dup /FontMatrix get /fm exch def /FontBBox get aload pop
	/ytop exch def pop /ybot exch def pop
	/ymid ytop ybot sub 2 div def
	0 ybot fm dtransform exch pop % botline
	dup neg exch % midline - this works better than (ytop-ybot)/2!
	0 ytop fm dtransform exch pop exch %topline
	% now in the order midline, topline, botline.
	3 array astore
} def
% select font
/F {
	dup /fontname exch def fontbox
	/thisfontbox exch def SF 
} def
% set point size
/S {/ptsize exch def SF} def
% actually set font
/SF {
	fontname findfont ptsize curscale div scalefont setfont
	thisfontbox aload pop
	1 1 3 {
		pop ptsize mul curscale div neg 3 1 roll
	} for
	halign astore pop
} def
% sets the scale to 72 / n, where n is on the stack, and stores the value
% in curscale for font scaling
/curscale 1 def
/getpagesize{newpath clippath pathbbox /pageheight exch def 
	/pagewidth exch def pop pop newpath} def
/flip{pageheight exch sub translate} def
/ss {/curscale exch 72 exch div dup dup scale def} def
/land {90 rotate} def
StartXpic
%%EndProlog
80 ss
0.5 W
/Times-Roman F
10 S
41 168 m
(Distance Graph) ce 0 t
17 88 m
(5) ce 0 t
201 80 m
(1) ce 0 t
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(alpha) ce 0 t
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329 120 l
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329 120 m
x
337 112 m
(ate) ce 0 t
257 40 m
329 104 l
0 D
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329 104 l
326 96 l
0 D
329 104 m
x
57 120 m
217 192 l
0 D
208 192 m
217 192 l
212 185 l
0 D
217 192 m
x
225 184 m
161 120 l
0 D
170 123 m
161 120 l
164 129 l
0 D
161 120 m
x
161 104 m
225 40 l
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223 49 m
225 40 l
217 43 l
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225 40 m
x
233 176 m
233 48 l
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237 57 m
233 48 l
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233 48 m
x
249 48 m
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246 168 m
249 176 l
253 168 l
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249 176 m
x
57 104 m
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217 32 l
208 32 l
0 D
217 32 m
x
57 112 m
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121 116 m
129 112 l
121 109 l
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129 112 m
x
241 32 m
(alfalfa) ce 0 t
145 112 m
(half) ce 0 t
41 112 m
(lethal) ce 0 t
233 200 m
0 0 -1.33333 2.66667 -4 8 s
-2.66667 5.33333 -1.33333 9.33333 4 12 s
5.33333 2.66667 10.6667 2.66667 16 0 s
5.33333 -2.66667 5.33333 -9.33333 0 -20 s
0 D
257 206 m
249 200 l
249 210 l
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249 200 m
x
249 24 m
0 0 1.33333 -2.66667 4 -8 s
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-5.33333 2.66667 -5.33333 9.33333 0 20 s
0 D
226 19 m
233 24 l
233 15 l
0 D
233 24 m
x
25 120 m
0 0 -2.66667 1.33333 -8 4 s
-5.33333 2.66667 -9.33333 1.33333 -12 -4 s
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265 8 m
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%%Trailer
showpage
% Trailer for xpic to PostScript converter
% $Header: x2ps.tra,v 1.2 89/07/02 15:59:53 moraes Exp $
xpic end restore

%%EndDocument
 @endspecial 650 4508 a @beginspecial 0 @llx 0 @lly 309.600006
@urx 203.399994 @ury 3096 @rwi @setspecial
%%BeginDocument: ovlp.ps
%!
%%Creator: tromp@wolf.cwi.nl (John Tromp,M231,4078,02207-16322)
%%Title: ovlp.x (xpic)
%%CreationDate: Thu Sep  9 12:20:15 1993
%%Pages: 1
%%BoundingBox: 0 0 309.6 203.4
%	(in inches) at 0 0, width 4.3, height 2.825
%%EndComments
% Prolog for xpic to PostScript converter
% Author: Mark Moraes
% $Header: x2ps.pro,v 1.2 88/03/19 16:50:09 moraes Exp 
%	%d D - change style SOLID, DOTTED, SHORT-DASH, LONG-DASH, DOT-DASH
%	%s F - change font to fontname
%	%d S - change size (font size in points)
%	(%s) rj %d t - text right just. (%d is TOPLINE, MIDLINE, BOTLINE)
%	(%s) lj %d t - text left just. (%d is TOPLINE, MIDLINE, BOTLINE)
%	(%s) ce %d t - text centered (%d is TOPLINE, MIDLINE, BOTLINE)
%	%d %d l - lineto
%	%d %d m - moveto
%	%d %d s - spline segment
%	x - flush line, spline
%	<wid> <ht> <x> <y> b - box
%	<wid> <ht> <x> <y> e - ellipse
%	%d ss - setscale
%	%d W - change linewidth
%	getpagesize - gets the values of PAGEHEIGHT and PAGEWIDTH
%	%d %d flip - translate by %d, PAGEHEIGHT - %d (this 
%		transforms to X windows coordinates)
save 50 dict begin /xpic exch def
/StartXpic {newpath 0 0 moveto [] 0 setdash 0 setgray 1 setlinecap} def
% Set defaults
/fontname /Times-Roman def
/ptsize 12 def
% halign has the values for MIDLINE, TOPLINE, BOTLINE
/halign 3 array def
/s {rcurveto} def
/x {stroke} def
/l {lineto} def
/m {moveto} def
/b {
	/ury exch def /urx exch def /lly exch def /llx exch def 
	llx lly moveto urx lly lineto urx ury lineto 
	llx ury lineto llx lly lineto stroke
} def
/mtrx matrix def
/e {
	/yc exch def /xc exch def /yrad exch def /xrad exch def
	xc xrad add yc moveto
	/savematrix mtrx currentmatrix def
	xc yc translate
	xrad yrad scale
	0 0 1 0 360 arc
	savematrix setmatrix stroke
} def
% The next three take the text string, and moveto the right horiz. position
% leaving the string on the stack.
/lj {} def
/rj {dup stringwidth pop neg 0 rmoveto} def
/ce {dup stringwidth pop 2 div neg 0 rmoveto} def
% And this is invoked after one of the three above, and 
% computes the vert. pos, and then displays the string.
/t {halign exch get 0 exch rmoveto show newpath} def
% Store an array of patterns in /styles - a pattern is an array consisting 
% of an array and an offset. Corresp to xpic patterns
% solid, dotted, short-dashed, long-dashed, dot-dashed
/styles [ [] 0 ] [ [1 3] 0 ] [ [4 4] 0 ] [ [8 4] 0 ] [ [1 4 4 4] 0 ]
	5 array astore def
% change style to arg.
/D {stroke styles exch get aload pop setdash newpath} def
/W {stroke 0.5 mul setlinewidth newpath} def
% fontbox takes a fontname off the stack, and returns an array
% containing the values of the bottom line of the bounding box, the
% mid line of the bounding box, and the top line of the bounding box
% of that font, taken from the baseline, scaled to a font of size 1
/fontbox {
	findfont dup /FontMatrix get /fm exch def /FontBBox get aload pop
	/ytop exch def pop /ybot exch def pop
	/ymid ytop ybot sub 2 div def
	0 ybot fm dtransform exch pop % botline
	dup neg exch % midline - this works better than (ytop-ybot)/2!
	0 ytop fm dtransform exch pop exch %topline
	% now in the order midline, topline, botline.
	3 array astore
} def
% select font
/F {
	dup /fontname exch def fontbox
	/thisfontbox exch def SF 
} def
% set point size
/S {/ptsize exch def SF} def
% actually set font
/SF {
	fontname findfont ptsize curscale div scalefont setfont
	thisfontbox aload pop
	1 1 3 {
		pop ptsize mul curscale div neg 3 1 roll
	} for
	halign astore pop
} def
% sets the scale to 72 / n, where n is on the stack, and stores the value
% in curscale for font scaling
/curscale 1 def
/getpagesize{newpath clippath pathbbox /pageheight exch def 
	/pagewidth exch def pop pop newpath} def
/flip{pageheight exch sub translate} def
/ss {/curscale exch 72 exch div dup dup scale def} def
/land {90 rotate} def
StartXpic
%%EndProlog
80 ss
0.5 W
/Times-Roman F
10 S
40 176 m
(Overlap Grpah) ce 0 t
16 88 m
(1) ce 0 t
40 112 m
(lethal) ce 0 t
144 112 m
(half) ce 0 t
240 192 m
(alpha) ce 0 t
240 32 m
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(wn)f(as)h(\\Monge)523 4014 y(conditions")34 b(in)h(the)g(con)n(text)f
(of)g(transp)r(ortation)g(problems)g([1)o(,)h(3)o(,)g(7].)g(In)f(this)h
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(ciated)e(graph.)g(No)n(w,)h(although)g(\014nding)g(a)523
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(eigh)n(ted)f(directed)h(graph)f(is)g(in)i(general)d(a)523
5152 y(hard)d(problem,)g(there)h FG(is)g FK(a)f(p)r(olynomial-time)g
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(goal)e(is)g(simply)h(to)g(\014nd)h(a)e(decomp)r(osition)523
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%%Page: 14 24
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FC(G)2305 917 y FA(S)2354 905 y FK(,)g(then)h(op)r(ens)f(eac)n(h)f
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(b)r(e)g(used)f(to)h(\014nd)g(optimal)f(assignmen)n(ts.)523
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1403 y(Ho\013man's)g(1963)d(pap)r(er)i([7].)606 1503
y(Consider)k(the)i(follo)n(wing)e(algorithm)h(for)f(\014nding)i(a)f
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FC(S)5 b FK(.)523 1802 y FH(Algorithm)31 b(Concat-Cycles)625
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FC(G)1749 1980 y FA(S)1826 1968 y FK(and)h(\014nd)g(a)g(minim)n(um)g(w)
n(eigh)n(t)g(assignmen)n(t)e FC(C)731 2067 y FK(on)g
FC(G)911 2079 y FA(S)959 2067 y FK(.)h(Let)g FC(C)34
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b(F)-7 b(or)34 b(eac)n(h)g(cycle)h FC(c)1330 2245 y FA(i)1393
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y FA(i)1689 2433 y FK(\))19 b(+)f FD(j)p FC(s)1885 2445
y FA(i)1908 2453 y Fy(1)1945 2433 y FD(j)27 b FK(b)n(y)h(Claim)f(2.4.)
625 2599 y(3.)41 b(Concatenate)24 b(together)g(the)i(strings)i(~)-46
b FC(s)1979 2611 y FA(i)2032 2599 y FK(and)25 b(pro)r(duce)g(the)h
(resulting)f(string)i(~)-45 b FC(s)25 b FK(as)731 2698
y(output.)523 2864 y Fv(Theorem)32 b FH(2.8)41 b FG(A)n(lgorithm)f
(Conc)l(at-Cycles)h(pr)l(o)l(duc)l(es)f(a)f(string)h(of)g(length)f(at)h
(most)523 2964 y FK(4)18 b FD(\001)h FG(OPT)p FK(\()p
FC(S)5 b FK(\))p FG(.)606 3130 y FK(Before)34 b(pro)n(ving)f(Theorem)h
(2.8,)h(w)n(e)f(\014rst)h(need)g(a)f(preliminary)g(lemma)h(giving)f(an)
523 3230 y(upp)r(er)j(b)r(ound)g(on)f(the)h(amoun)n(t)f(of)h(o)n(v)n
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(also)g(implied)h(b)n(y)f(the)h(results)f(in)h([4].)523
3495 y Fv(Lemma)k FH(2.9)40 b FG(L)l(et)24 b FC(c)f FG(and)i
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(minimum)e(weight)i(assignment)f FC(C)30 b FG(with)523
3595 y FC(s)23 b FD(2)g FC(c)29 b FG(and)h FC(s)928 3565
y Fw(0)974 3595 y FD(2)23 b FC(c)1088 3565 y Fw(0)1112
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FC(s)h FG(and)g FC(s)2402 3565 y Fw(0)2454 3595 y FG(is)h(less)f(than)g
FC(w)r FK(\()p FC(c)p FK(\))18 b(+)e FC(w)r FK(\()p FC(c)3276
3565 y Fw(0)3300 3595 y FK(\))p FG(.)523 3761 y Fv(Pr)n(oof.)27
b FK(Let)h FC(x)c FK(=)f FG(strings)7 b FK(\()p FC(c)p
FK(\))28 b(and)g FC(x)1730 3731 y Fw(0)1777 3761 y FK(=)23
b FG(strings)7 b FK(\()p FC(c)2181 3731 y Fw(0)2204 3761
y FK(\).)29 b(Since)f FC(C)34 b FK(is)27 b(a)h(minim)n(um)g(w)n(eigh)n
(t)523 3861 y(assignmen)n(t,)g(w)n(e)h(kno)n(w)f FC(x)e
FD(6)p FK(=)e FC(x)1526 3830 y Fw(0)1550 3861 y FK(.)29
b(Otherwise,)g(b)n(y)f(Claim)h(2.5,)f(w)n(e)h(could)g(\014nd)g(a)g
(ligh)n(ter)523 3960 y(assignmen)n(t)23 b(b)n(y)h(com)n(bining)g(the)g
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FA(k)2076 4359 y FK(for)g(some)g FC(t)e FD(2)i FC(x)g
FK(and)g(large)f(enough)h FC(k)523 4473 y FK(and)c FC(s)724
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b FC(u)1795 4870 y FB(1+)p FA(w)r FB(\()p FA(c)1985 4853
y Ft(0)2007 4870 y FB(\))p FA(;w)r FB(\()p FA(c)p FB(\)+)p
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FB(\()p FA(c)2814 5036 y Ft(0)2835 5053 y FB(\))2865
5038 y FC(:)523 5187 y FK(This)32 b(sho)n(ws)e(that)i
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FC(c)1923 5157 y Fw(0)1948 5187 y FK(\))f FC(<)f(w)r
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y(the)c(fact)g(that)g FG(c)l(ar)l(d)9 b FK(\()p FC(x)p
FK(\))28 b(is)g(the)g(minim)n(um)g(p)r(erio)r(dicit)n(y)-7
b(.)987 b Fu(2)p eop end
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1104 y FC(l)955 1116 y FA(i)1001 1104 y FD(\000)18 b
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FC(s)i FK(and)f FC(t)h FK(as)f(ha)n(ving)523 3849 y(the)42
b(maxim)n(um)f(o)n(v)n(erlap)e(\(where)i FC(t)g FK(ma)n(y)f(equal)h
FC(s)p FK(\),)g(it)h(c)n(ho)r(oses)e(the)h(directed)g(edge)523
3949 y(from)g FG(last)8 b FK(\()p FC(s)p FK(\))41 b(to)g
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(us,)h(MGREED)n(Y)f(constructs/joins)523 4048 y(paths,)24
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(collection)g(of)h(disjoin)n(t)f(cycles)523 4148 y FC(M)32
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b(will)h(call)f FC(M)31 b FK(the)22 b(assignmen)n(t)g(created)f(b)n(y)
523 4248 y(MGREED)n(Y.)29 b(No)n(w)f(think)h(of)f(MGREED)n(Y)h(as)f
(taking)g(a)g(list)h(of)f(all)h(the)g(edges)f(sorted)523
4347 y(in)j(the)f(decreasing)f(order)g(of)h(their)g(o)n(v)n(erlaps)e
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4447 y(and)j(going)e(do)n(wn)i(the)g(list)g(deciding)f(for)h(eac)n(h)f
(edge)g(whether)h(to)f(include)i(it)f(or)f(not.)523 4547
y(Let)27 b(us)g(sa)n(y)f(that)h(an)g(edge)g FC(e)f FG(dominates)35
b FK(another)26 b(edge)h FC(f)36 b FK(if)27 b FC(e)g
FK(precedes)f FC(f)36 b FK(in)27 b(this)h(list)523 4646
y(and)i(shares)e(its)i(head)g(\(or)f(tail\))h(with)h(the)f(head)g(\(or)
f(tail,)h(resp)r(ectiv)n(ely\))f(of)h FC(f)9 b FK(.)30
b(By)f(the)523 4746 y(de\014nition)36 b(of)f(MGREED)n(Y,)g(it)g
(includes)g(an)g(edge)g FC(f)44 b FK(if)35 b(and)g(only)g(if)h(it)f
(has)g(not)g(y)n(et)523 4845 y(included)28 b(an)f(edge)h(dominating)f
FC(f)9 b FK(.)523 5007 y Fv(Theorem)32 b FH(2.10)41 b
FG(The)33 b(assignment)f(cr)l(e)l(ate)l(d)g(by)h(algorithm)g(MGREED)n
(Y)f(is)h(an)f(opti-)523 5107 y(mal)e(assignment.)523
5268 y Fv(Pr)n(oof.)25 b FK(Note)g(that)h(the)f(o)n(v)n(erlap)f(w)n
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(t)523 5368 y(add)h(up)f(to)h(the)g(total)f(length)h(of)g(all)f
(strings.)g(Accordingly)-7 b(,)24 b(an)i(assignmen)n(t)e(is)i(optimal)p
eop end
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b(this)f(is)h(not)f(the)h(case,)f(and)g(let)h FC(e)f
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n(t)h(edge)g FC(f)40 b FK(that)32 b(dominates)523 1303
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b FC(N)9 b FK(,)523 1403 y(con)n(tradicting)30 b(our)h(c)n(hoice)f(of)h
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FK(=)g FC(k)j FD(!)e FC(j)36 b FK(is)31 b(in)h FC(M)d
FD(n)21 b FC(N)9 b FK(.)523 1503 y(The)30 b(t)n(w)n(o)f
FC(N)39 b FK(edges)29 b FC(i)d FD(!)h FC(j)35 b FK(and)30
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FC(e)g FK(are)f(not)g(in)523 1602 y FC(M)9 b FK(,)32
b(and)g(th)n(us)h(are)e(dominated)h(b)n(y)g FC(e)p FK(.)g(Since)h
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FC(;)14 b FG(ov)9 b FK(\()p FC(k)s(;)14 b(l)r FK(\))p
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FC(N)38 b FK(these)523 1802 y(t)n(w)n(o)31 b(edges)g(with)h
FC(e)e FK(=)f FC(k)k FD(!)d FC(j)36 b FK(and)c FC(i)d
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FC(N)2999 1771 y Fw(0)3053 1802 y FK(that)h(has)523 1901
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b(This)g(w)n(ould)523 2001 y(con)n(tradict)27 b(our)f(c)n(hoice)h(of)h
FC(N)9 b FK(.)1790 b Fu(2)606 2139 y FK(Since)29 b(algorithm)f(MGREED)n
(Y)g(\014nds)h(an)f(optimal)h(assignmen)n(t,)f(the)h(string)f(it)h
(pro-)523 2239 y(duces)j(is)h(no)f(longer)f(than)h(the)h(string)f(pro)r
(duced)g(b)n(y)g(algorithm)f(Concat-Cycles.)g(\(In)523
2338 y(fact,)d(it)g(could)f(b)r(e)h(shorter)f(since)g(it)h(breaks)e
(eac)n(h)h(cycle)g(in)h(the)g(optim)n(um)g(p)r(osition.\))523
2651 y FL(2.4)116 b(Impro)m(ving)36 b(to)i(3)26 b Fr(\001)h
FL(OPT)o(\()p Fq(S)7 b FL(\))523 2850 y FK(Recall)32
b(that)h(in)f(the)h(last)f(step)h(of)f(algorithm)f(MGREED)n(Y,)h(w)n(e)
g(simply)h(concatenate)523 2950 y(all)c(the)h(strings)f(in)g(set)h
FC(T)40 b FK(without)30 b(an)n(y)f(compression.)f(In)n(tuitiv)n(ely)-7
b(,)30 b(if)g(w)n(e)f(instead)g(try)523 3050 y(to)i(o)n(v)n(erlap)e
(the)i(strings)f(in)h FC(T)12 b FK(,)30 b(w)n(e)h(migh)n(t)f(b)r(e)i
(able)e(to)h(ac)n(hiev)n(e)f(a)g(b)r(ound)h(b)r(etter)h(than)523
3149 y(4)22 b FD(\001)g FK(OPT)o(\()p FC(S)5 b FK(\).)33
b(Let)h(TGREED)n(Y)f(denote)g(the)g(algorithm)f(that)h(op)r(erates)f
(in)i(the)f(same)523 3249 y(w)n(a)n(y)28 b(as)g(MGREED)n(Y)g(except)h
(that)g(in)g(the)g(last)g(step,)f(it)i(merges)d(the)i(strings)f(in)h
FC(T)40 b FK(b)n(y)523 3349 y(running)30 b(GREED)n(Y)g(on)g(them.)g(W)
-7 b(e)31 b(can)f(sho)n(w)f(that)h(TGREED)n(Y)g(indeed)h(ac)n(hiev)n
(es)d(a)523 3448 y(b)r(etter)g(b)r(ound:)g(it)g(pro)r(duces)f(a)g(sup)r
(erstring)g(of)g(length)h(at)g(most)f(3)18 b FD(\001)g
FK(OPT)o(\()p FC(S)5 b FK(\).)523 3603 y Fv(Theorem)32
b FH(2.11)41 b FG(A)n(lgorithm)22 b(TGREED)n(Y)f(pr)l(o)l(duc)l(es)h(a)
g(sup)l(erstring)f(of)i(length)e(at)h(most)523 3703 y
FK(3)c FD(\001)h FG(OPT)p FK(\()p FC(S)5 b FK(\))p FG(.)523
3858 y Fv(Pr)n(oof.)38 b FK(Let)h FC(S)46 b FK(=)41 b
FD(f)p FC(s)1292 3870 y FB(1)1328 3858 y FC(;)14 b(:)g(:)g(:)g(;)g(s)
1552 3870 y FA(m)1615 3858 y FD(g)38 b FK(b)r(e)h(a)f(set)h(of)f
(strings)g(and)g FC(s)h FK(b)r(e)g(the)g(sup)r(erstring)523
3957 y(obtained)34 b(b)n(y)h(TGREED)n(Y)f(on)g FC(S)5
b FK(.)35 b(Let)f FC(n)h FK(=)f(OPT)o(\()p FC(S)5 b FK(\))35
b(b)r(e)g(the)g(length)g(of)f(a)h(shortest)523 4057 y(sup)r(erstring)27
b(of)g FC(S)5 b FK(.)28 b(W)-7 b(e)28 b(sho)n(w)e(that)i
FD(j)p FC(s)p FD(j)23 b(\024)g FK(3)p FC(n)p FK(.)606
4157 y(Let)c FC(T)30 b FK(b)r(e)19 b(the)f(set)h(of)g(all)f(\\self-o)n
(v)n(erlapping")d(strings)j(obtained)g(b)n(y)g(MGREED)n(Y)h(on)f
FC(S)523 4256 y FK(and)23 b FC(C)29 b FK(b)r(e)24 b(the)f(assignmen)n
(t)f(created)g(b)n(y)h(MGREED)n(Y.)g(F)-7 b(or)22 b(eac)n(h)h
FC(x)g FD(2)g FC(T)12 b FK(,)23 b(let)g FC(c)3053 4268
y FA(x)3118 4256 y FK(denote)523 4356 y(the)j(cycle)e(in)i
FC(C)31 b FK(corresp)r(onding)24 b(to)h(string)f FC(x)p
FK(,)i(and)f(let)h FC(w)2347 4368 y FA(x)2412 4356 y
FK(=)d FC(w)r FK(\()p FC(c)2629 4368 y FA(x)2671 4356
y FK(\))j(b)r(e)f(its)h(w)n(eigh)n(t.)e(F)-7 b(or)523
4456 y(an)n(y)21 b(set)h FC(R)g FK(of)g(strings,)e(de\014ne)i
FD(jj)p FC(R)q FD(jj)h FK(=)1758 4393 y Fn(P)1845 4480
y FA(x)p Fw(2)p FA(R)1996 4456 y FD(j)p FC(x)p FD(j)f
FK(to)g(b)r(e)g(the)g(total)f(length)h(of)g(the)g(strings)523
4555 y(in)30 b(set)g FC(R)q FK(.)g(Also)f(let)h FC(w)g
FK(=)1362 4493 y Fn(P)1450 4580 y FA(x)p Fw(2)p FA(T)1598
4555 y FC(w)1657 4567 y FA(x)1700 4555 y FK(.)g(Since)g(CYC\()p
FC(G)2251 4567 y FA(S)2300 4555 y FK(\))d FD(\024)f FK(TSP\()p
FC(G)2710 4567 y FA(S)2758 4555 y FK(\))h FD(\024)g FK(OPT)o(\()p
FC(S)5 b FK(\),)30 b(w)n(e)523 4655 y(ha)n(v)n(e)c FC(w)g
FD(\024)d FC(n)p FK(.)606 4754 y(By)33 b(Lemma)h(2.9,)f(the)h
(compression)e(ac)n(hiev)n(ed)g(in)i(a)f(shortest)g(sup)r(erstring)g
(of)g FC(T)45 b FK(is)523 4854 y(less)26 b(than)g(2)p
FC(w)r FK(,)h(i.e.,)f FD(jj)p FC(T)12 b FD(jj)j(\000)g
FC(n)1473 4866 y FA(T)1548 4854 y FD(\024)23 b FK(2)p
FC(w)r FK(.)j(By)g(the)h(results)e(in)i([15)o(,)f(16)o(],)h(w)n(e)f
(kno)n(w)f(that)h(the)523 4954 y(compression)32 b(ac)n(hiev)n(ed)g(b)n
(y)h(GREED)n(Y)g(on)g(set)h FC(T)44 b FK(is)34 b(at)f(least)g(half)h
(the)f(compression)523 5053 y(ac)n(hiev)n(ed)26 b(in)i(an)n(y)f(sup)r
(erstring)g(of)g FC(T)12 b FK(.)27 b(That)h(is,)602 5208
y FD(jj)p FC(T)12 b FD(jj)17 b(\000)h(j)p FC(s)p FD(j)24
b(\025)e FK(\()p FD(jj)p FC(T)12 b FD(jj)18 b(\000)g
FC(n)1387 5220 y FA(T)1439 5208 y FK(\))p FC(=)p FK(2)23
b(=)f FD(jj)p FC(T)12 b FD(jj)18 b(\000)g FC(n)1969 5220
y FA(T)2040 5208 y FD(\000)g FK(\()p FD(jj)p FC(T)12
b FD(jj)18 b(\000)g FC(n)2459 5220 y FA(T)2511 5208 y
FK(\))p FC(=)p FK(2)k FD(\025)h(jj)p FC(T)12 b FD(jj)18
b(\000)g FC(n)3041 5220 y FA(T)3111 5208 y FD(\000)g
FC(w)r(:)523 5363 y FK(So,)27 b FD(j)p FC(s)p FD(j)d(\024)e
FC(n)907 5375 y FA(T)978 5363 y FK(+)c FC(w)r FK(.)p
eop end
%%Page: 17 27
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518 y FA(x)2441 506 y FK(that)h(is)g(a)f(pre\014x)g(of)h
FC(x)p FK(.)h(Let)523 606 y FC(S)579 576 y Fw(0)633 606
y FK(=)30 b FD(f)p FC(s)809 618 y FA(i)832 626 y Fx(x)873
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606 y FD(j)p FC(x)31 b FD(2)g FC(T)12 b FD(g)p FK(,)30
b(and)i FC(n)3220 576 y Fw(00)3293 606 y FK(=)523 706
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805 y(By)e(Claim)g(2.4,)g(a)f(sup)r(erstring)h(for)f
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y FA(\031)3246 905 y FD(j)24 b FK(+)523 942 y Fn(P)611
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1025 y FA(\031)2175 1005 y FK(are)f(the)i(sup)r(erstrings)e(obtained)h
(b)n(y)523 1104 y(o)n(v)n(erlapping)g(the)j(mem)n(b)r(ers)g(of)f
FC(S)1629 1074 y Fw(0)1684 1104 y FK(and)g FC(S)1905
1074 y Fw(00)1947 1104 y FK(,)h(resp)r(ectiv)n(ely)-7
b(,)31 b(in)h(the)g(order)e(giv)n(en)h(b)n(y)g FC(\031)s
FK(.)523 1204 y(Observ)n(e)26 b(that)i FC(S)1078 1174
y Fw(0)1124 1204 y FD(\022)23 b FC(S)32 b FK(implies)c
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1386 y FD(\024)g FC(n)1907 1352 y Fw(0)1948 1386 y FK(+)18
b FC(w)26 b FD(\024)d FC(n)18 b FK(+)g FC(w)r(:)523 1569
y FK(Com)n(bined)27 b(with)h FD(j)p FC(s)p FD(j)c(\024)e
FC(n)1352 1581 y FA(T)1423 1569 y FK(+)c FC(w)r FK(,)28
b(this)g(giv)n(es)f FD(j)p FC(s)p FD(j)c(\024)f FC(n)d
FK(+)f(2)p FC(w)25 b FD(\024)d FK(3)p FC(n:)637 b Fu(2)523
1926 y FL(2.5)116 b(GREED)m(Y)38 b(ac)m(hiev)m(es)g(linear)f(appro)m
(ximation)523 2125 y FK(One)32 b(w)n(ould)f(exp)r(ect)h(that)g(an)g
(analysis)e(similar)h(to)h(that)g(of)g(MGREED)n(Y)f(w)n(ould)h(also)523
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(out)g(not)g(to)f(b)r(e)i(the)f(case.)f(The)h(analysis)523
2325 y(of)f(GREED)n(Y)f(is)g(sev)n(erely)f(complicated)i(b)n(y)f(the)h
(fact)g(that)f(it)h(con)n(tin)n(ues)f(pro)r(cessing)f(the)523
2424 y(\\self-o)n(v)n(erlapping")27 b(strings.)j(MGREED)n(Y)h(w)n(as)f
(esp)r(ecially)g(designed)h(to)f(a)n(v)n(oid)g(these)523
2524 y(complications,)22 b(b)n(y)i(separating)d(suc)n(h)i(strings.)g
(Let)g FC(GR)q(E)5 b(E)g(D)r(Y)19 b FK(\()p FC(S)5 b
FK(\))24 b(denote)f(the)h(length)523 2623 y(of)31 b(the)h(sup)r
(erstring)e(pro)r(duced)h(b)n(y)g(GREED)n(Y)h(on)f(a)f(set)i
FC(S)5 b FK(.)31 b(It)h(is)f(tempting)h(to)f(claim)523
2723 y(that)1154 2823 y FC(GR)q(E)5 b(E)g(D)r(Y)19 b
FK(\()p FC(S)k FD([)c(f)p FC(s)p FD(g)p FK(\))j FD(\024)h
FC(GR)q(E)5 b(E)g(D)r(Y)19 b FK(\()p FC(S)5 b FK(\))19
b(+)f FD(j)p FC(s)p FD(j)p FC(:)523 2972 y FK(If)34 b(this)h(w)n(ere)d
(true,)i(a)g(simple)g(argumen)n(t)f(w)n(ould)g(extend)h(the)h(4)22
b FD(\001)g FK(OPT)o(\()p FC(S)5 b FK(\))35 b(result)e(for)523
3072 y(MGREED)n(Y)41 b(to)g(GREED)n(Y.)f(But)i(the)f(follo)n(wing)f
(coun)n(terexample)f(dispro)n(v)n(es)g(this)523 3171
y(seemingly)27 b(inno)r(cen)n(t)h(claim.)f(Let)1044 3354
y FC(S)h FK(=)22 b FD(f)p FC(ca)1332 3320 y FA(m)1395
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y FC(c)1659 3320 y FA(m)1722 3354 y FC(;)g(c)1795 3320
y FA(m)1858 3354 y FC(b)1894 3320 y FA(m)p FB(+1)2040
3354 y FC(;)g(b)2113 3320 y FA(m)2176 3354 y FC(c)p FD(g)p
FC(;)g(s)22 b FK(=)h FC(b)2476 3320 y FA(m)p FB(+1)2623
3354 y FC(a)2667 3320 y FA(m)p FB(+1)2814 3354 y FC(:)523
3537 y FK(No)n(w)38 b FC(GR)q(E)5 b(E)g(D)r(Y)19 b FK(\()p
FC(S)5 b FK(\))42 b(=)g FD(j)p FC(ca)1494 3507 y FA(m)p
FB(+1)1641 3537 y FC(c)1677 3507 y FA(m)1740 3537 y FC(b)1776
3507 y FA(m)p FB(+1)1922 3537 y FC(c)p FD(j)g FK(=)g(3)p
FC(m)25 b FK(+)h(4,)38 b(whereas)g FC(GR)q(E)5 b(E)g(D)r(Y)18
b FK(\()p FC(S)31 b FD([)523 3636 y(f)p FC(s)p FD(g)p
FK(\))22 b(=)h FD(j)p FC(b)847 3606 y FA(m)910 3636 y
FC(c)946 3606 y FA(m)1009 3636 y FC(b)1045 3606 y FA(m)p
FB(+1)1191 3636 y FC(a)1235 3606 y FA(m)p FB(+1)1382
3636 y FC(c)1418 3606 y FA(m)1481 3636 y FC(a)1525 3606
y FA(m)1588 3636 y FD(j)h FK(=)e(6)p FC(m)c FK(+)g(2)23
b FC(>)f FK(\(3)p FC(m)c FK(+)g(4\))h(+)f(\(2)p FC(m)g
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n(e)h(will)h(nev)n(ertheless)e(sho)n(w)h(that)523 3902
y Fv(Theorem)32 b FH(2.12)41 b FG(GREED)n(Y)29 b(pr)l(o)l(duc)l(es)h(a)
g(string)g(of)h(length)f(at)f(most)h FK(4)18 b FD(\001)g
FG(OPT)q FK(\()p FC(S)5 b FK(\))p FG(.)606 4068 y FK(Before)19
b(pro)n(ving)f(the)j(theorem)e(formally)-7 b(,)19 b(w)n(e)h(giv)n(e)f
(a)h(sk)n(etc)n(h)f(of)h(the)g(basic)f(idea)h(b)r(ehind)523
4168 y(the)35 b(pro)r(of.)e(If)i(w)n(e)f(w)n(an)n(t)g(to)g(relate)f
(the)i(merges)e(done)h(b)n(y)g(GREED)n(Y)g(to)g(an)g(optimal)523
4267 y(assignmen)n(t,)e(w)n(e)g(ha)n(v)n(e)g(to)h(k)n(eep)f(trac)n(k)g
(of)h(what)f(happ)r(ens)h(when)g(GREED)n(Y)g(violates)523
4367 y(the)d(maxim)n(um)f(o)n(v)n(erlap)e(principle,)i(i.e.)h(when)f
(some)g(self-o)n(v)n(erlap)e(is)i(b)r(etter)h(than)f(the)523
4467 y(o)n(v)n(erlap)k(in)i(GREED)n(Y's)g(merge.)f(One)h(thing)g(to)g
(try)g(is)f(to)h(c)n(harge)f(GREED)n(Y)g(some)523 4566
y(extra)e(cost)g(that)h(re\015ects)f(that)h(an)f(optimal)h(assignmen)n
(t)e(on)i(the)g(new)f(set)h(of)g(strings)523 4666 y(\(with)20
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(the)f(optimal)h(assignmen)n(t)523 4765 y(on)g(the)h(former)e(set)i
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(form)g(a)g(cycle\).)g(If)h(w)n(e)523 4865 y(could)26
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(ha)n(v)n(e)f(a)g(b)r(ound)i(for)e(GREED)n(Y.)523 4965
y(Unfortunately)-7 b(,)33 b(this)g(approac)n(h)e(fails)h(b)r(ecause)g
(the)h(self-o)n(v)n(erlapping)d(string)i(ma)n(y)g(b)r(e)523
5064 y(merged)24 b(b)n(y)h(GREED)n(Y)g(in)n(to)f(a)h(larger)e(string)h
(whic)n(h)h(itself)h(b)r(ecomes)e(self-o)n(v)n(erlapping,)523
5164 y(and)e(this)h(nesting)g(could)f(go)g(arbitrarily)e(deep.)j(Our)f
(pro)r(of)g(concen)n(trates)f(on)h(the)h(inner-)523 5264
y(most)28 b(self-o)n(v)n(erlapping)e(strings)i(only)-7
b(.)28 b(These)g(so)g(called)g(culprits)g(form)g(a)g(linear)g(order)523
5363 y(in)23 b(the)h(\014nal)f(sup)r(erstring.)f(W)-7
b(e)24 b(a)n(v)n(oid)d(the)j(complications)e(of)h(higher)g(lev)n(el)f
(self-o)n(v)n(erlaps)p eop end
%%Page: 18 28
TeXDict begin 18 27 bop 1599 299 a FF(2.5.)52 b(GREED)n(Y)25
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(more)f(clev)n(er)f(w)n(a)n(y)-7 b(,)26 b(w)n(e)523 1005
y(can)e(ev)n(en)g(eliminate)g(the)g(e\013ect)h(of)f(the)h(extra)e
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FK(OPT)m(\()p FC(S)5 b FK(\))523 1104 y(b)r(ound)28 b(as)f(w)n(e)g
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1303 y(as)25 b(taking)h(a)g(list)g(of)g(all)g(edges)f(sorted)g(b)n(y)h
(o)n(v)n(erlap,)e(and)i(going)f(do)n(wn)g(the)i(list)f(deciding)523
1403 y(for)g(eac)n(h)f(edge)g(whether)h(to)g(include)h(it)f(or)g(not.)g
(Call)g(an)f(edge)h FG(b)l(etter)35 b FK(\()p FG(worse)6
b FK(\))27 b(if)g(it)f(ap-)523 1503 y(p)r(ears)i(b)r(efore)g(\(after\))
g(another)f(in)i(this)g(list.)f(Better)g(edges)g(ha)n(v)n(e)f(at)h
(least)g(the)h(o)n(v)n(erlap)523 1602 y(of)c(w)n(orse)f(ones.)h(Recall)
g(that)h(an)f(edge)g(dominates)g(another)g(i\013)h(it)g(is)f(b)r(etter)
h(and)f(shares)523 1702 y(its)j(head)f(or)g(tail)g(with)i(the)f(other)f
(one.)606 1802 y(A)n(t)h(the)g(end,)g(GREED)n(Y)f(has)g(formed)g(a)h
(Hamiltonian)f(path)1571 1980 y FC(s)1610 1992 y FB(1)1670
1980 y FD(!)c FC(s)1815 1992 y FB(2)1876 1980 y FD(!)g(\001)14
b(\001)g(\001)23 b(!)g FC(s)2247 1992 y FA(m)523 2158
y FK(of)37 b(`greedy')e(edges.)h(\(w.l.o.g.,)g(the)h(strings)f(are)f
(ren)n(um)n(b)r(ered)h(to)g(re\015ect)h(their)f(order)523
2258 y(in)i(the)g(sup)r(erstring)f(pro)r(duced)g(b)n(y)g(GREED)n(Y.\))h
(F)-7 b(or)37 b(con)n(v)n(enience)f(w)n(e)h(will)h(usually)523
2358 y(abbreviate)26 b FC(s)970 2370 y FA(i)1026 2358
y FK(to)h FC(i)p FK(.)h(GREED)n(Y)f(do)r(es)g(not)h(include)g(an)f
(edge)g FC(f)36 b FK(i\013)625 2520 y(1.)41 b FC(f)36
b FK(is)27 b(dominated)h(b)n(y)f(an)g(already)g(c)n(hosen)f(edge)h
FC(e)p FK(,)h(or)625 2685 y(2.)41 b FC(f)36 b FK(is)27
b(not)h(dominated)f(but)i(it)f(w)n(ould)f(form)g(a)g(cycle.)523
2847 y(Let)39 b(us)h(call)e(the)i(latter)f(\\bad)g(bac)n(k)f(edges";)g
(a)h(bad)g(bac)n(k)f(edge)h FC(f)51 b FK(=)42 b FC(j)48
b FD(!)42 b FC(i)d FK(nec-)523 2947 y(essarily)c(has)i
FC(i)h FD(\024)g FC(j)5 b FK(.)37 b(Eac)n(h)e(bad)i(bac)n(k)f(edge)g
FC(f)47 b FK(=)38 b FC(j)43 b FD(!)c FC(i)d FK(corresp)r(onds)f(to)i(a)
f(string)523 3047 y FD(h)p FC(s)594 3059 y FA(i)622 3047
y FC(;)14 b(s)698 3059 y FA(i)p FB(+1)809 3047 y FC(;)g(:)g(:)g(:)g(;)g
(s)1033 3059 y FA(j)1068 3047 y FD(i)33 b FK(that,)f(at)h(some)f(p)r
(oin)n(t)g(in)h(the)g(execution)f(of)g(GREED)n(Y,)h(has)f(more)523
3146 y(\(self)6 b(\))25 b(o)n(v)n(erlap)d(than)i(the)g(pair)f(that)i
(is)e(merged.)h(When)g(GREED)n(Y)g(considers)e FC(f)9
b FK(,)24 b(it)g(has)523 3246 y(already)30 b(c)n(hosen)g(all)h(\(b)r
(etter\))h(edges)f(on)g(the)h(greedy)e(path)h(from)g
FC(i)g FK(to)g FC(j)5 b FK(,)32 b(but)g(not)f(y)n(et)523
3346 y(the)26 b(\(w)n(orse\))e(edges)g FC(i)14 b FD(\000)g
FK(1)21 b FD(!)i FC(i)i FK(and)g FC(j)j FD(!)c FC(j)18
b FK(+)c(1.)24 b(The)i(bad)f(bac)n(k)f(edge)h FC(f)34
b FK(is)25 b(said)f(to)h FG(sp)l(an)523 3445 y FK(the)34
b(closed)f(in)n(terv)-5 b(al)33 b FC(I)1265 3457 y FA(f)1341
3445 y FK(=)f([)p FC(i;)14 b(j)5 b FK(].)33 b(The)h(ab)r(o)n(v)n(e)e
(observ)-5 b(ations)32 b(pro)n(vide)g(a)h(pro)r(of)g(of)g(the)523
3545 y(follo)n(wing)27 b(lemma.)523 3709 y Fv(Lemma)32
b FH(2.13)40 b FG(L)l(et)25 b FC(e)g FG(and)h FC(f)34
b FG(b)l(e)25 b(two)h(b)l(ad)g(b)l(ack)g(e)l(dges.)h(The)f(close)l(d)h
(intervals)f FC(I)3060 3721 y FA(e)3121 3709 y FG(and)g
FC(I)3314 3721 y FA(f)523 3809 y FG(ar)l(e)33 b(either)g(disjoint,)h
(or)f(one)f(c)l(ontains)h(the)f(other.)i(If)f FC(I)2344
3821 y FA(e)2407 3809 y FD(\033)28 b FC(I)2536 3821 y
FA(f)2612 3809 y FG(then)k FC(e)g FG(is)h(worse)g(than)523
3908 y FC(f)38 b FG(\(thus,)30 b(ov)10 b FK(\()p FC(e)p
FK(\))23 b FD(\024)f FG(ov)10 b FK(\()p FC(f)f FK(\))p
FG(\).)606 4073 y FK(Th)n(us,)28 b(the)g(in)n(terv)-5
b(als)27 b(of)h(the)g(bad)g(bac)n(k)f(edges)g(are)g(nested)h(and)g(bad)
g(bac)n(k)f(edges)g(do)523 4172 y(not)c(cross)f(eac)n(h)h(other.)g
FG(Culprits)30 b FK(are)23 b(the)g(minimal)h(\(innermost\))f(suc)n(h)g
(in)n(terv)-5 b(als.)23 b(Eac)n(h)523 4272 y(culprit)f([)p
FC(i;)14 b(j)5 b FK(])22 b(corresp)r(onds)e(to)i(a)g
FG(culprit)j(string)k FD(h)p FC(s)2134 4284 y FA(i)2162
4272 y FC(;)14 b(s)2238 4284 y FA(i)p FB(+1)2350 4272
y FC(;)g(:)g(:)g(:)f(;)h(s)2573 4284 y FA(j)2608 4272
y FD(i)p FK(.)23 b(Note)f(that,)g(b)r(ecause)523 4372
y(of)i(the)g(minimalit)n(y)g(of)f(the)h(culprits,)g(if)g
FC(f)32 b FK(=)22 b FC(j)28 b FD(!)23 b FC(i)h FK(is)f(the)h(bac)n(k)f
(edge)g(of)h(a)f(culprit)h([)p FC(i;)14 b(j)5 b FK(],)523
4471 y(and)29 b FC(e)g FK(is)h(another)e(bad)h(bac)n(k)g(edge)g(that)g
(shares)f(head)i(or)e(tail)h(with)h FC(f)9 b FK(,)29
b(then)h FC(I)3102 4483 y FA(e)3164 4471 y FD(\033)c
FC(I)3291 4483 y FA(f)3335 4471 y FK(,)523 4571 y(and)h(therefore)g
FC(f)36 b FK(dominates)28 b FC(e)p FK(.)606 4671 y(Call)40
b(the)h(w)n(orst)f(edge)g(b)r(et)n(w)n(een)g(ev)n(ery)f(t)n(w)n(o)h
(successiv)n(e)f(culprits)i(on)f(the)h(greedy)523 4770
y(path)36 b(a)f FG(we)l(ak)j(link)9 b FK(.)36 b(Note)g(that)f(w)n(eak)g
(links)h(are)e(also)h(w)n(orse)f(than)h(all)h(edges)f(in)g(the)523
4870 y(t)n(w)n(o)g(adjacen)n(t)f(culprits)i(as)e(w)n(ell)h(as)g(their)g
(bac)n(k)g(edges.)g(If)g(w)n(e)g(remo)n(v)n(e)f(all)h(the)h(w)n(eak)523
4969 y(links,)e(the)f(greedy)g(path)g(is)h(partitioned)f(in)n(to)g(a)g
(set)h(of)f(paths,)h(called)f FG(blo)l(cks)7 b FK(.)34
b(Ev)n(ery)523 5069 y(blo)r(c)n(k)40 b(consists)g(of)h(a)f(nonempt)n(y)
h(culprit)g(as)f(the)h(middle)g(segmen)n(t,)f(and)h(\(p)r(ossibly)523
5169 y(empt)n(y\))20 b(left)h(and)f(righ)n(t)f FG(extensions)7
b FK(.)19 b(The)h(set)g(of)g(strings)f(\(no)r(des\))h
FC(S)25 b FK(is)19 b(th)n(us)h(partitioned)523 5268 y(in)n(to)39
b(three)f(sets)h FC(S)1151 5280 y FA(l)1176 5268 y FC(;)14
b(S)1264 5280 y FA(m)1327 5268 y FC(;)g(S)1415 5280 y
FA(r)1491 5268 y FK(of)38 b(left,)i(middle,)f(and)g(righ)n(t)f
(strings.)g(The)h(example)f(in)523 5368 y(Figure)30 b(2.3)g(has)h(7)f
(substrings,)g(of)h(whic)n(h)g(2)f(b)n(y)h(itself)g(and)g(the)g(merge)f
(of)h(4,)g(5,)f(and)h(6)p eop end
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 @endspecial 912 872 a(FIGURE)26 b(2.3.)h(Culprits)f(and)f(w)n(eak)h
(links)g(in)g(Greedy)f(merge)h(path.)523 1136 y FK(form)c(the)i
(culprits)e(\(indicated)h(b)n(y)g(thic)n(k)n(er)f(lines\).)h(Bad)f(bac)
n(k)g(edges)g(are)f(2)i FD(!)g FK(2,)f(6)h FD(!)g FK(4,)523
1235 y(and)33 b(6)g FD(!)g FK(1.)g(The)h(w)n(eak)f(link)h(3)e
FD(!)h FK(4)g(is)h(the)g(w)n(orst)e(edge)h(b)r(et)n(w)n(een)h(culprits)
g([2])f(and)523 1335 y([4)p FC(;)14 b FK(5)p FC(;)g FK(6].)23
b(The)h(blo)r(c)n(ks)g(in)g(this)h(example)f(are)f(th)n(us)i([1)p
FC(;)14 b FK(2)p FC(;)g FK(3])22 b(and)j([4)p FC(;)14
b FK(5)p FC(;)g FK(6)p FC(;)g FK(7],)22 b(and)i(w)n(e)g(ha)n(v)n(e)523
1435 y FC(S)574 1447 y FA(l)622 1435 y FK(=)f FD(f)p
FK(1)p FD(g)p FC(;)14 b(S)924 1447 y FA(m)1009 1435 y
FK(=)22 b FD(f)p FK(2)p FC(;)14 b FK(4)p FC(;)g FK(5)p
FC(;)g FK(6)p FD(g)p FC(;)g(S)1547 1447 y FA(r)1604 1435
y FK(=)22 b FD(f)p FK(3)p FC(;)14 b FK(7)p FD(g)p FK(.)606
1534 y(The)28 b(follo)n(wing)e(lemma)i(sho)n(ws)e(that)i(a)g(bad)f(bac)
n(k)g(edge)g(m)n(ust)h(b)r(e)g(from)f(a)g(middle)i(or)523
1634 y(righ)n(t)e(no)r(de)h(to)f(a)g(middle)h(or)f(left)h(no)r(de.)523
1800 y Fv(Lemma)k FH(2.14)40 b FG(L)l(et)29 b FC(f)i
FK(=)23 b FC(j)28 b FD(!)23 b FC(i)29 b FG(b)l(e)g(a)g(b)l(ad)h(b)l
(ack)g(e)l(dge.)g(No)l(de)f FC(i)g FG(is)g(either)h(a)f(left)h(no)l(de)
f(or)523 1899 y(the)k(\014rst)g(no)l(de)h(of)g(a)g(culprit.)g(No)l(de)f
FC(j)39 b FG(is)33 b(either)h(a)g(right)g(no)l(de)g(or)f(the)h(last)f
(no)l(de)h(of)g(a)523 1999 y(culprit.)523 2165 y Fv(Pr)n(oof.)24
b FK(Let)h FC(c)e FK(=)g([)p FC(k)s(;)14 b(l)r FK(])24
b(b)r(e)i(the)f(leftmost)g(culprit)h(in)f FC(I)2272 2177
y FA(f)2315 2165 y FK(.)g(No)n(w)g(either)g FC(i)d FK(=)h
FC(k)28 b FK(is)d(the)g(\014rst)523 2265 y(no)r(de)g(of)f
FC(c)p FK(,)g(or)g FC(i)f(<)f(k)28 b FK(is)c(in)h(the)f(left)i
(extension)e(of)g FC(c)p FK(,)g(or)g FC(i)f(<)f(k)28
b FK(is)c(in)h(the)g(righ)n(t)e(extension)523 2364 y(of)36
b(the)h(culprit)g FC(c)1091 2334 y Fw(0)1150 2364 y FK(to)f(the)h(left)
g(of)g FC(c)p FK(.)f(In)g(the)h(latter)f(case)g(ho)n(w)n(ev)n(er,)e
FC(I)2835 2376 y FA(f)2915 2364 y FK(includes)i(the)523
2464 y(w)n(eak)31 b(link,)g(whic)n(h)h(b)n(y)f(de\014nition)h(is)g(w)n
(orse)e(than)h(all)h(edges)e(b)r(et)n(w)n(een)i(the)g(culprits)f
FC(c)3334 2434 y Fw(0)523 2564 y FK(and)25 b FC(c)p FK(,)g(including)h
(the)g(edge)e FC(i)14 b FD(\000)g FK(1)22 b FD(!)h FC(i)p
FK(.)i(This)g(con)n(tradicts)f(the)i(observ)-5 b(ation)24
b(preceding)523 2663 y(Lemma)j(2.13.)g(A)h(similar)e(argumen)n(t)h
(holds)g(for)g FC(s)2135 2675 y FA(j)2170 2663 y FK(.)1103
b Fu(2)606 2813 y FK(Let)28 b FC(C)814 2825 y FA(m)906
2813 y FK(b)r(e)g(the)h(assignmen)n(t)e(on)g(the)i(set)f
FC(S)2031 2825 y FA(m)2122 2813 y FK(of)g(middle)g(strings)g(\(no)r
(des\))g(that)g(has)523 2912 y(one)22 b(cycle)h(for)f(eac)n(h)g
(culprit,)h(consisting)f(of)h(the)g(greedy)f(edges)g(together)g(with)h
(the)h(bac)n(k)523 3012 y(edge)g(of)g(the)h(culprit.)g(If)g(w)n(e)f
(consider)f(the)i(application)f(of)g(the)h(algorithm)e(MGREED)n(Y)523
3112 y(on)32 b(the)g(subset)f(of)h(strings)f FC(S)1471
3124 y FA(m)1534 3112 y FK(,)h(it)g(is)g(easy)f(to)g(see)h(that)g(the)g
(algorithm)e(will)i(actually)523 3211 y(construct)24
b(the)i(assignmen)n(t)d FC(C)1509 3223 y FA(m)1573 3211
y FK(.)i(Theorem)f(2.10)f(then)j(implies)f(the)g(follo)n(wing)f(lemma.)
523 3377 y Fv(Lemma)32 b FH(2.15)40 b FC(C)1103 3389
y FA(m)1196 3377 y FG(is)30 b(an)g(optimal)h(assignment)f(on)g(the)g
(set)f FC(S)2571 3389 y FA(m)2664 3377 y FG(of)h(midd)t(le)i(strings.)
606 3543 y FK(Let)h(the)g(graph)f FC(G)1214 3555 y FA(l)1272
3543 y FK(=)f(\()p FC(V)1448 3555 y FA(l)1474 3543 y
FC(;)14 b(E)1572 3555 y FA(l)1598 3543 y FK(\))33 b(consist)f(of)h(the)
g(left/middle)h(part)e(of)h(all)f(blo)r(c)n(ks)g(in)523
3643 y(the)d(greedy)f(path,)i(i.e.)f FC(V)1336 3655 y
FA(l)1387 3643 y FK(=)c FC(S)1528 3655 y FA(l)1573 3643
y FD([)20 b FC(S)1699 3655 y FA(m)1791 3643 y FK(and)29
b FC(E)2015 3655 y FA(l)2070 3643 y FK(is)g(the)g(set)g(of)g(non-w)n
(eak)f(greedy)g(edges)523 3743 y(b)r(et)n(w)n(een)20
b(no)r(des)f(of)h FC(V)1196 3755 y FA(l)1222 3743 y FK(.)f(Let)h
FC(M)1486 3755 y FA(l)1531 3743 y FK(b)r(e)g(a)f(maxim)n(um)g(o)n(v)n
(erlap)f(assignmen)n(t)h(on)g FC(V)2932 3755 y FA(l)2958
3743 y FK(,)h(as)f(created)523 3842 y(b)n(y)31 b(MGREED)n(Y)f(on)h(the)
g(ordered)f(sublist)h(of)g(edges)f(in)h FC(V)2418 3854
y FA(l)2464 3842 y FD(\002)21 b FC(V)2598 3854 y FA(l)2624
3842 y FK(.)31 b(Let)g FC(V)2878 3854 y FA(r)2943 3842
y FK(=)d FC(S)3087 3854 y FA(m)3171 3842 y FD([)21 b
FC(S)3298 3854 y FA(r)3335 3842 y FK(,)523 3942 y(and)30
b(de\014ne)h(similarly)e(the)i(graph)e FC(G)1721 3954
y FA(r)1786 3942 y FK(=)e(\()p FC(V)1958 3954 y FA(r)1995
3942 y FC(;)14 b(E)2093 3954 y FA(r)2130 3942 y FK(\))31
b(and)f(the)h(optimal)f(assignmen)n(t)f FC(M)3321 3954
y FA(r)523 4041 y FK(on)39 b(the)g(righ)n(t/middle)f(strings.)g(Let)h
FC(l)1790 4053 y FA(c)1862 4041 y FK(b)r(e)g(the)h(sum)f(of)f(the)i
(lengths)e(of)h(all)g(culprit)523 4141 y(strings.)32
b(De\014ne)h FC(l)1107 4153 y FA(l)1163 4141 y FK(=)1259
4079 y Fn(P)1346 4166 y FA(i)p Fw(2)p FA(S)1455 4175
y Fx(l)1497 4141 y FG(d)9 b FK(\()p FC(s)1619 4153 y
FA(i)1647 4141 y FC(;)14 b(s)1723 4153 y FA(i)p FB(+1)1835
4141 y FK(\))33 b(as)e(the)i(total)f(length)h(of)f(all)g(left)h
(extensions)523 4253 y(and)i FC(l)717 4265 y FA(r)788
4253 y FK(=)888 4191 y Fn(P)976 4278 y FA(i)p Fw(2)p
FA(S)1085 4286 y Fx(r)1136 4253 y FG(d)9 b FK(\()p FC(s)1258
4223 y FA(R)1258 4275 y(i)1313 4253 y FC(;)14 b(s)1389
4223 y FA(R)1389 4275 y(i)p Fw(\000)p FB(1)1501 4253
y FK(\))35 b(as)g(the)g(total)f(length)h(of)g(all)g(righ)n(t)f
(extensions.)g(\(Here)523 4364 y FC(x)570 4334 y FA(R)664
4364 y FK(denotes)39 b(the)g(rev)n(ersal)e(of)i(string)f
FC(x)p FK(.\))i(The)f(length)g(of)g(the)g(string)f(pro)r(duced)h(b)n(y)
523 4464 y(GREED)n(Y)30 b(is)g FC(l)1027 4476 y FA(l)1072
4464 y FK(+)20 b FC(l)1182 4476 y FA(c)1235 4464 y FK(+)g
FC(l)1345 4476 y FA(r)1402 4464 y FD(\000)f FC(o)1526
4476 y FA(w)1580 4464 y FK(,)30 b(where)g FC(o)1916 4476
y FA(w)2000 4464 y FK(is)g(the)g(summed)g(blo)r(c)n(k)g(o)n(v)n(erlap)e
(\(i.e.)i(the)523 4564 y(sum)e(of)f(the)h(o)n(v)n(erlaps)e(of)h(the)h
(w)n(eak)f(links\).)606 4663 y(Denoting)33 b(the)h(o)n(v)n(erlap)1414
4601 y Fn(P)1501 4688 y FA(e)p Fw(2)p FA(E)1647 4663
y FG(ov)10 b FK(\()p FC(e)p FK(\))34 b(of)f(a)g(set)h(of)f(edges)g
FC(E)38 b FK(as)33 b FG(ov)10 b FK(\()p FC(E)5 b FK(\),)34
b(de\014ne)f(the)523 4763 y(cost)27 b(of)h(a)f(set)h(of)f(edges)g
FC(E)33 b FK(on)27 b(a)g(set)h(of)f(strings)g(\(no)r(des\))h
FC(V)46 b FK(as)1495 4945 y FG(c)l(ost)8 b FK(\()p FC(E)d
FK(\))23 b(=)g FD(jj)p FC(V)c FD(jj)g(\000)f FG(ov)9
b FK(\()p FC(E)c FK(\))p FC(:)523 5128 y FK(Note)26 b(that)g(the)g
(distance)f(plus)h(o)n(v)n(erlap)e(of)i(a)f(string)g
FC(s)h FK(to)f(another)g(equals)g FD(j)p FC(s)p FD(j)p
FK(.)h(Because)523 5228 y(an)i(assignmen)n(t)f(\(e.g.)h
FC(M)1333 5240 y FA(l)1386 5228 y FK(or)g FC(M)1570 5240
y FA(r)1606 5228 y FK(\))h(has)e(an)h(edge)g(from)g(eac)n(h)f(no)r(de,)
h(its)g(cost)g(equals)g(its)523 5327 y(distance)g(w)n(eigh)n(t.)g
(Since)g FC(V)1399 5339 y FA(l)1454 5327 y FK(and)g FC(V)1664
5339 y FA(r)1730 5327 y FK(are)f(subsets)h(of)h FC(S)k
FK(and)28 b FC(M)2580 5339 y FA(l)2634 5327 y FK(and)g
FC(M)2877 5339 y FA(r)2942 5327 y FK(are)f(optimal)p
eop end
%%Page: 20 30
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% Prolog for xpic to PostScript converter
% Author: Mark Moraes
% $Header: x2ps.pro,v 1.2 88/03/19 16:50:09 moraes Exp 
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%	%d %d m - moveto
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%	<wid> <ht> <x> <y> e - ellipse
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%%EndDocument
 @endspecial 820 929 a(FIGURE)25 b(2.4.)i(Left/middle)f(and)g
(middle/righ)n(t)g(parts)g(with)g(w)n(eak)g(links.)523
1191 y FK(assignmen)n(ts,)g(w)n(e)h(ha)n(v)n(e)f FG(c)l(ost)8
b FK(\()p FC(M)1578 1203 y FA(l)1603 1191 y FK(\))23
b FD(\024)g FC(n)k FK(and)g FG(c)l(ost)8 b FK(\()p FC(M)2243
1203 y FA(r)2280 1191 y FK(\))23 b FD(\024)g FC(n)p FK(.)k(F)-7
b(or)27 b FC(E)2733 1203 y FA(l)2786 1191 y FK(and)g
FC(E)3008 1203 y FA(r)3072 1191 y FK(w)n(e)g(ha)n(v)n(e)523
1290 y(that)h FG(c)l(ost)8 b FK(\()p FC(E)942 1302 y
FA(l)968 1290 y FK(\))23 b(=)g FC(l)1136 1302 y FA(l)1179
1290 y FK(+)18 b FC(l)1287 1302 y FA(c)1349 1290 y FK(and)27
b FG(c)l(ost)8 b FK(\()p FC(E)1749 1302 y FA(r)1786 1290
y FK(\))24 b(=)e FC(l)1954 1302 y FA(r)2009 1290 y FK(+)c
FC(l)2117 1302 y FA(c)2151 1290 y FK(.)606 1390 y(W)-7
b(e)28 b(ha)n(v)n(e)e(established)i(the)g(follo)n(wing)e
(\(in\)equalities:)786 1565 y FC(l)811 1577 y FA(l)855
1565 y FK(+)18 b FC(l)963 1577 y FA(c)1015 1565 y FK(+)g
FC(l)1123 1577 y FA(r)1242 1565 y FK(=)83 b(\()p FC(l)1447
1577 y FA(l)1491 1565 y FK(+)18 b FC(l)1599 1577 y FA(c)1632
1565 y FK(\))h(+)f(\()p FC(l)1823 1577 y FA(c)1876 1565
y FK(+)g FC(l)1984 1577 y FA(r)2020 1565 y FK(\))h FD(\000)f
FC(l)2179 1577 y FA(c)1242 1690 y FK(=)83 b FG(c)l(ost)8
b FK(\()p FC(E)1629 1702 y FA(l)1655 1690 y FK(\))18
b(+)g FG(c)l(ost)8 b FK(\()p FC(E)2027 1702 y FA(r)2065
1690 y FK(\))18 b FD(\000)g FC(l)2223 1702 y FA(c)1242
1814 y FK(=)83 b FD(jj)p FC(V)1484 1826 y FA(l)1510 1814
y FD(jj)18 b(\000)g FG(ov)10 b FK(\()p FC(E)1840 1826
y FA(l)1866 1814 y FK(\))19 b(+)f FD(jj)p FC(V)2094 1826
y FA(r)2131 1814 y FD(jj)h(\000)f FG(ov)9 b FK(\()p FC(E)2461
1826 y FA(r)2499 1814 y FK(\))18 b FD(\000)g FC(l)2657
1826 y FA(c)1242 1939 y FK(=)83 b FG(c)l(ost)8 b FK(\()p
FC(M)1649 1951 y FA(l)1674 1939 y FK(\))19 b(+)f FG(ov)9
b FK(\()p FC(M)2010 1951 y FA(l)2036 1939 y FK(\))18
b FD(\000)g FG(ov)10 b FK(\()p FC(E)2352 1951 y FA(l)2378
1939 y FK(\))19 b(+)1390 2063 y FG(c)l(ost)8 b FK(\()p
FC(M)1649 2075 y FA(r)1685 2063 y FK(\))19 b(+)f FG(ov)10
b FK(\()p FC(M)2022 2075 y FA(r)2058 2063 y FK(\))19
b FD(\000)f FG(ov)9 b FK(\()p FC(E)2374 2075 y FA(r)2412
2063 y FK(\))18 b FD(\000)h FC(l)2571 2075 y FA(c)1242
2188 y FD(\024)83 b FK(2)p FC(n)18 b FK(+)g FG(ov)9 b
FK(\()p FC(M)1785 2200 y FA(l)1811 2188 y FK(\))18 b
FD(\000)g FG(ov)10 b FK(\()p FC(E)2127 2200 y FA(l)2153
2188 y FK(\))19 b(+)f FG(ov)9 b FK(\()p FC(M)2489 2200
y FA(r)2526 2188 y FK(\))19 b FD(\000)f FG(ov)9 b FK(\()p
FC(E)2842 2200 y FA(r)2879 2188 y FK(\))19 b FD(\000)f
FC(l)3038 2200 y FA(c)3072 2188 y FC(:)606 2363 y FK(W)-7
b(e)28 b(pro)r(ceed)g(b)n(y)f(b)r(ounding)h(the)h(o)n(v)n(erlap)c
(di\013erences)j(in)g(the)h(ab)r(o)n(v)n(e)d(equation.)h(Our)523
2462 y(basic)36 b(idea)f(is)i(to)f(c)n(harge)e(the)j(o)n(v)n(erlap)d
(of)i(eac)n(h)f(edge)h(of)g FC(M)45 b FK(to)36 b(an)g(edge)f(of)i
FC(E)k FK(or)35 b(a)523 2562 y(w)n(eak)f(link)i(or)f(the)g(bac)n(k)g
(edge)g(of)g(a)h(culprit)f(in)h(a)f(w)n(a)n(y)f(suc)n(h)h(that)h(ev)n
(ery)e(edge)h(of)h FC(E)523 2662 y FK(and)27 b(ev)n(ery)e(w)n(eak)h
(link)h(is)g(c)n(harged)e(at)i(most)g(once)f(and)h(the)g(bac)n(k)f
(edge)h(of)g(eac)n(h)f(culprit)523 2761 y(is)31 b(c)n(harged)e(at)h
(most)h(t)n(wice.)g(This)g(is)f(ac)n(hiev)n(ed)g(through)g(com)n
(bining)g(the)h(left/middle)523 2861 y(and)23 b(middle/righ)n(t)g
(parts)g(carefully)g(as)g(sho)n(wn)f(b)r(elo)n(w.)h(F)-7
b(or)23 b(con)n(v)n(enience,)g(w)n(e)g(will)g(refer)523
2960 y(to)32 b(the)g(union)f(op)r(eration)g(for)g(m)n(ultisets)h(\()p
FG(i.e.)p FK(,)i(allo)n(wing)c(duplicates\))i(as)f(the)h
FG(disjoint)523 3060 y(union)p FK(.)606 3160 y(Let)22
b FC(V)41 b FK(b)r(e)22 b(the)g(disjoin)n(t)g(union)f(of)h
FC(V)1733 3172 y FA(l)1781 3160 y FK(and)f FC(V)1984
3172 y FA(r)2021 3160 y FK(,)h(let)g FC(E)27 b FK(b)r(e)22
b(the)g(disjoin)n(t)g(union)g(of)f FC(E)3176 3172 y FA(l)3224
3160 y FK(and)523 3259 y FC(E)584 3271 y FA(r)621 3259
y FK(,)26 b(and)f(let)h FC(G)d FK(=)g(\()p FC(V)5 b(;)14
b(E)5 b FK(\))26 b(b)r(e)g(the)g(disjoin)n(t)f(union)h(of)f
FC(G)2300 3271 y FA(l)2352 3259 y FK(and)g FC(G)2576
3271 y FA(r)2613 3259 y FK(.)h(Th)n(us)f(eac)n(h)g(string)f(in)523
3359 y FC(S)574 3371 y FA(l)619 3359 y FD([)c FC(S)745
3371 y FA(r)812 3359 y FK(o)r(ccurs)29 b(once,)g(while)h(eac)n(h)f
(string)g(in)g FC(S)2080 3371 y FA(m)2173 3359 y FK(o)r(ccurs)g(t)n
(wice)g(in)h FC(G)p FK(.)g(W)-7 b(e)30 b(mo)r(dify)h
FC(E)523 3459 y FK(to)g(tak)n(e)f(adv)-5 b(an)n(tage)29
b(of)h(the)h(blo)r(c)n(k)g(o)n(v)n(erlaps.)d(Add)j(eac)n(h)f(w)n(eak)g
(link)g(to)h FC(E)36 b FK(as)30 b(an)g(edge)523 3558
y(from)j(the)g(last)g(no)r(de)g(in)g(the)h(corresp)r(onding)d
(middle/righ)n(t)h(path)h(of)g FC(G)2888 3570 y FA(r)2959
3558 y FK(to)g(the)g(\014rst)523 3658 y(no)r(de)26 b(of)g(the)g
(corresp)r(onding)f(left/middle)h(path)g(of)g FC(G)2268
3670 y FA(l)2294 3658 y FK(.)g(This)g(pro)r(cedure)f(yields)h(a)g(new)
523 3758 y(set)31 b(of)f(edges)g FC(E)1044 3727 y Fw(0)1068
3758 y FK(.)h(Its)f(o)n(v)n(erlap)f(equals)h FG(ov)10
b FK(\()p FC(E)1982 3727 y Fw(0)2005 3758 y FK(\))29
b(=)f FG(ov)9 b FK(\()p FC(E)2341 3770 y FA(l)2367 3758
y FK(\))21 b(+)f FG(ov)10 b FK(\()p FC(E)2688 3770 y
FA(r)2725 3758 y FK(\))21 b(+)f FC(o)2903 3770 y FA(w)2957
3758 y FK(.)31 b(A)g(picture)523 3857 y(of)d(\()p FC(V)5
b(;)14 b(E)806 3827 y Fw(0)829 3857 y FK(\))28 b(for)f(our)g(previous)g
(example)g(is)g(giv)n(en)g(in)h(Figure)f(2.4.)606 3957
y(Let)39 b FC(M)48 b FK(b)r(e)40 b(the)g(disjoin)n(t)f(union)g(of)h
FC(M)1912 3969 y FA(l)1976 3957 y FK(and)f FC(M)2230
3969 y FA(r)2267 3957 y FK(,)g(an)g(assignmen)n(t)f(on)h(graph)f
FC(G)p FK(.)523 4056 y(Its)30 b(o)n(v)n(erlap)d(equals)i
FG(ov)10 b FK(\()p FC(M)f FK(\))26 b(=)g FG(ov)10 b FK(\()p
FC(M)1756 4068 y FA(l)1781 4056 y FK(\))20 b(+)f FG(ov)10
b FK(\()p FC(M)2120 4068 y FA(r)2157 4056 y FK(\).)30
b(Ev)n(ery)d(edge)j(of)f FC(M)38 b FK(connects)30 b(t)n(w)n(o)523
4156 y FC(V)571 4168 y FA(l)631 4156 y FK(no)r(des)k(or)f(t)n(w)n(o)g
FC(V)1190 4168 y FA(r)1261 4156 y FK(no)r(des;)h(th)n(us,)g(all)g
(edges)f(of)h FC(M)43 b FK(satisfy)34 b(the)g(h)n(yp)r(othesis)f(of)h
(the)523 4256 y(follo)n(wing)27 b(lemma.)523 4419 y Fv(Lemma)32
b FH(2.16)40 b FG(L)l(et)c FC(N)46 b FG(b)l(e)36 b(any)h(assignment)g
(on)f FC(V)19 b FG(.)37 b(L)l(et)g FC(e)e FK(=)g FC(t)g
FD(!)h FC(h)g FG(b)l(e)h(an)f(e)l(dge)i(of)523 4518 y
FC(N)27 b FD(n)18 b FC(E)743 4488 y Fw(0)796 4518 y FG(that)30
b(is)g(not)g(in)f FC(V)1351 4530 y FA(r)1407 4518 y FD(\002)18
b FC(V)1538 4530 y FA(l)1564 4518 y FG(.)30 b(Then)g
FC(e)g FG(is)g(dominate)l(d)h(by)f(either)621 4678 y(1.)43
b(an)29 b(adjac)l(ent)i FC(E)1244 4648 y Fw(0)1297 4678
y FG(e)l(dge,)g(or)621 4841 y(2.)43 b(a)30 b(culprit's)g(b)l(ack)h(e)l
(dge)f(with)h(which)g(it)f(shar)l(es)g(the)g(he)l(ad)h
FC(h)e FG(and)i FC(h)23 b FD(2)g FC(V)3004 4853 y FA(r)3041
4841 y FG(,)30 b(or)621 5004 y(3.)43 b(a)30 b(culprit's)g(b)l(ack)h(e)l
(dge)f(with)h(which)g(it)f(shar)l(es)g(the)g(tail)g FC(t)g
FG(and)g FC(t)23 b FD(2)h FC(V)2924 5016 y FA(l)2950
5004 y FG(.)523 5164 y Fv(Pr)n(oof.)i FK(Supp)r(ose)h(\014rst)f(that)h
FC(e)f FK(corresp)r(onds)f(to)h(a)g(bad)h(bac)n(k)e(edge.)h(By)h(Lemma)
f(2.14,)523 5264 y FC(h)g FK(corresp)r(onds)e(to)h(a)h(left)g(no)r(de)g
(or)f(to)h(the)g(\014rst)g(no)r(de)g(of)f(a)h(culprit.)g(In)g(the)g
(latter)g(case,)523 5363 y FC(e)h FK(is)g(dominated)h(b)n(y)f(the)g
(bac)n(k)g(edge)g(of)g(the)g(culprit)h(\(see)f(the)h(commen)n(t)f
(after)g(Lemma)p eop end
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518 y FA(r)2680 506 y FK(\(and)g(case)e(2)i(holds\),)523
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606 y FK(.)j(Similarly)-7 b(,)28 b(either)g FC(t)h FK(is)f(the)h(last)f
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606 y FK(\(and)h(case)e(3)523 706 y(holds\))32 b(or)f(else)h
FC(t)f FD(2)f FC(V)1239 718 y FA(r)1277 706 y FK(.)i(Since)g
FC(e)g FK(is)g(not)g(in)g FC(V)2013 718 y FA(r)2072 706
y FD(\002)21 b FC(V)2206 718 y FA(l)2232 706 y FK(,)32
b(it)h(follo)n(ws)e(then)h(that)h(case)e(2)h(or)523 805
y(case)25 b(3)h(holds.)g(\(Note)g(that)g(if)h FC(e)f
FK(is)f(in)i(fact)f(the)g(bac)n(k)g(edge)f(of)h(some)g(culprit,)g(then)
g(b)r(oth)523 905 y(cases)h(2)g(and)g(3)g(hold.\))606
1005 y(Supp)r(ose)36 b(that)f FC(e)g FK(do)r(es)h(not)f(corresp)r(ond)f
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1104 y(dominated)d(b)n(y)g(some)g(greedy)f(edge)g(since)h(it)h(w)n(as)e
(not)i(c)n(hosen)e(b)n(y)h(GREED)n(Y.)g(If)g(the)523
1204 y(greedy)27 b(edge)h(dominating)g FC(e)g FK(is)g(in)g
FC(E)1732 1174 y Fw(0)1784 1204 y FK(then)h(w)n(e)f(ha)n(v)n(e)f(case)g
(1.)h(If)h(it)g(is)f(not)g(in)g FC(E)3121 1174 y Fw(0)3145
1204 y FK(,)g(then)523 1303 y(either)h FC(h)h FK(is)f(the)h(\014rst)g
(no)r(de)f(of)h(a)f(culprit)h(in)g FC(V)2029 1315 y FA(r)2095
1303 y FK(or)f FC(t)h FK(is)f(the)h(last)f(no)r(de)h(of)g(a)f(culprit)g
(in)523 1403 y FC(V)571 1415 y FA(l)597 1403 y FK(,)d(and)g(in)h(b)r
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(edge)h(of)g(the)h(culprit.)f(Th)n(us,)g(w)n(e)523 1503
y(ha)n(v)n(e)g(case)h(2)g(or)g(3.)2170 b Fu(2)606 1652
y FK(While)35 b(Lemma)g(2.16)e(ensures)h(that)h(eac)n(h)f(edge)g(of)h
FC(M)43 b FK(is)35 b(b)r(ounded)g(in)g(o)n(v)n(erlap,)d(it)523
1752 y(ma)n(y)27 b(b)r(e)i(that)f(some)g(edges)f(of)h
FC(E)1589 1722 y Fw(0)1641 1752 y FK(are)f(double)h(c)n(harged.)e(W)-7
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1851 y(decreasing)21 b(its)i(o)n(v)n(erlap)e(and)h(without)h(in)n(v)-5
b(alidating)22 b(Lemma)h(2.16)e(in)n(to)h(an)h(assignmen)n(t)523
1951 y FC(M)613 1921 y Fw(0)664 1951 y FK(suc)n(h)k(that)h(eac)n(h)f
(edge)g(of)g FC(E)1567 1921 y Fw(0)1618 1951 y FK(is)h(dominated)f(b)n
(y)h(one)f(of)g(its)h(adjacen)n(t)f FC(M)3012 1921 y
Fw(0)3063 1951 y FK(edges.)523 2117 y Fv(Lemma)32 b FH(2.17)40
b FG(L)l(et)25 b FC(N)34 b FG(b)l(e)26 b(any)g(assignment)g(on)f
FC(V)45 b FG(such)25 b(that)h FC(N)18 b FD(n)9 b FC(E)2723
2087 y Fw(0)2772 2117 y FG(do)l(es)26 b(not)f(c)l(ontain)523
2217 y(any)41 b(e)l(dges)g(in)g FC(V)1081 2229 y FA(r)1145
2217 y FD(\002)26 b FC(V)1284 2229 y FA(l)1310 2217 y
FG(.)41 b(Then)g(ther)l(e)g(is)g(an)g(assignment)f FC(N)2568
2187 y Fw(0)2632 2217 y FG(on)h FC(V)59 b FG(satisfying)42
b(the)523 2316 y(fol)t(lowing)32 b(pr)l(op)l(erties.)621
2482 y(1.)43 b FC(N)807 2452 y Fw(0)848 2482 y FD(n)18
b FC(E)974 2452 y Fw(0)1027 2482 y FG(has)31 b(also)f(no)g(e)l(dges)h
(in)e FC(V)1830 2494 y FA(r)1886 2482 y FD(\002)18 b
FC(V)2017 2494 y FA(l)2043 2482 y FG(,)621 2648 y(2.)43
b(ov)9 b FK(\()p FC(N)928 2618 y Fw(0)951 2648 y FK(\))24
b FD(\025)e FG(ov)10 b FK(\()p FC(N)f FK(\))p FG(,)621
2814 y(3.)43 b(e)l(ach)30 b(e)l(dge)h(in)f FC(E)1268
2784 y Fw(0)1309 2814 y FD(n)18 b FC(N)1445 2784 y Fw(0)1498
2814 y FG(is)30 b(dominate)l(d)h(by)f(one)g(of)h(its)f(two)g(adjac)l
(ent)g FC(N)3027 2784 y Fw(0)3080 2814 y FG(e)l(dges.)523
2980 y Fv(Pr)n(oof.)f FK(Since)h FC(N)39 b FK(already)28
b(has)i(the)g(\014rst)f(t)n(w)n(o)g(prop)r(erties,)g(it)i(su\016ces)e
(to)h(argue)e(that)523 3080 y(if)35 b FC(N)43 b FK(violates)33
b(prop)r(ert)n(y)h(3,)g(then)h(w)n(e)e(can)h(construct)g(another)g
(assignmen)n(t)f FC(N)3148 3050 y Fw(0)3205 3080 y FK(that)523
3180 y(satis\014es)27 b(prop)r(erties)g(1)g(and)g(2,)g(and)h(has)f
(more)g(edges)f(in)i(common)f(with)i FC(E)2966 3150 y
Fw(0)2989 3180 y FK(.)606 3279 y(Let)d FC(e)d FK(=)g
FC(k)j FD(!)d FC(j)31 b FK(b)r(e)c(an)e(edge)h(in)h FC(E)1718
3249 y Fw(0)1757 3279 y FD(\000)15 b FC(N)35 b FK(that)26
b(dominates)g(b)r(oth)h(adjacen)n(t)e FC(N)35 b FK(edges,)523
3379 y FC(f)f FK(=)24 b FC(i)h FD(!)g FC(j)5 b FK(,)29
b(and)f FC(g)g FK(=)d FC(k)i FD(!)f FC(l)r FK(.)i(By)g(Lemma)h(2.7,)f
(replacing)f(edges)h FC(f)38 b FK(and)28 b FC(g)k FK(of)d
FC(N)37 b FK(with)523 3479 y FC(e)30 b FK(and)g FC(i)d
FD(!)h FC(l)k FK(pro)r(duces)d(an)h(assignmen)n(t)f FC(N)1955
3448 y Fw(0)2009 3479 y FK(with)i(at)f(least)g(as)f(large)g(o)n(v)n
(erlap.)f(T)-7 b(o)30 b(see)523 3578 y(that)c(the)g(new)g(edge)f
FC(i)e FD(!)g FC(l)k FK(of)e FC(N)1576 3548 y Fw(0)1614
3578 y FD(n)14 b FC(E)1736 3548 y Fw(0)1785 3578 y FK(is)26
b(not)g(in)f FC(V)2155 3590 y FA(r)2207 3578 y FD(\002)14
b FC(V)2334 3590 y FA(l)2360 3578 y FK(,)26 b(observ)n(e)e(that)i(if)g
FC(i)d FD(2)g FC(V)3133 3590 y FA(r)3196 3578 y FK(then)523
3678 y FC(j)28 b FD(2)23 b FC(V)711 3690 y FA(r)776 3678
y FK(b)r(ecause)k(of)h(the)g(edge)f FC(f)k FK(=)23 b
FC(i)g FD(!)g FC(j)32 b FK(\()p FC(N)c FD(n)17 b FC(E)2146
3648 y Fw(0)2197 3678 y FK(do)r(es)28 b(not)f(ha)n(v)n(e)f(edges)h(in)h
FC(V)3090 3690 y FA(r)3146 3678 y FD(\002)17 b FC(V)3276
3690 y FA(l)3302 3678 y FK(\),)523 3778 y(whic)n(h)31
b(implies)g(that)g FC(k)j FK(is)d(in)g FC(V)1544 3790
y FA(r)1612 3778 y FK(b)r(ecause)f(of)h(the)g FC(E)2232
3747 y Fw(0)2286 3778 y FK(edge)g FC(e)d FK(=)g FC(k)j
FD(!)e FC(j)36 b FK(\()p FC(E)2993 3747 y Fw(0)3047 3778
y FK(do)r(es)31 b(not)523 3877 y(ha)n(v)n(e)f(edges)h(in)g
FC(V)1092 3889 y FA(l)1139 3877 y FD(\002)20 b FC(V)1272
3889 y FA(r)1310 3877 y FK(\),)31 b(whic)n(h)h(implies)f(that)h(also)e
FC(l)g FD(2)g FC(V)2465 3889 y FA(r)2533 3877 y FK(b)r(ecause)h(of)g
(the)h FC(N)40 b FK(edge)523 3977 y FC(g)26 b FK(=)c
FC(k)k FD(!)d FC(l)r FK(.)2395 b Fu(2)523 4126 y Fv(Pr)n(oof.)19
b FH(\(of)k(Theorem)g(2.12.\))18 b FK(By)i(Lemmas)f(2.16)g(and)h(2.17,)
f(w)n(e)g(can)h(construct)f(from)523 4226 y(the)28 b(assignmen)n(t)f
FC(M)36 b FK(another)27 b(assignmen)n(t)g FC(M)2034 4196
y Fw(0)2085 4226 y FK(with)h(at)g(least)f(as)g(large)f(total)i(o)n(v)n
(erlap,)523 4325 y(and)c(suc)n(h)g(that)h(w)n(e)f(can)g(c)n(harge)f
(the)h(o)n(v)n(erlap)f(of)h(eac)n(h)g(edge)g(of)g FC(M)2635
4295 y Fw(0)2682 4325 y FK(to)g(an)h(edge)e(of)i FC(E)3236
4295 y Fw(0)3284 4325 y FK(or)523 4425 y(to)c(the)g(bac)n(k)g(edge)f
(of)h(a)g(culprit.)g(Ev)n(ery)f(edge)g(of)h FC(E)2128
4395 y Fw(0)2173 4425 y FK(is)g(c)n(harged)e(for)i(at)g(most)g(one)f
(edge)h(of)523 4525 y FC(M)613 4495 y Fw(0)636 4525 y
FK(,)j(while)h(the)g(bac)n(k)e(edge)h(of)g(eac)n(h)g(culprit)g(is)g(c)n
(harged)f(for)h(at)g(most)g(t)n(w)n(o)g(edges)f(of)i
FC(M)3312 4495 y Fw(0)3335 4525 y FK(:)523 4624 y(for)e(the)g
FC(M)874 4594 y Fw(0)920 4624 y FK(edge)g(en)n(tering)f(the)h(\014rst)g
(culprit)g(no)r(de)h(in)f FC(V)2326 4636 y FA(r)2386
4624 y FK(and)g(the)h(edge)e(coming)h(out)g(of)523 4724
y(the)k(last)g(culprit)f(no)r(de)h(in)g FC(V)1432 4736
y FA(l)1458 4724 y FK(.)g(Therefore,)e FG(ov)10 b FK(\()p
FC(M)f FK(\))23 b FD(\024)g FG(ov)9 b FK(\()p FC(M)2472
4694 y Fw(0)2496 4724 y FK(\))23 b FD(\024)f FG(ov)10
b FK(\()p FC(E)2826 4694 y Fw(0)2850 4724 y FK(\))17
b(+)f(2)p FC(o)3062 4736 y FA(c)3095 4724 y FK(,)27 b(where)523
4824 y FC(o)563 4836 y FA(c)625 4824 y FK(is)i(the)f(summed)h(o)n(v)n
(erlap)d(of)i(all)g(culprit)h(bac)n(k)e(edges.)h(Denote)g(b)n(y)g
FC(w)2851 4836 y FA(c)2914 4824 y FK(the)g(summed)523
4923 y(w)n(eigh)n(t)f(of)g(all)g(culprit)h(cycles,)f(i.e.,)h(the)g(w)n
(eigh)n(t)e(of)i(the)g(\(optimal\))g(assignmen)n(t)e
FC(C)3179 4935 y FA(m)3270 4923 y FK(on)523 5023 y FC(S)574
5035 y FA(m)673 5023 y FK(from)36 b(Lemma)g(2.15.)f(Then)h
FC(l)1640 5035 y FA(c)1711 5023 y FK(=)h FC(w)1872 5035
y FA(c)1930 5023 y FK(+)24 b FC(o)2059 5035 y FA(c)2093
5023 y FK(.)36 b(As)h(in)f(the)h(pro)r(of)e(of)h(Theorem)g(2.8,)523
5122 y(w)n(e)d(ha)n(v)n(e)f FC(o)888 5134 y FA(c)944
5122 y FD(\000)22 b FK(2)p FC(w)1132 5134 y FA(c)1199
5122 y FD(\024)32 b FC(n)h FK(and)g FC(w)1605 5134 y
FA(c)1672 5122 y FD(\024)f FC(n)p FK(.)i(\(Note)f(that)h(the)g(o)n(v)n
(erlap)d(of)i(a)g(culprit)h(bac)n(k)523 5222 y(edge)28
b(is)h(less)f(than)h(the)h(length)f(of)f(the)h(longest)g(string)f(in)h
(the)g(culprit)g(cycle.\))g(Putting)p eop end
%%Page: 22 32
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1199 y FA(c)1254 1312 y FD(\024)82 b FK(4)p FC(n:)3296
1494 y Fu(2)523 1851 y FL(2.6)116 b(Whic)m(h)38 b(algorithm)e(is)i(the)
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b(arious)25 b(b)r(ounds)h(for)g(the)h(algorithms)e(GREED)n(Y,)i(MGREED)
n(Y,)f(and)523 2150 y(TGREED)n(Y,)39 b(one)f(ma)n(y)g(w)n(onder)g(what)
h(this)g(implies)g(ab)r(out)g(their)g(relativ)n(e)f(p)r(erfor-)523
2250 y(mance.)18 b(First)g(of)h(all)f(w)n(e)g(note)g(that)h(MGREED)n(Y)
f(can)g(nev)n(er)f(do)h(b)r(etter)h(than)g(TGREED)n(Y)523
2350 y(since)h(the)g(latter)g(applies)g(the)g(GREED)n(Y)g(algorithm)e
(to)i(an)g(in)n(termediate)g(set)g(of)g(strings)523 2449
y(that)28 b(the)g(former)f(merely)g(concatenates.)606
2549 y(Do)r(es)22 b(the)g(3)p FC(n)f FK(b)r(ound)h(for)f(TGREED)n(Y)h
(then)g(mean)g(that)g(it)g(is)g(the)g(b)r(est)g(of)g(the)g(three?)523
2648 y(This)e(pro)n(v)n(es)d(not)j(alw)n(a)n(ys)e(to)h(b)r(e)h(the)g
(case.)f(In)h(the)g(example)f FD(f)p FC(c)p FK(\()p FC(ab)p
FK(\))2645 2618 y FA(k)2686 2648 y FC(;)14 b FK(\()p
FC(ab)p FK(\))2867 2618 y FA(k)q FB(+1)2991 2648 y FC(a;)g
FK(\()p FC(ba)p FK(\))3216 2618 y FA(k)3257 2648 y FC(c)p
FD(g)p FK(,)523 2748 y(GREED)n(Y)30 b(pro)r(duces)f(the)h(shortest)g
(sup)r(erstring)f FC(c)p FK(\()p FC(ab)p FK(\))2345 2718
y FA(k)q FB(+1)2470 2748 y FC(ac)h FK(of)f(length)i FC(n)26
b FK(=)h(2)p FC(k)22 b FK(+)e(5,)523 2848 y(whereas)i(TGREED)n(Y)h
(\014rst)f(separates)g(the)h(middle)h(string)e(to)h(end)g(up)h(with)f
(something)523 2947 y(lik)n(e)k FC(c)p FK(\()p FC(ab)p
FK(\))855 2917 y FA(k)896 2947 y FC(ac)p FK(\()p FC(ab)p
FK(\))1120 2917 y FA(k)q FB(+1)1245 2947 y FC(a)g FK(of)h(length)g(4)p
FC(k)21 b FK(+)d(6.)606 3047 y(P)n(erhaps)29 b(then)i(GREED)n(Y)f(is)h
(alw)n(a)n(ys)e(b)r(etter)i(than)f(TGREED)n(Y,)h(despite)g(the)g(fact)
523 3147 y(that)20 b(w)n(e)e(cannot)h(pro)n(v)n(e)f(as)g(go)r(o)r(d)h
(an)g(upp)r(er)g(b)r(ound)h(for)e(it.)i(This)f(turns)g(out)g(not)h(to)f
(b)r(e)g(the)523 3246 y(case)f(either,)g(as)g(sho)n(wn)f(b)n(y)h(the)h
(follo)n(wing)e(example.)h(On)g(input)i FD(f)p FC(cab)2694
3216 y FA(k)2733 3246 y FC(;)14 b(ab)2850 3216 y FA(k)2891
3246 y FC(ab)2971 3216 y FA(k)3011 3246 y FC(a;)g(b)3128
3216 y FA(k)3168 3246 y FC(dab)3291 3216 y FA(k)q Fw(\000)p
FB(1)3417 3246 y FD(g)p FK(,)523 3346 y(TGREED)n(Y)28
b(separates)f(the)i(middle)f(string,)g(merges)f(the)i(other)f(t)n(w)n
(o,)g(and)g(next)g(com-)523 3445 y(bines)f(these)g(to)f(pro)r(duce)g
(the)i(shortest)e(sup)r(erstring)f FC(cab)2366 3415 y
FA(k)2407 3445 y FC(dab)2530 3415 y FA(k)2570 3445 y
FC(ab)2650 3415 y FA(k)2691 3445 y FC(a)h FK(of)h(length)g(3)p
FC(k)19 b FK(+)d(6,)523 3545 y(whereas)h(GREED)n(Y)h(merges)f(the)i
(\014rst)f(t)n(w)n(o,)g(lea)n(ving)f(nothing)h(b)r(etter)h(than)g
FC(cab)3050 3515 y FA(k)3090 3545 y FC(ab)3170 3515 y
FA(k)3210 3545 y FC(ab)3290 3515 y FA(k)3331 3545 y FC(dab)3454
3515 y FA(k)q Fw(\000)p FB(1)523 3645 y FK(of)28 b(length)f(4)p
FC(k)21 b FK(+)d(5.)606 3744 y(Another)35 b(greedy)f(t)n(yp)r(e)h(of)g
(algorithm)f(that)i(ma)n(y)e(come)h(to)g(mind)h(is)f(one)g(that)g(ar-)
523 3844 y(bitrarily)29 b(pic)n(ks)g(an)n(y)g(of)h(the)g(strings)f(and)
h(then)g(rep)r(eatedly)g(merges)e(on)i(the)g(righ)n(t)f(the)523
3944 y(string)f(with)g(maxim)n(um)h(o)n(v)n(erlap.)d(This)i(algorithm,)
f(call)h(it)h(NAIVE,)g(turns)f(out)g(to)g(b)r(e)523 4043
y(disastrous)e(on)h(examples)g(lik)n(e)684 4226 y FD(f)p
FC(abcde;)14 b(bcde)p FK(#)p FC(a;)g(cde)p FK(#)p FC(a)p
FK(#)p FC(b;)g(de)p FK(#)p FC(a)p FK(#)p FC(b)p FK(#)p
FC(c;)g(e)p FK(#)p FC(a)p FK(#)p FC(b)p FK(#)p FC(c)p
FK(#)p FC(d;)g FK(#)p FC(a)p FK(#)p FC(b)p FK(#)p FC(c)p
FK(#)p FC(d)p FK(#)p FC(e)p FD(g)p FC(:)523 4408 y FK(Instead)26
b(of)h(pro)r(ducing)f(the)g(optimal)h FC(abcde)p FK(#)p
FC(a)p FK(#)p FC(b)p FK(#)p FC(c)p FK(#)p FC(d)p FK(#)p
FC(e)p FK(,)f(NAIVE)h(migh)n(t)f(pro)r(duce)523 4508
y(#)p FC(a)p FK(#)p FC(b)p FK(#)p FC(c)p FK(#)p FC(d)p
FK(#)p FC(e)p FK(#)p FC(a)p FK(#)p FC(b)p FK(#)p FC(c)p
FK(#)p FC(de)p FK(#)p FC(a)p FK(#)p FC(b)p FK(#)p FC(cde)p
FK(#)p FC(a)p FK(#)p FC(bcde)p FK(#)p FC(abcde)18 b FK(b)n(y)g(pic)n
(king)g(#)p FC(a)p FK(#)p FC(b)p FK(#)p FC(c)p FK(#)p
FC(d)p FK(#)p FC(e)523 4608 y FK(as)30 b(a)g(starting)g(p)r(oin)n(t.)h
(It)g(is)f(clear)g(that)h(in)g(this)f(w)n(a)n(y)g(sup)r(erstrings)f(ma)
n(y)h(b)r(e)h(pro)r(duced)523 4707 y(whose)c(length)h(gro)n(ws)d
(quadratically)h(in)i(the)g(optim)n(um)g(length)g FC(n)p
FK(.)523 5015 y FL(2.7)116 b(Lo)m(w)m(er)37 b(b)s(ound)523
5214 y FK(W)-7 b(e)34 b(sho)n(w)f(here)h(that)g(the)g(sup)r(erstring)f
(problem)h(is)g FG(MAX)h(SNP-har)l(d)p FK(.)g(This)f(implies)523
5313 y(that)h(if)h(there)f(is)g(a)g(p)r(olynomial)f(time)i(appro)n
(ximation)d(sc)n(heme)i(for)f(the)i(sup)r(erstring)p
eop end
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b(b)r(ound)153 b(23)523 506 y FK(problem,)31 b(then)g(there)g(is)g(one)
g(also)f(for)g(a)h(wide)g(class)g(of)g(optimization)f(problems,)h(in-)
523 606 y(cluding)24 b(sev)n(eral)e(v)-5 b(arian)n(ts)22
b(of)i(maxim)n(um)f(satis\014abilit)n(y)-7 b(,)23 b(the)h(no)r(de)g(co)
n(v)n(er)e(and)h(indep)r(en-)523 706 y(den)n(t)30 b(set)f(problems)g
(in)g(b)r(ounded-degree)g(graphs,)f(max)h(cut,)h(etc.)f(This)h(is)f
(considered)523 805 y(rather)e(unlik)n(ely)-7 b(.)1073
775 y FB(1)606 905 y FK(Let)24 b FC(A)p FK(,)g FC(B)j
FK(b)r(e)d(t)n(w)n(o)f(optimization)g(\(maximization)g(or)g
(minimization\))h(problems.)e(W)-7 b(e)523 1005 y(sa)n(y)34
b(that)i FC(A)f FG(L-r)l(e)l(duc)l(es)g FK(\(for)g FG(line)l(arly)k(r)l
(e)l(duc)l(es)p FK(\))c(to)g FC(B)40 b FK(if)35 b(there)h(are)e(t)n(w)n
(o)g(p)r(olynomial)523 1104 y(time)28 b(algorithms)e
FC(f)37 b FK(and)27 b FC(g)j FK(and)e(constan)n(ts)e
FC(\013)i FK(and)g FC(\014)f(>)c FK(0)k(suc)n(h)g(that:)625
1247 y(1.)41 b(Giv)n(en)28 b(an)g(instance)h FC(a)f FK(of)h
FC(A)p FK(,)g(algorithm)e FC(f)37 b FK(pro)r(duces)28
b(an)g(instance)h FC(b)f FK(of)g FC(B)33 b FK(suc)n(h)731
1347 y(that)d(the)h(cost)f(of)g(the)h(optim)n(um)g(solution)e(of)i
FC(b)p FK(,)f FC(opt)p FK(\()p FC(b)p FK(\),)g(is)h(at)f(most)g
FC(\013)21 b FD(\001)f FC(opt)p FK(\()p FC(a)p FK(\),)731
1447 y(and)625 1601 y(2.)41 b(Giv)n(en)36 b(an)n(y)g(solution)g
FC(y)k FK(of)c FC(b)p FK(,)h(algorithm)e FC(g)40 b FK(pro)r(duces)c(in)
h(p)r(olynomial)f(time)h(a)731 1701 y(solution)27 b FC(x)h
FK(of)f FC(a)h FK(suc)n(h)f(that)h FD(j)p FC(cost)p FK(\()p
FC(x)p FK(\))20 b FD(\000)e FC(opt)p FK(\()p FC(a)p FK(\))p
FD(j)23 b(\024)g FC(\014)t FD(j)p FC(cost)p FK(\()p FC(y)s
FK(\))c FD(\000)f FC(opt)p FK(\()p FC(b)p FK(\))p FD(j)p
FK(.)606 1844 y(Some)35 b(basic)f(facts)h(ab)r(out)g(L-reductions)f
(are:)g(First,)h(the)h(comp)r(osition)e(of)h(t)n(w)n(o)f(L-)523
1944 y(reductions)c(is)g(also)f(an)h(L-reduction.)g(Second,)g(if)h
(problem)e FC(A)i FK(L-reduces)e(to)h(problem)523 2043
y FC(B)37 b FK(and)c FC(B)j FK(can)d(b)r(e)g(appro)n(ximated)e(in)i(p)r
(olynomial)f(time)h(with)g(relativ)n(e)f(error)f FC(\017)h
FK(\(i.e.,)523 2143 y(within)d(a)e(factor)h(of)g(1)18
b(+)g FC(\017)28 b FK(or)f(1)18 b FD(\000)h FC(\017)28
b FK(dep)r(ending)g(on)g(whether)g FC(B)k FK(is)c(a)f(minimization)i
(or)523 2243 y(maximization)21 b(problem\))h(then)g FC(A)g
FK(can)g(b)r(e)g(appro)n(ximated)e(with)j(relativ)n(e)d(error)g
FC(\013\014)t(\017)p FK(.)j(In)523 2342 y(particular,)j(if)i
FC(B)k FK(has)27 b(a)g(p)r(olynomial)f(time)i(appro)n(ximation)e(sc)n
(heme,)h(then)h(so)e(do)r(es)h FC(A)p FK(.)523 2442 y(The)j(class)f
(MAX)i(SNP)f(is)f(a)h(class)f(of)h(optimization)f(problems)h(de\014ned)
g(syn)n(tactically)523 2542 y(in)22 b([11].)g(It)g(is)g(kno)n(wn)g
(that)g(ev)n(ery)f(problem)h(in)g(this)h(class)e(can)h(b)r(e)g(appro)n
(ximated)f(within)523 2641 y FG(some)28 b FK(constan)n(t)f(factor.)g(A)
h(problem)g(is)f(MAX)i(SNP-hard)d(if)i(ev)n(ery)f(problem)g(in)h(MAX)
523 2741 y(SNP)g(can)f(b)r(e)h(L-reduced)f(to)g(it.)523
2895 y Fv(Theorem)32 b FH(2.18)41 b FG(The)30 b(sup)l(erstring)g(pr)l
(oblem)h(is)f(MAX)f(SNP-har)l(d.)523 3050 y Fv(Pr)n(oof.)19
b FK(The)g(reduction)g(is)g(from)g(a)f(sp)r(ecial)h(case)g(of)g(the)g
(TSP)g(with)h(triangle)e(inequalit)n(y)-7 b(.)523 3150
y(Let)35 b(TSP\(1,2\))f(b)r(e)h(the)g(TSP)g(restricted)f(to)g
(instances)h(where)f(all)g(the)i(distances)e(are)523
3249 y(either)24 b(1)f(or)g(2.)h(W)-7 b(e)24 b(can)f(consider)g(an)h
(instance)f(to)h(this)g(problem)g(as)f(b)r(eing)h(sp)r(eci\014ed)g(b)n
(y)523 3349 y(a)d(graph)f FC(H)7 b FK(;)21 b(the)g(edges)g(of)g
FC(H)28 b FK(are)20 b(precisely)g(those)h(that)h(ha)n(v)n(e)e(length)h
(1)g(while)g(the)h(edges)523 3449 y(that)i(are)e(not)h(in)h
FC(H)30 b FK(ha)n(v)n(e)22 b(length)h(2.)g(W)-7 b(e)24
b(need)f(here)g(the)h(v)n(ersion)e(of)h(the)g(TSP)g(where)g(w)n(e)523
3548 y(seek)28 b(the)i(shortest)e(Hamiltonian)h(path)g(\(instead)g(of)g
(cycle\),)g(and,)g(more)f(imp)r(ortan)n(tly)-7 b(,)523
3648 y(w)n(e)40 b(need)f(the)i(additional)e(restriction)g(that)h(the)g
(graph)f FC(H)47 b FK(b)r(e)40 b(of)g(b)r(ounded)g(degree)523
3747 y(\(the)31 b(precise)f(b)r(ound)i(is)e(not)h(imp)r(ortan)n(t\).)g
(It)g(w)n(as)e(sho)n(wn)h(in)h([12)o(])g(that)g(the)g(TSP\(1,2\))523
3847 y(problem)c(\(ev)n(en)h(for)f(this)g(restricted)g(v)n(ersion\))g
(is)g(MAX)i(SNP-hard.)606 3947 y(Let)g FC(H)36 b FK(b)r(e)29
b(a)f(graph)g(of)h(b)r(ounded)g(degree)f FC(D)j FK(sp)r(ecifying)e(an)f
(instance)h(of)f(TSP\(1,2\).)523 4046 y(The)19 b(hardness)g(result)g
(holds)g(for)f(b)r(oth)i(the)g(symmetric)f(and)g(the)h(asymmetric)e
(TSP)h(\(i.e.,)523 4146 y(for)i(b)r(oth)h(undirected)g(and)f(directed)h
(graphs)e FC(H)7 b FK(\).)22 b(W)-7 b(e)22 b(let)g FC(H)28
b FK(b)r(e)22 b(a)f(directed)h(graph)e(here.)523 4246
y(Without)25 b(loss)e(of)g(generalit)n(y)-7 b(,)23 b(assume)g(that)h
(eac)n(h)f(v)n(ertex)g(of)h FC(H)30 b FK(has)24 b(outdegree)e(at)i
(least)523 4345 y(2.)i(The)g(reduction)g(is)h(similar)e(to)h(the)h(one)
f(of)g([5])g(used)h(to)f(sho)n(w)f(the)i(NP-completeness)523
4445 y(of)36 b(the)h(sup)r(erstring)f(decision)f(problem.)h(W)-7
b(e)37 b(ha)n(v)n(e)e(to)i(pro)n(v)n(e)d(here)i(that)h(it)g(is)f(an)g
(L-)523 4544 y(reduction.)c(F)-7 b(or)32 b(ev)n(ery)g(v)n(ertex)f
FC(v)36 b FK(of)d FC(H)39 b FK(w)n(e)33 b(ha)n(v)n(e)e(t)n(w)n(o)h
(letters)g FC(v)k FK(and)d FC(v)2870 4514 y Fw(0)2893
4544 y FK(.)g(In)g(addition)523 4644 y(there)g(is)g(one)g(more)f
(letter)i(#.)f(Corresp)r(onding)f(to)h(eac)n(h)f(v)n(ertex)g
FC(v)37 b FK(w)n(e)c(ha)n(v)n(e)f(a)g(string)523 4744
y FC(v)s FK(#)p FC(v)678 4714 y Fw(0)702 4744 y FK(,)25
b(called)f(the)h FG(c)l(onne)l(ctor)33 b FK(for)24 b
FC(v)s FK(.)h(F)-7 b(or)24 b(eac)n(h)g(v)n(ertex)g FC(v)s
FK(,)h(en)n(umerate)e(the)i(edges)f(out)h(of)523 4843
y FC(v)g FK(in)e(an)e(arbitrary)f(cyclic)i(order)f(as)g(\()p
FC(v)s(;)14 b(w)1839 4855 y FB(0)1877 4843 y FK(\))p
FC(;)g(:)g(:)g(:)g(;)g FK(\()p FC(v)s(;)g(w)2265 4855
y FA(d)p Fw(\000)p FB(1)2390 4843 y FK(\))22 b(\(*\).)g(Corresp)r
(onding)e(to)i(the)523 4943 y FC(i)p FK(th)29 b(edge)f(\()p
FC(v)s(;)14 b(w)1020 4955 y FA(i)1049 4943 y FK(\))29
b(out)g(of)g FC(v)j FK(w)n(e)d(ha)n(v)n(e)e(a)i(string)f
FC(p)2093 4955 y FA(i)2121 4943 y FK(\()p FC(v)s FK(\))e(=)e
FC(v)2386 4913 y Fw(0)2410 4943 y FC(w)2469 4955 y FA(i)p
Fw(\000)p FB(1)2582 4943 y FC(v)2625 4913 y Fw(0)2649
4943 y FC(w)2708 4955 y FA(i)2736 4943 y FK(,)29 b(where)f(subscript)
523 5043 y(arithmetic)g(is)f(mo)r(dulo)h FC(d)p FK(.)f(W)-7
b(e)28 b(will)g(sa)n(y)f(that)h(these)f(strings)g(are)f
FG(asso)l(ciate)l(d)k FK(with)e FC(v)s FK(.)p 523 5162
499 5 v 627 5244 a Fm(1)661 5276 y FF(In)i(fact,)h(Arora)g(et)f(al.)h
([2)q(])f(ha)n(v)n(e)g(recen)n(tly)g(sho)n(wn)g(that)g(MAX)g(SNP-hard)f
(problems)i(do)523 5359 y(not)26 b(ha)n(v)n(e)f(p)r(olynomial)h(time)g
(appro)n(ximation)h(sc)n(hemes,)f(unless)g(P)g(=)f(NP)-6
b(.)p eop end
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b(b)r(ound)153 b(24)606 506 y FK(Let)38 b FC(n)f FK(b)r(e)h(the)f(n)n
(um)n(b)r(er)h(of)f(v)n(ertices)f(and)h FC(m)h FK(the)g(n)n(um)n(b)r
(er)f(of)g(edges)g(of)g FC(H)7 b FK(.)37 b(If)h(all)523
606 y(v)n(ertices)27 b(ha)n(v)n(e)g(degree)h(at)g(most)g
FC(D)j FK(then)d FC(m)d FD(\024)e FC(D)r(n)p FK(.)29
b(Let)f FC(k)k FK(b)r(e)c(the)h(minim)n(um)g(n)n(um)n(b)r(er)523
706 y(of)i(edges)g(whose)g(addition)g(to)g FC(H)38 b
FK(su\016ces)31 b(to)h(form)f(a)g(Hamiltonian)g(path.)g(Th)n(us,)g(the)
523 805 y(optimal)e(cost)f(of)h(the)h(TSP)e(instance)h(is)g
FC(n)19 b FD(\000)g FK(1)g(+)g FC(k)s FK(.)29 b(W)-7
b(e)29 b(shall)f(argue)g(that)h(the)h(length)523 905
y(of)c(the)g(shortest)f(common)g(sup)r(erstring)g(is)h(2)p
FC(m)14 b FK(+)g(3)p FC(n)g FK(+)h FC(k)i FK(+)e(1.)25
b(It)h(will)g(follo)n(w)f(then)h(that)523 1005 y(the)i(reduction)f(is)h
(linear)f(since)g FC(m)h FK(is)f(linear)g(in)h FC(n)p
FK(.)606 1104 y(Consider)20 b(the)i(distance-w)n(eigh)n(ted)d(graph)h
FC(G)2039 1116 y FA(S)2109 1104 y FK(for)g(this)h(set)g(of)g(strings,)g
(and)f(let)i FC(G)3214 1116 y FB(2)3272 1104 y FK(b)r(e)523
1204 y(its)h(subgraph)e(with)i(only)f(edges)g(of)h(minimal)g(w)n(eigh)n
(t)f(\(2\).)g(Clearly)-7 b(,)22 b FC(G)2749 1216 y FB(2)2809
1204 y FK(has)g(exactly)g(one)523 1303 y(comp)r(onen)n(t)30
b(for)g(eac)n(h)f(v)n(ertex)g(of)i FC(H)7 b FK(,)30 b(whic)n(h)g
(consists)f(of)h(a)g(cycle)g(of)g(the)h(asso)r(ciated)e
FC(p)523 1403 y FK(strings,)18 b(and)h(a)f(connector)g(that)h(has)f(an)
h(edge)f(to)h(eac)n(h)f(of)h(them.)g(W)-7 b(e)19 b(need)g(only)g
(consider)523 1503 y(`standard')25 b(sup)r(erstrings)f(in)i(whic)n(h)f
(all)g(strings)g(asso)r(ciated)f(with)i(some)f(v)n(ertex)f(form)h(a)523
1602 y(subgraph)k(of)g FC(G)1047 1614 y FB(2)1085 1602
y FK(,)h(so)f(that)h(only)g(the)g(last)f FC(p)h FK(string)f(has)g(an)h
(outgoing)e(edge)i(of)f(w)n(eigh)n(t)523 1702 y(more)34
b(than)g(2)h(\(3)f(or)g(4\).)g(Namely)-7 b(,)34 b(if)i(some)d(v)n
(ertex)h(fails)g(this)h(requiremen)n(t,)f(then)h(at)523
1802 y(least)30 b(t)n(w)n(o)f(of)h(its)g(asso)r(ciated)f(strings)g(ha)n
(v)n(e)g(outgoing)f(edges)i(of)f(w)n(eigh)n(t)h(more)f(than)h(2,)523
1901 y(th)n(us)d(w)n(e)g(do)f(not)h(increase)f(the)h(length)g(b)n(y)g
(putting)g(all)g(its)g FC(p)f FK(strings)g(directly)h(after)g(its)523
2001 y(connector)g(in)h(a)f(standard)g(w)n(a)n(y)-7 b(.)27
b(A)h(standard)f(sup)r(erstring)g(naturally)g(corresp)r(onds)f(to)523
2100 y(an)h(ordering)f(of)i(v)n(ertices)e FC(v)1401 2112
y FB(1)1439 2100 y FC(;)14 b(v)1516 2112 y FB(2)1553
2100 y FC(;)g(:)g(:)g(:)g(;)g(v)1778 2112 y FA(n)1823
2100 y FK(.)606 2200 y(F)-7 b(or)20 b(the)g(con)n(v)n(erse)e(there)i
(remains)f(a)h(c)n(hoice)f(of)h(whic)n(h)g(string)g FC(q)j
FK(succeeds)c(a)h(connector)523 2300 y FC(v)563 2312
y FA(i)591 2300 y FK(#)p FC(v)703 2270 y Fw(0)700 2321
y FA(i)728 2300 y FK(.)30 b(If)g FC(H)36 b FK(has)29
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y FK(to)g FC(v)1869 2312 y FA(i)p FB(+1)2010 2300 y FK(and)g(the)g
(`next')g(edge)f(out)g(of)h FC(v)3031 2312 y FA(i)3088
2300 y FK(\(in)g(\(*\)\))523 2399 y(go)r(es)j(to,)g(sa)n(y)g
FC(v)1032 2411 y FA(j)1067 2399 y FK(,)g(then)h(c)n(ho)r(osing)f
FC(q)j FK(=)c FC(v)1875 2369 y Fw(0)1872 2421 y FA(i)1900
2399 y FC(v)1940 2411 y FA(i)p FB(+1)2052 2399 y FC(v)2095
2369 y Fw(0)2092 2421 y FA(i)2120 2399 y FC(v)2160 2411
y FA(j)2229 2399 y FK(results)h(in)h(a)f(w)n(eigh)n(t)f(of)i(3)f(on)g
(the)523 2499 y(edge)19 b(from)h(the)g(last)f FC(p)h
FK(string)f(to)h(the)g(next)g(connector)f FC(v)2286 2511
y FA(i)p FB(+1)2398 2499 y FK(#)p FC(v)2510 2469 y Fw(0)2507
2521 y FA(i)p FB(+1)2619 2499 y FK(,)h(whereas)e(this)i(w)n(eigh)n(t)
523 2599 y(w)n(ould)26 b(otherwise)e(b)r(e)j(4.)e(If)i
FC(H)32 b FK(do)r(esn't)27 b(ha)n(v)n(e)d(this)i(edge,)g(then)g(the)h
(c)n(hoice)e(of)h FC(q)j FK(do)r(esn't)523 2698 y(matter.)c(Let)h(us)g
(call)f(a)g(sup)r(erstring)f(`Standard')h(if)h(in)g(addition)g(to)f(b)r
(eing)h(standard,)e(it)523 2798 y(also)j(satis\014es)f(this)i(latter)f
(requiremen)n(t)g(for)g(all)h(v)n(ertices.)606 2897 y(No)n(w)34
b(supp)r(ose)g(that)h(the)g(addition)f(of)g FC(k)k FK(edges)c(to)g
FC(H)41 b FK(giv)n(es)33 b(a)h(Hamiltonian)h(path)523
2997 y FC(v)563 3009 y FB(1)601 2997 y FC(;)14 b(v)678
3009 y FB(2)715 2997 y FC(;)g(:)g(:)g(:)g(;)g(v)940 3009
y FA(n)p Fw(\000)p FB(1)1070 2997 y FC(;)g(v)1147 3009
y FA(n)1192 2997 y FK(.)37 b(Then)f(w)n(e)h(can)f(construct)g(a)g
(corresp)r(onding)e(Standard)i(sup)r(er-)523 3097 y(string.)26
b(If)h(the)g(out-degree)e(of)h FC(v)1543 3109 y FA(i)1598
3097 y FK(is)g FC(d)1723 3109 y FA(i)1751 3097 y FK(,)h(then)g(its)f
(length)h(will)g(b)r(e)2623 3034 y Fn(P)2711 3055 y FA(n)2711
3122 y(i)p FB(=1)2823 3097 y FK(\(2)16 b(+)g(2)p FC(d)3079
3109 y FA(i)3122 3097 y FK(+)g(1\))g(+)523 3196 y FC(k)21
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FC(k)k FK(+)c(1.)606 3296 y(Con)n(v)n(ersely)-7 b(,)23
b(supp)r(ose)h(w)n(e)g(are)g(giv)n(en)g(a)g(common)g(sup)r(erstring)g
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3396 y FC(k)23 b FK(+)c(1.)29 b(This)h(can)f(then)i(b)r(e)f(turned)g
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3495 y(If)e FC(v)645 3507 y FB(1)682 3495 y FC(;)14 b(v)759
3507 y FB(2)797 3495 y FC(;)g(:)g(:)g(:)f(;)h(v)1021
3507 y FA(n)1093 3495 y FK(is)26 b(the)g(corresp)r(onding)f(order)g(of)
h(v)n(ertices,)f(it)i(follo)n(ws)e(that)h FC(H)33 b FK(cannot)523
3595 y(b)r(e)28 b(missing)f(more)g(than)h FC(k)i FK(of)e(the)g(edges)f
(\()p FC(v)1939 3607 y FA(i)1967 3595 y FC(;)14 b(v)2044
3607 y FA(i)p FB(+1)2156 3595 y FK(\).)1085 b Fu(2)606
3744 y FK(Since)41 b(the)g(strings)f(in)g(the)h(ab)r(o)n(v)n(e)f
(L-reduction)f(ha)n(v)n(e)h(b)r(ounded)h(length)g(\(4\),)f(the)523
3844 y(reduction)28 b(applies)h(also)e(to)i(the)g(maximization)f(v)n
(ersion)f(of)h(the)i(sup)r(erstring)d(problem)523 3944
y([15)o(,)h(16)o(].)g(That)f(is,)h(maximizing)f(the)h(total)f
(compression)f(is)i(also)e(MAX)i(SNP-hard.)523 4251 y
FL(2.8)116 b(Op)s(en)39 b(problems)523 4450 y FK(W)-7
b(e)28 b(end)g(the)g(c)n(hapter)f(with)h(sev)n(eral)d(op)r(en)j
(questions)f(raised)g(from)g(this)h(researc)n(h:)606
4550 y(\(1\))j(Obtain)g(an)f(algorithm)g(whic)n(h)h(ac)n(hiev)n(es)e(a)
h(p)r(erformance)g(b)r(etter)h(than)g(3)g(times)523 4649
y(the)d(optim)n(um.)606 4749 y(\(2\))38 b(Pro)n(v)n(e)e(or)i(dispro)n
(v)n(e)e(the)j(conjecture)e(that)i(GREED)n(Y)f(ac)n(hiev)n(es)e(2)i
(times)g(the)523 4848 y(optim)n(um.)p eop end
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2520 y FK([1])41 b(N.)29 b(Alon,)f(S.)h(Cosares,)d(D.)j(Ho)r(c)n(h)n
(baum,)f(and)h(R.)f(Shamir.)g(An)h(algorithm)e(for)h(the)694
2607 y(detection)36 b(and)h(construction)e(of)i(Monge)e(Sequences.)h
FG(Line)l(ar)j(A)n(lgebr)l(a)f(and)h(its)694 2693 y(Applic)l(ations)31
b(114/115,)i(669{680,)h(1989.)565 2848 y FK([2])41 b(A.)28
b(Arora,)e(C.)i(Lund,)g(R.)g(Mot)n(w)n(ani,)e(M.)i(Sudan,)g(and)f(M.)h
(Szegedy)-7 b(.)27 b(Pro)r(of)g(v)n(eri\014-)694 2935
y(cation)j(and)g(hardness)f(of)h(appro)n(ximation)f(problems.)g
FG(Pr)l(o)l(c.)k(33r)l(d)h(IEEE)f(Symp.)694 3017 y(F)-6
b(ound.)30 b(Comp.)h(Sci.)p FK(,)e(1992,)c(14-23.)565
3167 y([3])41 b(E.)18 b(Barnes)g(and)g(A.)h(Ho\013man.)g(On)g(transp)r
(ortation)e(problems)h(with)i(upp)r(er)f(b)r(ounds)694
3254 y(on)31 b(leading)h(rectangles.)e FG(SIAM)k(Journal)g(on)f(A)n
(lgebr)l(aic)i(and)g(Discr)l(ete)e(Metho)l(ds)694 3332
y(6,)d(487{496,)j(1985.)565 3482 y FK([4])41 b(N.)35
b(Fine)h(and)f(H.)g(Wilf.)h(Uniqueness)f(theorems)g(for)f(p)r(erio)r
(dic)h(functions.)h FG(Pr)l(o)l(c.)694 3569 y(A)n(mer.)30
b(Math.)h(So)l(c.)d FK(16,)f(1965,)e(109-114.)565 3719
y([5])41 b(J.)33 b(Gallan)n(t,)f(D.)i(Maier,)f(J.)g(Storer.)f(On)h
(\014nding)g(minimal)h(length)f(sup)r(erstrings.)694
3806 y FG(Journal)c(of)i(Computer)f(and)g(System)g(Scienc)l(es)g(20,)h
(50{58,)h(1980.)565 3956 y FK([6])41 b(M.)32 b(Garey)f(and)g(D.)i
(Johnson.)e(Computers)g(and)h(In)n(tractabilit)n(y)-7
b(.)31 b FG(F)-6 b(r)l(e)l(eman,)34 b(New)694 4043 y(Y)-6
b(ork,)30 b(1979.)565 4193 y FK([7])41 b(A.)33 b(Ho\013man.)g(On)g
(simple)g(transp)r(ortation)f(problems.)g FG(Convexity:)k(Pr)l(o)l(c)l
(e)l(e)l(dings)694 4280 y(of)30 b(symp)l(osia)g(in)g(pur)l(e)f
(mathematics,)h(V)-6 b(ol)29 b(7,)h(A)n(meric)l(an)g(Mathematic)l(al)h
(So)l(ciety,)694 4358 y(317{327,)i(1963.)565 4508 y FK([8])41
b(A.)22 b(Lesk)f(\(Edited\).)h(Computational)f(Molecular)f(Biology)-7
b(,)20 b(Sources)g(and)i(Metho)r(ds)694 4595 y(for)27
b(Sequence)g(Analysis.)g FG(Oxfor)l(d)j(University)g(Pr)l(ess,)h(1988.)
565 4745 y FK([9])41 b(M.)30 b(Li.)g(T)-7 b(o)n(w)n(ards)28
b(a)i(DNA)h(sequencing)f(theory)-7 b(.)29 b FG(31st)k(IEEE)g(Symp.)f
(on)h(F)-6 b(ounda-)694 4832 y(tions)29 b(of)i(Computer)f(Scienc)l(e,)h
(125{134,)i(1990.)523 4982 y FK([10])41 b(C.)k(P)n(apadimitriou)e(and)i
(K.)g(Steiglitz.)g FG(Combinatorial)j(Optimization:)g(A)n(lgo-)694
5069 y(rithms)30 b(and)g(Complexity.)g FK(Pren)n(tice-Hall,)c(1982.)523
5219 y([11])41 b(C.)36 b(P)n(apadimitriou)f(and)h(M)g(Y)-7
b(annak)i(akis.)35 b(Optimization,)i(appro)n(ximation)d(and)694
5306 y(complexit)n(y)d(classes.)f FG(20th)35 b(A)n(CM)f(Symp.)g(on)g
(The)l(ory)h(of)f(Computing)p FK(,)f(229{234,)694 5384
y(1988.)p eop end
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b(problems)154 b(26)523 506 y FK([12])41 b(C.)19 b(P)n(apadimitriou)f
(and)i(M)g(Y)-7 b(annak)i(akis.)18 b(The)i(tra)n(v)n(eling)e(salesman)g
(problem)h(with)694 593 y(distances)27 b(one)g(and)g(t)n(w)n(o.)g
FG(Mathematics)32 b(of)e(Op)l(er)l(ations)g(R)l(ese)l(ar)l(ch.)f
FK(T)-7 b(o)27 b(app)r(ear.)523 746 y([13])41 b(H.)24
b(P)n(eltola,)e(H.)j(So)r(derlund,)e(J.)h(T)-7 b(arhio,)23
b(and)g(E.)h(Ukk)n(onen.)f(Algorithms)g(for)g(some)694
833 y(string)k(matc)n(hing)h(problems)g(arising)f(in)i(molecular)e
(genetics.)h FG(Information)j(Pr)l(o-)694 920 y(c)l(essing)f(83)g(\(Pr)
l(o)l(c.)h(IFIP)f(Congr)l(ess,)h(1983\))g(53{64.)523
1077 y FK([14])41 b(J.)20 b(Storer.)f FG(Data)k(c)l(ompr)l(ession:)i
(metho)l(ds)f(and)f(the)l(ory.)h(Computer)g(Scienc)l(e)f(Pr)l(ess,)694
1160 y(1988.)523 1297 y FK([15])41 b(J.)35 b(T)-7 b(arhio)34
b(and)h(E.)g(Ukk)n(onen.)g(A)g(Greedy)g(appro)n(ximation)f(algorithm)g
(for)g(con-)694 1384 y(structing)24 b(shortest)g(common)g(sup)r
(erstrings.)g FG(The)l(or)l(etic)l(al)29 b(Computer)e(Scienc)l(e)h(57)
694 1462 y(131{145)k(1988)523 1615 y FK([16])41 b(J.)23
b(T)-7 b(urner.)22 b(Appro)n(ximation)g(algorithms)g(for)h(the)h
(shortest)e(common)h(sup)r(erstring)694 1701 y(problem.)k
FG(Information)k(and)f(Computation)g(83)h(1{20)g(1989.)523
1855 y FK([17])41 b(L.)23 b(G.)g(V)-7 b(alian)n(t.)23
b(A)h(Theory)e(of)h(the)g(Learnable.)f FG(Comm.)27 b(A)n(CM)37
b FK(27\(11\))22 b(1134{1142)694 1937 y(1984.)p eop end
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2549 y FL(3.1)116 b(In)m(tro)s(duction)523 2748 y FK(Consider)37
b(a)h(prop)r(ert)n(y)f(that)i(pixels)f(in)h(a)f(frame)f(bu\013er)i(ma)n
(y)e(or)h(ma)n(y)f(not)i(p)r(ossess,)523 2848 y(suc)n(h)33
b(as)g(ha)n(ving)g(a)g(particular)f(color,)g(or,)h(more)g(generally)-7
b(,)32 b(a)h(color)f(from)i(a)f(selected)523 2947 y(range.)j(W)-7
b(e)37 b(can)g(iden)n(tify)g(regions)f(of)h(connected)f(pixels)h(that)h
(share)d(this)j(prop)r(ert)n(y)-7 b(.)523 3047 y(Eac)n(h)33
b(region)f(is)i(delimited)h(b)n(y)e(a)h(set)f(of)h(pixels)g(lac)n(king)
f(the)h(prop)r(ert)n(y)-7 b(,)33 b(constituting)523 3147
y(its)j FG(b)l(oundary)7 b FK(.)36 b(W)-7 b(e)36 b(are)f(concerned)f
(with)i FG(4-c)l(onne)l(cte)l(d)2357 3116 y FB(1)2429
3147 y FK(regions,)e(where)h(eac)n(h)g(pixel)523 3246
y(connects)c(to)g(its)g(4)f(horizon)n(tally)g(and)h(v)n(ertically)e
(adjacen)n(t)i(neigh)n(b)r(ours.)f(In)h(con)n(trast,)523
3346 y(the)26 b(b)r(oundary)f(of)h(a)f(region)f(consists)h(of)h(a)f(n)n
(um)n(b)r(er)h(of)f(8-connected)g(comp)r(onen)n(ts,)g(one)523
3445 y(external)c(comp)r(onen)n(t,)h(and)g(zero)f(or)h(more)f(in)n
(ternal)g(ones.)h(Giv)n(en)g(a)g(designated)f(region,)523
3545 y(a)31 b(\014lling)g(algorithm)f(is)i(to)f(apply)g(to)g(eac)n(h)g
(pixel)g(in)g(the)h(region)e(a)h(certain)g(op)r(eration,)523
3645 y(suc)n(h)c(as)g(setting)h(the)g(color)e(to)h(a)h(sp)r(eci\014ed)f
(v)-5 b(alue,)28 b(exactly)f(once.)523 3894 y Fl(3.1.1)99
b(Pr)-5 b(op)g(erties)34 b(and)h(Op)-5 b(er)g(ations)523
4043 y FK(There)23 b(is)h(one,)g(v)n(ery)f(imp)r(ortan)n(t,)g
(restriction)g(w)n(e)h(put)g(on)g(the)g(p)r(ossible)g(com)n(bination)f
(of)523 4143 y(prop)r(ert)n(y)29 b(and)g(op)r(eration,)g(namely)g(that)
h(the)g(op)r(eration)f FH(in)m(v)-5 b(alidates)29 b FK(the)h(prop)r
(ert)n(y)-7 b(.)523 4242 y(So,)29 b(the)h(op)r(eration)e(\\mak)n(e)g
(green")g(is)i(allo)n(w)n(ed)e(for)h(the)g(prop)r(ert)n(y)g(\\red",)f
(but)i(not)f(for)523 4342 y(the)f(prop)r(ert)n(y)e(\\red)h(or)g
(green".)606 4442 y(W)-7 b(e)32 b(p)r(ose)f(this)g(restriction)g(with)g
(the)h(aim)f(of)h(\014nding)f(a)g(constan)n(t)g(space)f(\014lling)h
(al-)523 4541 y(gorithm.)25 b(T)-7 b(o)25 b(see)g(wh)n(y)f(this)i(is)f
(necessary)-7 b(,)24 b(assume)h(there)g(is)g(a)g(third)g(colour,)f
(blue,)i(and)523 4641 y(consider)32 b(some)g(arbitrary)e(green)i
(region)f(with)i(a)f(blue)h(b)r(oundary)-7 b(.)32 b(With)i(a)e(prop)r
(ert)n(y)523 4741 y(\\red)i(or)f(green")g(de\014ning)i(the)g(regions,)e
(the)i(op)r(eration)e(\\mak)n(e)g(green")g(is)i(of)f(course)523
4840 y(ine\013ectiv)n(e)29 b(on)f(our)g(green-only)f(region.)h(If)h
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4940 y(ho)n(w)n(ev)n(er,)37 b(they)i(w)n(ould)f(need)h(to)g(b)r(e)h
(found)f(b)n(y)f(the)i(\014lling)f(algorithm)e(and)i(colored)523
5039 y(green.)31 b(This)i(reduces)e(the)i(task)e(of)h(the)h(\014lling)f
(algorithm)f(to)h(one)g(of)g(visiting)g(all)g(the)p 523
5159 499 5 v 627 5241 a Fm(1)661 5273 y FF(Note)f(that)f(w)n(e)i(div)n
(erge)e(here)h(from)h(the)e(graph-theoretic)h(in)n(terpretation)g(of)g
(ha)n(ving)g(so)523 5356 y(man)n(y)25 b(disjoin)n(t)i(paths)e(b)r(et)n
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(\014nding)g(an)g(algorithm)f(for)h(unrestricted)g(\014lling)523
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b(.)606 905 y(F)g(urthermore,)23 b(is)h(it)h(widely)f(b)r(eliev)n(ed)h
(that)f(no)g(\014nite)h(\\bug")e(automaton)2973 875 y
FB(2)3034 905 y FK(\(p)r(ossibly)523 1005 y(equipp)r(ed)35
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y(tra)n(v)n(ersal.)24 b(This)j(means)f(that)h(the)g(restriction)f(w)n
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b(the)h(prop)r(ert)n(y)f(for)g(the)h(pixel)g(whose)f(address)731
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(denote)g(the)g(set)g(of)g(pixels)g(that)g(remain)f(to)h(b)r(e)g
(\014lled,)h(The)523 3747 y(b)r(oundary)e(of)g FC(R)q
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(the)h(pixels)f(in)g(the)h(region.')e(This)h(distinction)g(seems)523
4644 y(a)j(natural)f(one)h(for)g(familiar)g(FF)-9 b(As,)27
b(whic)n(h)h(can)e(b)r(e)i(written)g(purely)f(in)g(terms)g(of)g
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f(tra)n(v)n(erse)e(a)i(designated)g(region.)g(The)g(correct)523
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h(m)n(uc)n(h)h(dep)r(ends)g(on)g(the)h(SETting)f(of)523
4943 y(pixels.)22 b(Therefore,)f(w)n(e)h(will)g(refrain)f(from)h(the)h
(ab)r(o)n(v)n(e)e(comp)r(ositional)g(view)h(and)g(simply)523
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b(a)g(\014nite)g(automaton)g(as)g(a)g(bug)f(w)n(alking)i(in)e(a)h
(graph,)g(at)g(eac)n(h)g(step)f(seeing)i(whic)n(h)523
5359 y(of)k(the)g(4)g(direction)g(are)g(p)r(ossible.)p
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b(for)g(all)f(practical)g(purp)r(oses)g(where)g(the)i(regions)d(will)i
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b(the)h(follo)n(wing)e(recursiv)n(e)g(\015o)r(o)r(d)i(\014ll)g
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g(pro)r(cedure.)e(See)i(Fishkin)f(and)g(Barsky)f([1])h(for)g(a)g
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(n)n(um)n(b)r(er)523 4035 y(of)g(times)g(a)g(pixel)f(in)i(the)f(region)
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(simple)h(FF)-9 b(A,)22 b(ev)n(ery)523 4134 y(pixel)32
b(prop)r(erly)f(inside)h(the)g(\014lled)g(region)e(is)i(visited)g(once)
f(from)h(eac)n(h)f(side)h(\(4)f(times\))523 4234 y(and)22
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(times.)i(The)f(pixels)g(on)f(the)i(b)r(ound-)523 4334
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(simplicities')h(sak)n(e,)e(are)h(excluded)g(from)523
4433 y(the)29 b(ab)r(o)n(v)n(e)f(a)n(v)n(erage.)e(Let)j(us)g(men)n
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(the)h(memory)e(usage.)g(While)i(there)f(are)f(no)g(explicit)i(data)e
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b(,)33 b(and)g(so)g(on.)g(Since)h(bac)n(k-trac)n(king)d(o)r(ccurs)h
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(out)h(ev)n(enly)f(in)h(all)g(four)f(directions,)523
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7 100 v 2023 1290 67 7 v 1854 1287 7 100 v 1791 1290
67 7 v 1887 1420 7 100 v 1791 1323 100 7 v 1655 1021
7 100 v 1558 1024 100 7 v 1691 991 V 1688 988 7 67 v
1691 1091 100 7 v 1688 1154 7 67 v 1655 1154 7 100 v
1558 1058 100 7 v 1160 1556 233 7 v 1389 1553 7 233 v
1426 1523 233 7 v 1422 1519 7 200 v 1426 1622 233 7 v
1422 1818 7 200 v 1160 1589 233 7 v 1389 1818 7 233 v
1027 1556 100 7 v 1123 1553 7 100 v 1160 1523 100 7 v
1157 1519 7 67 v 1160 1622 100 7 v 1157 1685 7 67 v 1027
1589 100 7 v 1123 1685 7 100 v 1293 1290 100 7 v 1389
1287 7 100 v 1356 1420 V 1293 1323 67 7 v 1456 1420 7
100 v 1459 1323 67 7 v 1426 1290 100 7 v 1422 1287 7
100 v 1691 1523 100 7 v 1688 1519 7 100 v 1558 1489 100
7 v 1655 1486 7 67 v 1558 1655 100 7 v 1655 1719 7 67
v 1691 1622 100 7 v 1688 1719 7 100 v 1293 1855 100 7
v 1389 1951 7 100 v 1356 1818 V 1293 1822 67 7 v 1456
1818 7 100 v 1459 1822 67 7 v 1426 1855 100 7 v 1422
1951 7 100 v 1957 2120 233 7 v 1954 2350 7 233 v 1987
2084 V 1990 2087 200 7 v 1887 2084 7 233 v 1691 2087
200 7 v 1691 2120 233 7 v 1920 2350 7 233 v 1954 2482
7 100 v 1957 2386 100 7 v 1987 2350 7 100 v 1990 2353
67 7 v 1887 2350 7 100 v 1824 2353 67 7 v 1920 2482 7
100 v 1824 2386 100 7 v 2219 2217 7 100 v 2223 2120 100
7 v 2090 2154 V 2186 2217 7 67 v 2090 2054 100 7 v 2186
2051 7 67 v 2219 2084 7 100 v 2223 2087 100 7 v 1987
1818 7 100 v 1990 1822 100 7 v 2020 1951 7 100 v 2023
1855 67 7 v 1854 1951 7 100 v 1791 1855 67 7 v 1887 1818
7 100 v 1791 1822 100 7 v 1655 2217 7 100 v 1558 2120
100 7 v 1691 2154 V 1688 2217 7 67 v 1691 2054 100 7
v 1688 2051 7 67 v 1655 2084 7 100 v 1558 2087 100 7
v 876 2889 a FF(FIGURE)25 b(3.1.)i(A)e(w)n(orst)i(case)g(example)e
(region)i(for)f(breadth)f(\014rst)h(\014ll.)523 3153
y FC(R)q FK(,)i(at)g(an)n(y)f(time,)i(denotes)e(the)i(set)f(of)g
(pixels)g(that)g(remain)g(to)f(b)r(e)i(\014lled.)f(Clearly)-7
b(,)27 b(if)i FC(R)523 3252 y FK(consists)e(of)g FC(c)g
FK(comp)r(onen)n(ts,)g(then)h(at)f(least)g FC(c)g FK(di\013eren)n(t)g
(pixels)g(m)n(ust)h(b)r(e)f(remem)n(b)r(ered,)523 3352
y(one)19 b(in)h(eac)n(h)f(comp)r(onen)n(t,)g(in)h(order)e(to)h(b)r(e)h
(able)f(to)h(complete)f(\014lling)g(all)h(of)f FC(R)q
FK(.)g(Whenev)n(er)523 3452 y(a)31 b(pixel)h(is)g(\014lled)g(that)g(is)
f(a)h(cutting)g(p)r(oin)n(t)g(of)f(its)h(comp)r(onen)n(t)g(\(a)f
(cutting)h(p)r(oin)n(t)g(in)g(a)523 3551 y(connected)g(graph)e(is)i(a)f
(p)r(oin)n(t)h(whose)f(remo)n(v)-5 b(al)30 b(lea)n(v)n(es)g(more)h
(than)g(one)h(comp)r(onen)n(t\),)523 3651 y(additional)24
b(pixels)h(m)n(ust)f(b)r(e)h(remem)n(b)r(ered.)f(On)h(the)g(other)f
(hand,)g(if)i(w)n(e)e(only)g(\014ll)h(pixels)523 3751
y(that)f(lea)n(v)n(e)e FC(R)i FK(connected,)f(then)h(w)n(e)f(just)g(mo)
n(v)n(e)g(to)g(an)n(y)f(of)h(its)h(un\014lled)g(neigh)n(b)r(ours)e(and)
523 3850 y(con)n(tin)n(ue,)i(without)g(ha)n(ving)g(to)g(remem)n(b)r(er)
f(an)n(y)h(pixel)g(other)g(than)g(where)g(w)n(e)f(are.)h(This)523
3950 y(reduces)29 b(the)g(problem)g(of)h(\014lling)f(to)g(that)h(of)f
(\014nding)h(a)f(non-cutting)g(pixel.)g(If)h(w)n(e)f(can)523
4049 y(\014nd)j(suc)n(h)f(a)h(pixel)f(in)h(time)g FC(T)12
b FK(\()p FC(n)p FK(\),)31 b(then)h(\014lling)g(all)f(the)h(pixels)g
(in)g FC(R)g FK(can)f(b)r(e)h(done)f(in)523 4149 y(time)d
FC(nT)12 b FK(\()p FC(n)p FK(\))27 b(with)h(no)g(additional)f(storage.)
523 4456 y FL(3.5)116 b(On)39 b(planar)e(em)m(b)s(eddings)523
4655 y FK(Let's)27 b(consider)f(the)h(problem)f(in)i(a)e(sligh)n(tly)g
(di\013eren)n(t,)h(more)g(general,)e(setting.)i(A)g(pla-)523
4755 y(nar)33 b(em)n(b)r(edding)h(of)g(a)f(\(planar\))g(graph)g(can)g
(b)r(e)i(succinctly)e(represen)n(ted)g(b)n(y)g(a)h(triple)523
4855 y(\()p FC(V)5 b(;)14 b(H)r(;)g(r)r FK(\).)22 b FC(V)40
b FK(is)20 b(the)h(set)g(of)f(v)n(ertices)g(in)g(the)h(graph.)f
FC(H)30 b FD(\022)22 b FC(V)i FD(\002)t FC(V)39 b FK(is)21
b(the)g(set)f(of)h FG(half-e)l(dges)p FK(.)523 4954 y(A)k(half-edge)f
FC(h)f FK(=)f(\()p FC(v)s(;)14 b(w)r FK(\))26 b(is)f(that)g(half)f(of)h
(the)g(edge)f FD(f)p FC(v)s(;)14 b(w)r FD(g)24 b FK(whic)n(h)h(is)f
(adjacen)n(t)g(to)h FC(v)s FK(.)g(Of)523 5054 y(course,)c(w)n(e)g
(require)g(that)h FC(H)29 b FK(b)r(e)22 b(symmetric;)g(a)f(half)h(edge)
g(\()p FC(v)s(;)14 b(w)r FK(\))23 b(cannot)f(exist)f(without)523
5154 y(a)27 b(corresp)r(onding)f(other)g(half)i(\()p
FC(w)r(;)14 b(v)s FK(\).)29 b(Finally)-7 b(,)27 b FC(r)f
FK(:)d FC(H)30 b FD(!)23 b FC(H)34 b FK(is)28 b(a)f(function)h(that)f
(orders)523 5253 y(all)h(half-edges)f(around)g(eac)n(h)h(v)n(ertex)f
(in)i(a)e(clo)r(c)n(k-wise)g(manner.)h(If)h(the)f(half-edges)f(ad-)523
5353 y(jacen)n(t)h(to)g(a)g(v)n(ertex)f FC(v)32 b FK(of)c(degree)f
FC(d)i FK(are,)e(in)i(clo)r(c)n(k-wise)e(order,)g FC(h)2630
5365 y FB(0)2667 5353 y FC(;)14 b(h)2752 5365 y FB(1)2789
5353 y FC(;)g(:)g(:)g(:)f(;)h(h)3021 5365 y FA(d)p Fw(\000)p
FB(1)3145 5353 y FK(,)28 b(then)p eop end
%%Page: 32 42
TeXDict begin 32 41 bop 2197 299 a FF(3.5.)52 b(On)25
b(planar)h(em)n(b)r(eddings)154 b(32)523 506 y FC(r)r
FK(\()p FC(h)642 518 y FA(i)671 506 y FK(\))23 b(=)g
FC(h)862 521 y FB(\()p FA(i)p FB(+1\))c(mo)r(d)f FA(d)1223
506 y FK(.)27 b(Half-edges)e(adjacen)n(t)g(to)h(the)h(same)e(v)n(ertex)
g(are)g(also)g(said)g(to)h(b)r(e)523 606 y(adjacen)n(t.)h(Note)g(that)g
(the)h(v)n(ertices)e(are)g(in)h(one-one)f(corresp)r(ondence)g(to)g(the)
i FG(orbits)g FK(of)523 706 y FC(r)r FK(.)h(Let)e(us)h(call)f(suc)n(h)g
(a)g(triple)h(\()p FC(V)5 b(;)14 b(H)r(;)g(r)r FK(\))29
b(a)e FG(labyrinth)p FK(.)606 805 y(W)-7 b(e)38 b(in)n(tro)r(duce)f
(some)h(more)e(terminology)h(that)h(will)g(b)r(e)g(useful)g(later.)f
(Lab)n(yrin)n(th)523 905 y FC(L)580 875 y Fw(0)626 905
y FK(=)23 b(\()p FC(V)813 875 y Fw(0)836 905 y FC(;)14
b(H)949 875 y Fw(0)972 905 y FC(;)g(r)1048 875 y Fw(0)1072
905 y FK(\))23 b(is)f(a)g FG(sublabyrinth)i FK(of)e FC(L)h
FK(=)f(\()p FC(V)5 b(;)14 b(H)r(;)g(r)r FK(\))24 b(i\013)f
FC(V)2470 875 y Fw(0)2516 905 y FD(\022)g FC(V)c FK(,)j
FC(H)2792 875 y Fw(0)2838 905 y FD(\022)h FC(H)15 b FD(\\)8
b FC(V)3140 875 y Fw(0)3172 905 y FD(\002)g FC(V)3311
875 y Fw(0)3335 905 y FK(,)523 1005 y(and)32 b FD(8)p
FC(h)e FD(2)g FC(H)974 974 y Fw(0)1028 1005 y FK(:)g
FC(r)1120 974 y Fw(0)1144 1005 y FK(\()p FC(h)p FK(\))g(=)g
FC(r)1420 974 y FA(i)1449 1005 y FK(\()p FC(h)p FK(\))p
FC(;)14 b(i)30 b(>)f FK(0)f(minimal)f(suc)n(h)h(that)g
FC(r)2550 974 y FA(i)2578 1005 y FK(\()p FC(h)p FK(\))i
FD(2)h FC(H)2882 974 y Fw(0)2905 1005 y FK(.)h(When)g
FC(L)e FK(=)523 1104 y(\()p FC(V)5 b(;)14 b(H)r(;)g(r)r
FK(\))31 b(is)e(a)h(lab)n(yrin)n(th)e(and)i FC(V)1601
1074 y Fw(0)1651 1104 y FD(\022)c FC(V)19 b FK(,)30 b(then)g
FC(L)p FD(j)p FC(V)2199 1074 y Fw(0)2252 1104 y FK(is)f(the)h(sublab)n
(yrin)n(th)f(\()p FC(V)3066 1074 y Fw(0)3089 1104 y FC(;)14
b(H)3202 1074 y Fw(0)3225 1104 y FC(;)g(r)3301 1074 y
Fw(0)3325 1104 y FK(\))523 1204 y(with)37 b FC(H)797
1174 y Fw(0)857 1204 y FK(=)g FC(H)30 b FD(\\)25 b FC(V)1205
1174 y Fw(0)1252 1204 y FD(\002)f FC(V)1408 1174 y Fw(0)1431
1204 y FK(.)36 b(Also,)g(with)h FC(v)j FD(2)e FC(V)18
b FK(,)37 b FC(L)23 b FD(n)h FC(v)39 b FK(denotes)d FC(L)p
FD(j)p FK(\()p FC(V)43 b FD(\000)24 b(f)p FC(v)s FD(g)p
FK(\).)35 b(An)523 1303 y FG(e)l(dge)c FK(in)g(a)f(lab)n(yrin)n(th)f
(is)h(a)g(set)g(of)g(t)n(w)n(o)g(corresp)r(onding)e(half-edges)h
FD(f)p FK(\()p FC(v)s(;)14 b(w)r FK(\))p FC(;)g FK(\()p
FC(w)r(;)g(v)s FK(\))p FD(g)p FK(.)32 b(A)523 1403 y
FG(p)l(ath)f FK(is)f(de\014ned)g(in)h(the)f(natural)f(w)n(a)n(y)-7
b(.)29 b(A)i(lab)n(yrin)n(th)e(is)h FG(c)l(onne)l(cte)l(d)g
FK(if)g(for)g(an)n(y)f(distinct)523 1503 y(v)n(ertices)40
b FC(v)s(;)14 b(w)44 b FK(there)d(is)f(a)h(path)g(from)g
FC(v)j FK(to)d FC(w)r FK(.)h(A)f FG(c)l(onne)l(cte)l(d)h(c)l(omp)l
(onent)p FK(,)f(or)f(just)523 1602 y(comp)r(onen)n(t,)28
b(of)h(a)f(lab)n(yrin)n(th)g FC(L)c FK(=)g(\()p FC(V)5
b(;)14 b(H)r(;)g(r)r FK(\),)30 b(is)e(a)g(maximal)g(connected)h(sublab)
n(yrin)n(th)523 1702 y FC(L)580 1672 y Fw(0)634 1702
y FK(=)i(\()p FC(V)829 1672 y Fw(0)852 1702 y FC(;)14
b(H)965 1672 y Fw(0)988 1702 y FC(;)g(r)1064 1672 y Fw(0)1088
1702 y FK(\).)33 b(F)-7 b(or)32 b(a)g(half-edge)f FC(h)g
FK(=)g(\()p FC(v)s(;)14 b(w)r FK(\))34 b(in)e(lab)n(yrin)n(th)g
FC(L)e FK(=)h(\()p FC(V)5 b(;)14 b(H)r(;)g(r)r FK(\),)34
b FC(C)6 b FK(\()p FC(h)p FK(\))523 1802 y(denotes)27
b(the)h(connected)g(comp)r(onen)n(t)f(of)h FC(L)18 b
FD(n)g FC(v)30 b FK(con)n(taining)d FC(w)r FK(.)523 2051
y Fl(3.5.1)99 b(Exploring)34 b(the)h(labyrinth)523 2200
y FK(An)25 b FG(explor)l(er)h FK(is)f(placed)g(in)g(the)g(lab)n(yrin)n
(th)f(on)h(an)g(arbitrary)d(half-edge,)j(equipp)r(ed)g(with)523
2300 y(a)37 b(\014nite)h(set)g(of)f(distinct)h(mark)n(ers.)e(A)n(t)h
(an)n(y)g(time,)h(mo)n(v)n(emen)n(t)f(is)g(p)r(ossible)g(in)h(three)523
2399 y(directions)523 2565 y FH(righ)m(t)k FK(Mo)n(v)n(e)26
b(from)h(a)g(half-edge)g FC(h)h FK(to)f FC(r)r FK(\()p
FC(h)p FK(\).)523 2731 y FH(left)41 b FK(Mo)n(v)n(e)27
b(from)g(a)g(half-edge)g FC(h)g FK(to)h FC(r)1757 2701
y Fw(\000)p FB(1)1847 2731 y FK(\()p FC(h)p FK(\).)523
2897 y FH(cross)41 b FK(Mo)n(v)n(e)27 b(from)g(a)g(half-edge)g(\()p
FC(v)s(;)14 b(w)r FK(\))29 b(to)f(its)f(coun)n(terpart)f(\()p
FC(w)r(;)14 b(v)s FK(\).)523 3064 y(Mark)n(er)k(manipulation)h(is)h(ac)
n(hiev)n(ed)f(through)g(the)h(primitiv)n(e)f(op)r(erations)g
FH(drop)h FK(mark)n(er)523 3163 y(\(on)29 b(curren)n(t)g(half-edge\),)g
FH(pic)m(k)h FK(mark)n(er)e(\(from)h(curren)n(t)g(or)f(adjacen)n(t)h
(half-edge\),)g(and)523 3263 y(the)23 b(tests)g FH(carrying)i
FK(mark)n(er,)c(and)i FH(here)f FK(mark)n(er)f(\(is)i(mark)n(er)e(on)h
(curren)n(t)g(half-edge?\).)523 3362 y(Note)h(that)g(mark)n(ers)e(are)h
(placed)h(at)g(half-edges,)f(not)g(v)n(ertices.)g(W)-7
b(e)24 b(further)f(allo)n(w)e(some)523 3462 y(more)31
b(in)n(v)n(olv)n(ed)e(op)r(erations)h(whose)h(implemen)n(tation)g(in)h
(terms)f(of)g(the)h(ab)r(o)n(v)n(e)e(primi-)523 3562
y(tiv)n(es)h(is)h(straigh)n(tforw)n(ard.)d(Of)i(these,)h(the)g
FH(near)g FK(test)g(c)n(hec)n(ks)f(whether)g(an)n(y)g(adjacen)n(t)523
3661 y(half-edge)40 b(con)n(tains)f(a)h(giv)n(en)g(mark)n(er,)f(the)i
FH(replace)f FK(op)r(eration)g(pic)n(ks)g(up)h(a)f(giv)n(en)523
3761 y(mark)n(er)29 b(at)i(an)g(adjacen)n(t)f(half-edge)h(to)g(drop)f
(another)g(mark)n(er)f(in)j(its)f(place,)g(and)f(the)523
3861 y FH(goto)e FK(op)r(eration)f(mo)n(v)n(es)g(to)h(an)g(adjacen)n(t)
f(half-edge)h(con)n(taining)f(a)h(giv)n(en)f(mark)n(er.)g(Fi-)523
3960 y(nally)-7 b(,)27 b(a)h FH(halt)f FK(op)r(eration)g(is)g(a)n(v)-5
b(ailable)27 b(to)g(signal)g(the)h(completion)f(of)h(a)f(task.)606
4060 y(The)21 b(explorer)f(is)i(a)e(\014nite)i(automaton)f(expressed)f
(as)h(a)g(pro)r(cedure)f(with)i(no)f(v)-5 b(ariables)523
4159 y(other)27 b(than)h(a)f(\014xed)h(set)f(of)h(mark)n(ers.)523
4408 y Fl(3.5.2)99 b(Finding)33 b(a)i(non-cutting)g(p)-5
b(oint)523 4558 y FK(The)34 b(problem)g(w)n(e're)f(in)n(terested)h(in)h
(is)f(that)g(of)g(\014nding)h(a)e(non-cutting)h(v)n(ertex.)g(The)523
4658 y(solution)23 b(is)g(an)g(explorer)e(whic)n(h,)i(started)g(on)g
(an)g(an)n(y)f(initial)h(half-edge)g(of)g(an)g(arbitrary)523
4757 y(but)38 b(\014nite)g(lab)n(yrin)n(th,)f(alw)n(a)n(ys)e(halts)i
(in)h(a)f(\014nite)h(n)n(um)n(b)r(er)f(of)g(steps)g(on)h(a)f(half-edge)
523 4857 y(adjacen)n(t)27 b(to)h(a)f(non-cutting)g(v)n(ertex.)g(Our)g
(prop)r(osed)f(initial)i(solution)f(is:)p eop end
%%Page: 33 43
TeXDict begin 33 42 bop 2197 299 a FF(3.5.)52 b(On)25
b(planar)h(em)n(b)r(eddings)154 b(33)p 606 427 1167 4
v 606 2071 4 1644 v 662 521 a FK(mark)n(ers)26 b FG(last)8
b FK(,)p FG(out)662 621 y FK(lo)r(op)768 721 y(if)28
b(carrying)e FG(last)36 b FK(then)844 820 y(drop-left)27
b FG(last)8 b FK(;)28 b(drop)f FG(out)768 920 y FK(elsif)h(near)f
FG(last)35 b FK(then)937 1020 y(if)28 b(at-left)g FG(out)35
b FK(then)1013 1119 y(mo)n(v)n(e-here)26 b FG(out)1013
1219 y FK(if)i(here)f FG(last)36 b FK(then)1089 1318
y(halt)1013 1418 y(endif)937 1518 y(else)27 b(left;)h(pic)n(k)g
FG(last)8 b FK(,)p FG(out)937 1617 y FK(endif)768 1717
y(else)27 b(left)768 1817 y(endif)768 1916 y(cross)662
2016 y(end)p 1769 2071 V 606 2074 1167 4 v 606 2180 a(The)c(follo)n
(wing)f(discussion)g(serv)n(es)f(b)r(oth)j(as)e(an)g(explanation)g(of)h
(the)h(w)n(orkings)d(of)h(the)523 2279 y(explorer,)k(and)i(as)e(a)i
(pro)r(of)f(of)g(its)h(correctness.)606 2379 y(What)20
b(the)f(explorer)f(do)r(es)h(is)g(to)g(p)r(erform)f(a)h(series)f(of)h
FG(c)l(onne)l(ctivity)k(tests)18 b FK(\(con-test)h(for)523
2479 y(short\))24 b FC(T)814 2491 y FB(0)851 2479 y FC(;)14
b(T)937 2491 y FB(1)974 2479 y FC(;)g(:)g(:)g(:)24 b
FK(at)h(v)n(ertices)e FC(v)1568 2491 y FB(0)1606 2479
y FC(;)14 b(v)1683 2491 y FB(1)1720 2479 y FC(;)g(:)g(:)g(:)p
FK(.)24 b(Connectivit)n(y)h(test)g FC(T)2598 2491 y FA(i)2649
2479 y FK(determines)g(whether)523 2578 y(v)n(ertex)30
b FC(v)818 2590 y FA(i)877 2578 y FK(is)h(a)f(cutting)h(p)r(oin)n(t)g
(or)f(not.)h(A)n(t)g(the)h(start)e(of)h FC(T)2443 2590
y FA(i)2470 2578 y FK(,)g(mark)n(er)e FG(last)39 b FK(is)31
b(dropp)r(ed)523 2678 y(on)36 b(a)g(half-edge)f(adjacen)n(t)h(to)g
FC(v)1577 2690 y FA(i)1605 2678 y FK(.)g(If)h(the)f(con-test)g(fails,)g
(i.e.)g FC(v)2617 2690 y FA(i)2681 2678 y FK(is)g(a)g(cutting)h(p)r
(oin)n(t,)523 2778 y(then)c FG(last)40 b FK(is)32 b(pic)n(k)n(ed)g(up)g
(and)g(dropp)r(ed)g(at)g(a)g(neigh)n(b)r(ouring)f(v)n(ertex)g
FC(v)2830 2790 y FA(i)p FB(+1)2942 2778 y FK(,)i(where)e(the)523
2877 y(pro)r(cess)21 b(is)i(rep)r(eated.)f(Let)h FC(h)1435
2889 y FB(0)1495 2877 y FK(b)r(e)f(the)h(half-edge)f(on)g(whic)n(h)h
(explorer)e(is)h(initially)h(placed)523 2977 y(and)j
FC(h)731 2989 y FA(i)p FB(+1)865 2977 y FK(=)d(\()p FC(v)1025
2989 y FA(i)p FB(+1)1137 2977 y FC(;)14 b(v)1214 2989
y FA(i)1242 2977 y FK(\))26 b(b)r(e)h(the)f(half-edge)f(on)h(whic)n(h)f
(the)i(explorer)d(arriv)n(es)g(at)h FC(v)3120 2989 y
FA(i)p FB(+1)3258 2977 y FK(for)523 3076 y(con-test)i
FC(T)886 3088 y FA(i)p FB(+1)997 3076 y FK(.)606 3176
y(W)-7 b(e)27 b(\014rst)f(consider)g(what)g(happ)r(ens)h(when)f
FG(last)35 b FK(is)26 b(dropp)r(ed)h(at)f(a)g(non-cutting)h(p)r(oin)n
(t)523 3276 y FC(v)k FK(=)d FC(v)727 3288 y FA(i)755
3276 y FK(.)j(This)f(case)g(is)h(illustrated)f(in)h(Figure)f(3.2\(a\).)
g(Since)h(remo)n(v)-5 b(al)29 b(of)h FC(v)k FK(lea)n(v)n(es)29
b(the)523 3375 y(lab)n(yrin)n(th)j(connected,)h(there)g(are)f(paths)h
(connecting)g(all)g(of)g(the)g(neigh)n(b)r(ours)f(of)h
FC(v)j FK(in)523 3475 y FC(L)24 b FD(n)g FC(v)s FK(,)36
b(indicated)h(b)n(y)f(the)h(dotted)g(lines.)f(These)g(paths)g(form,)g
(in)h(this)g(case,)e(3)h(faces)523 3575 y FC(F)576 3587
y FB(1)614 3575 y FC(;)14 b(F)704 3587 y FB(2)741 3575
y FC(;)g(F)831 3587 y FB(3)869 3575 y FK(,)31 b(one)f(b)r(et)n(w)n(een)
h(ev)n(ery)e(2)h(adjacen)n(t)h(half-edges)e(of)i FC(v)j
FK(\(in)d(case)f FC(v)k FK(lies)d(on)f(the)523 3674 y(b)r(oundary)24
b(of)i(the)f(lab)n(yrin)n(th,)f(then)i(one)f(of)g(these)g(faces)g(is)g
(the)g(in\014nite)h(external)f(face.\))523 3774 y(Supp)r(ose)34
b FC(h)903 3786 y FA(i)930 3774 y FK(,)h(the)f(half-edge)f(on)h(whic)n
(h)g(explorer)e(arriv)n(es)g(at)h FC(v)s FK(,)i(is)f(the)g(b)r(ottom)g
(one.)523 3874 y(Explorer)e(will)i(drop)f(mark)n(er)f
FG(last)42 b FK(at)34 b(the)g(next)g(left)h(half-edge,)e(and)g(mark)n
(er)g FG(out)41 b FK(at)523 3973 y FC(h)571 3985 y FB(0)608
3973 y FK(.)34 b(Then)g(it)g(will)g(cross,)f(i.e.)h(lea)n(v)n(e)e
FC(v)37 b FK(through)c(the)i FG(out)41 b FK(edge.)33
b(Since)h(it's)h(no)e(longer)523 4073 y(carrying)25 b
FG(last)8 b FK(,)28 b(it)f(will)g(just)h(alternate)e
FH(left)h FK(and)g FH(cross)g FK(un)n(til)h(it)f(gets)g(bac)n(k)f(to)h
FC(v)s FK(.)h(That)523 4172 y(is,)g(it)h(will)g(follo)n(w)f(a)g
FG(right-hand)j(walk)f FK(along)d(the)i(b)r(oundary)f(of)g(face)g
FC(F)2829 4184 y FB(1)2867 4172 y FK(,)g(returning)g(to)523
4272 y FC(v)33 b FK(on)c(the)h(half-edge)f(righ)n(t)g(of)g
FG(out)8 b FK(.)29 b(Next,)h(the)g(tests)g FH(near)g
FG(last)38 b FK(and)29 b FH(at-left)g FG(out)38 b FK(will)523
4372 y(succeed)33 b(\(the)i(at-left)e(is)h(of)f(course)g(from)g(the)h
(viewp)r(oin)n(t)g(of)f FC(v)s FK(\),)h(and)g FG(out)41
b FK(will)34 b(mo)n(v)n(e)523 4471 y(one)28 b(half-edge)f(to)h(the)h
(righ)n(t.)e(This)i(pro)r(cess)e(will)h(rep)r(eat)g(un)n(til)g
FG(out)36 b FK(mo)n(v)n(es)27 b(to)h FG(halt)9 b FK(,)28
b(at)523 4571 y(whic)n(h)23 b(p)r(oin)n(t)g(explorer)e(will)j(halt.)f
(This)g(sho)n(ws)e(that)j(non-cutting)e(p)r(oin)n(ts)h(are)f(correctly)
523 4671 y(iden)n(ti\014ed.)606 4770 y(Next)34 b(consider)f(the)i(case)
e(where)g FC(v)k FK(=)c FC(v)1938 4782 y FA(i)2000 4770
y FK(is)h(a)g(cutting)g(p)r(oin)n(t,)g(illustrated)g(in)g(Fig-)523
4870 y(ure)21 b(3.2\(b\).)h(So)f FC(L)6 b FD(n)g FC(v)24
b FK(is)e(not)f(connected)h(and)f(consists)g(of)g(m)n(ultiple)h(comp)r
(onen)n(ts.)f(Again)523 4969 y(supp)r(ose)29 b(that)h
FC(h)1067 4981 y FA(i)1124 4969 y FK(is)f(the)h(b)r(ottom)g(half-edge.)
e(The)i(dotted)g(lines)f(indicate)h(that)f FC(C)6 b FK(\()p
FC(h)3297 4981 y FA(i)3325 4969 y FK(\))523 5069 y(con)n(tains)29
b(t)n(w)n(o)f(more)h(neigh)n(b)r(ours)f(of)h FC(v)s FK(.)h(Eac)n(h)f
(other)f(comp)r(onen)n(t)i(of)f FC(L)19 b FD(n)g FC(v)33
b FK(is)c(in)n(ternal)523 5169 y(to)k(one)g(of)f(the)i(faces)e
(determined)h(b)n(y)g FC(C)6 b FK(\()p FC(h)1947 5181
y FA(i)1975 5169 y FK(\).)34 b(Our)e(explorer)45 b(starting)32
b(from)h FC(h)3145 5181 y FA(i)3172 5169 y FK(,)h(will)523
5268 y(tra)n(v)n(erse)e(faces)i(just)i(as)d(in)i(case)f(\(a\),)h(un)n
(til)g(a)f(non-empt)n(y)g(face)g(is)h(tra)n(v)n(ersed.)d(In)j(\(b\))523
5368 y(this)h(is)f(when)h(the)g(top)f(face)g(has)g(just)h(b)r(een)g
(tra)n(v)n(ersed)d(and)i(explorer)f(returns)h(to)g FC(v)s
FK(.)p eop end
%%Page: 34 44
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 @endspecial 1398 1428 a(FIGURE)25 b(3.2.)i(Connectivit)n(y)f(tests)523
1692 y FK(No)n(w)k(the)h(test)g FH(near)g FG(last)39
b FK(will)31 b(succeed)f(but)h FH(at-left)g FG(out)38
b FK(will)31 b(fail.)g(There)f(are)f(other)523 1791 y(half-edges)e(b)r
(et)n(w)n(een)i(the)g(one)f(where)g(explorer)e(returned)i(to)h
FC(v)i FK(and)e(the)f(one)g(where)g(it)523 1891 y(left)35
b FC(v)j FK(\()p FG(out)8 b FK(\),)35 b(constituting)g(one)f(or)g(more)
g(comp)r(onen)n(ts)g(di\013eren)n(t)g(from)h FC(C)6 b
FK(\()p FC(h)3097 1903 y FA(i)3125 1891 y FK(\).)35 b(The)523
1991 y(explorer)24 b(no)n(w)i(pic)n(ks)f(up)i FG(last)34
b FK(and)26 b(crosses)e(the)i(half-edge)f(on)h(its)g(left,)h(mo)n(ving)
e(to)h FC(v)3246 2003 y FA(i)p FB(+1)523 2090 y FK(to)i(start)f
(another)f(con-test.)523 2339 y Fl(3.5.3)99 b(T)-7 b(ermination)523
2489 y FK(F)g(or)28 b(con)n(v)n(enience,)f(let's)h(write)g
FC(C)1599 2501 y FA(i)1656 2489 y FK(for)f FC(C)6 b FK(\()p
FC(h)1928 2501 y FA(i)1956 2489 y FK(\).)29 b(The)f(crucial)g(observ)-5
b(ation)27 b(is)h(that)h FC(C)3246 2501 y FA(i)3302 2489
y FK(is)523 2588 y(a)g(prop)r(er)f(sublab)n(yrin)n(th)g(of)h
FC(C)1500 2600 y FA(i)p FB(+1)1612 2588 y FK(.)h(Since)f(the)g(lab)n
(yrin)n(th)g(is)g(\014nite,)g(so)g(is)g(the)g(sequence)523
2688 y(of)f(nested)f(comp)r(onen)n(ts:)1570 2788 y FC(C)1629
2800 y FB(1)1689 2788 y FD(\032)c FC(C)1836 2800 y FB(2)1897
2788 y FD(\032)f(\001)14 b(\001)g(\001)23 b(\032)g FC(C)2251
2800 y FA(r)2288 2788 y FC(;)523 2937 y FK(with)i FC(r)g
FD(\024)e FC(n)g FK(=)g FD(j)p FC(V)18 b FD(j)p FK(,)25
b(and)e FC(T)1386 2949 y FA(r)1447 2937 y FK(is)h(successful.)f(This)h
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3186 y Fl(3.5.4)99 b(Impr)-5 b(oving)33 b(Time)h(Complexity)523
3336 y FK(Next)26 b(let's)f(consider)g(the)h(time)g(complexit)n(y)-7
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FC(C)764 3447 y FA(i)819 3435 y FK(is)f(visited)h(in)h(addition)e(to)h
(v)n(ertex)f FC(v)1981 3447 y FA(i)2036 3435 y FK(itself.)i(F)-7
b(urthermore,)26 b(eac)n(h)g(half-edge)g(is)523 3535
y(visited)k(exactly)f(once)g(for)g(eac)n(h)f(adjacen)n(t)h(face)h(tra)n
(v)n(ersed)d(during)i FC(T)2766 3547 y FA(i)2793 3535
y FK(,)h(hence)g(at)f(most)523 3635 y(t)n(wice.)34 b(T)-7
b(ogether)34 b(with)h(the)f(fact)h(that)g(the)f(n)n(um)n(b)r(er)h(of)f
(edges)g(in)g(a)g(planar)g(graph)f(is)523 3734 y(linear)27
b(in)h(the)g(n)n(um)n(b)r(er)f(of)h(v)n(ertices,)e(this)i(giv)n(es)e
(an)i(upp)r(er)g(b)r(ound)g(of)f FC(O)r FK(\()p FC(n)2930
3704 y FB(2)2968 3734 y FK(\).)606 3834 y(But)h(w)n(e)f(can)g(do)h(b)r
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%%Page: 35 45
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FH(replace)f FK(op)r(eration)523 2489 y(includes)32 b(a)f(p)r(ossible)h
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(mark)n(ers)f(w)n(ere)523 2588 y(dropp)r(ed)25 b(earlier)f(in)i(order)e
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(During)g FC(T)3223 2600 y FA(i)p FB(+1)3335 2588 y FK(,)523
2688 y(the)f(only)e(w)n(a)n(y)g(for)h(our)f(initial)i(explorer)d(to)j
(tra)n(v)n(erse)c(comp)r(onen)n(t)j FC(C)2705 2700 y
FA(i)2757 2688 y FK(again)f(is)h(to)g(arriv)n(e)523 2787
y(at)i FC(v)k FK(on)c(the)h(half-edge)f(righ)n(t)g(of)g
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523 2887 y(follo)n(w)i(the)g(exact)g(same)g(path)g(that)h(also)e(led)i
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(immediately)h(to)523 3086 y FG(cut)7 b FK(.)36 b(This)g(only)f(sp)r
(eeds)h(up)g(the)g(con-test)g(and)f(do)r(esn't)h(in)n(v)-5
b(alidate)36 b(its)g(correctness.)523 3186 y(The)27 b(e\013ect)h(of)f
(this)h(shortcut)f(is)g(the)g(same)g(as)f(if)i(all)f(half-edges)f(from)
h FG(short)36 b FK(up)27 b(to)g FG(cut)523 3286 y FK(w)n(ere)d(remo)n
(v)n(ed,)g(as)g(indicated)h(b)n(y)g(the)g(dashed)g(lines.)g(With)h(the)
g(shortcut)e(mec)n(hanism,)523 3385 y FC(T)572 3397 y
FA(i)p FB(+1)713 3385 y FK(only)29 b(visits)h FC(C)1174
3397 y FA(i)p FB(+1)1306 3385 y FD(n)19 b FC(C)1426 3397
y FA(i)1454 3385 y FK(,)30 b(plus)g FC(v)1725 3397 y
FA(i)1753 3385 y FK(.)g(Hence,)g(successiv)n(e)e(con-tests)h(only)h(o)n
(v)n(erlap)d(in)523 3485 y(single)22 b(p)r(oin)n(ts,)g(and)h(using)f
(the)g(same)g(planarit)n(y)f(argumen)n(t,)h(w)n(e)g(obtain)g(an)g
FC(O)r FK(\()p FC(n)p FK(\))i(upp)r(er)523 3584 y(b)r(ound)k(on)f(the)h
(time)g(complexit)n(y)f(of)h(the)g(impro)n(v)n(ed)e(explorer.)523
3892 y FL(3.6)116 b(The)38 b(constan)m(t)f(space)i(FF)-13
b(A)523 4091 y FK(Ha)n(ving)23 b(solv)n(ed)g(the)i(subproblem)f(of)g
(\014nding)g(a)g(non-cutting)g(p)r(oin)n(t,)h(w)n(e)f(no)n(w)f(return)h
(to)523 4190 y(\015o)r(o)r(d)29 b(\014lling.)g(The)f(ab)r(o)n(v)n(e)g
(explorer)f(is)i(easily)f(translated)f(in)n(to)i(an)f(FF)-9
b(A.)30 b(The)f(curren)n(t)523 4290 y(p)r(osition)24
b(as)g(w)n(ell)g(as)f(eac)n(h)h(mark)n(er)e(is)i(represen)n(ted)f(b)n
(y)h(a)g(pair)g(\()p FC(p;)14 b(d)p FK(\))25 b(of)f(a)g(pixel)g
(address)523 4390 y(and)31 b(direction.)f(Carried)g(mark)n(ers)f(ha)n
(v)n(e)h(a)g(sp)r(ecial)h(v)-5 b(alue.)31 b(A)n(t)g(the)g(start)g(of)g
(the)g(lo)r(op,)523 4489 y(all)h(4)h(neigh)n(b)r(ouring)e(pixels)i(are)
e(tested)j(with)f Fk(get\(\))p FK(.)e(If)i(the)g(n)n(um)n(b)r(er)f(of)h
(neigh)n(b)r(ours)523 4589 y(b)r(elonging)27 b(to)h(the)h(region)e(is)h
(0,)f(then)i(the)f(FF)-9 b(A)29 b(is)f(\014nished)g(after)g(setting)g
(the)h(curren)n(t)523 4689 y(pixel.)23 b(If)h(the)f(n)n(um)n(b)r(er)g
(is)g(1)g(then)g(a)g(sligh)n(t)g(optimization)f(is)h(p)r(ossible;)g
(since)g(the)h(curren)n(t)523 4788 y(pixel)i(has)g(only)f(one)h(neigh)n
(b)r(our)f(in)h(the)g(region)f(it)i(is)e(a)h(non-cutting)g(p)r(oin)n(t)
g(and)g(can)f(b)r(e)523 4888 y(set,)j(without)g(ev)n(en)g(disturbing)f
(the)h(curren)n(t)f(con-test)h(\(but)h(pic)n(k)e(up)h(the)h
FG(last)36 b FK(mark)n(er)523 4988 y(if)30 b(it)h(happ)r(ens)e(to)h(b)r
(e)g(lo)r(cated)g(at)f(the)h(curren)n(t)f(pixel\).)h(Otherwise,)f(the)i
(FF)-9 b(A)30 b(b)r(eha)n(v)n(es)523 5087 y(similar)d(to)i(the)f(lab)n
(yrin)n(th)g(explorer.)e(It)j(uses)f(constan)n(t)f(space,)h(and,)g
(since)g(it)g(can)g(\014nd)523 5187 y(no)f(more)g(than)h
FC(n)f FK(non-cutting)h(p)r(oin)n(ts,)f(runs)h(in)f(time)h
FC(O)r FK(\()p FC(n)2455 5157 y FB(2)2493 5187 y FK(\).)p
eop end
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(the)h(w)n(orst-case)e(running)i(time)h(b)n(y)f(a)g(factor)f
FC(n)p FK(,)h(memory-use)523 905 y(can)k(b)r(e)h(reduced)f(to)h(a)f
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(whether)h(\002\()p FC(n)1588 974 y FB(2)1625 1005 y
FK(\))g(is)f(a)g(lo)n(w)n(er)f(b)r(ound)i(on)f(the)h(space-time)f
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(extremes.)p eop end
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3412 y(Common)f(to)h(all)f(ph)n(ysical)g(devices)g(is)g(the)h(switc)n
(hing)f(energy)g([14)o(])g(consumed)h(when)523 3512 y(a)d(wire)f(or)h
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3811 y(energy)c(consumption,)g(describ)r(ed)h(in)g(the)g(next)g
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(del)i(for)e(the)i(asymptotic)f(analysis)f(of)523 4010
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FG(AND)7 b FK(,)523 4209 y FG(Comp)l(ar)l(e)35 b FK(and)27
b FG(A)l(ddition)34 b FK(functions.)28 b(Lengauer)e(and)h(Mehlhorn)f
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4279 y FB(2)1801 4309 y FK(=)h FC(O)r FK(\()p FC(n)2050
4279 y FB(2)2088 4309 y FK(\))f(require)g(\012\()p FC(AT)12
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(Thompson)g(mo)r(del)g([17)o(].)g(Aggarw)n(al)e(et)i(al)523
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(Mehlhorn)f(to)h(obtain)f(an)h(\012\()p FC(n)3020 4478
y FB(2)3057 4508 y FK(\))g(energy)523 4608 y(b)r(ound)39
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(Leo)f([12)o(])h(sho)n(w)n(ed)f(that,)h(for)f(a)523 4707
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(requires)e(\012\()p FC(A)p FK(\))i(a)n(v)n(erage)c(switc)n(hing)523
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h(parit)n(y)e(circuit.)i(T)n(y)n(agi)d([18)o(])j(studied)f(the)h(a)n(v)
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(structures)g(suc)n(h)h(as)f(Programmable)e(Logic)523
5006 y(Arra)n(ys)f(\(PLA\).)606 5106 y(USM)34 b(measures)d(the)j
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(a)e(circuit.)h(Race)523 5205 y(conditions)25 b(\(ak)-5
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b(whic)n(h)f(are)f(de\014ned)h(and)f(discussed)g(in)h([7])g(and)f([6].)
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FC(W)39 b FK(the)27 b(set)523 4201 y(of)33 b(wires.)f(A)h
FG(le)l(gal)i(state)p FK(,)d(\(hereafter)g(also)g(called)g
FG(state)h FK(or)e FG(stable)k(state)p FK(\))e(of)g(a)f(circuit)523
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FC(C)6 b FK(.)35 b(That)g(is,)f(the)i(v)-5 b(alue)34
b(of)h(a)f(no)r(de)h(equals)f(that)h(of)g(all)523 4500
y(outgoing)g(wires,)g(and)g(the)i(v)-5 b(alue)35 b(of)h(a)g(non-input)g
(no)r(de)f(equals)g(the)i(corresp)r(onding)523 4599 y(function)j(of)g
(the)g(v)-5 b(alues)40 b(on)f(the)h(input)h(wires.)e(Num)n(b)r(ers)g
(are)g(equated)h(with)g(their)523 4699 y(binary)35 b(represen)n
(tations.)f(Ob)n(viously)-7 b(,)35 b(the)h(v)-5 b(alue)36
b(of)f(an)n(y)g(no)r(de)h(or)f(wire)h(in)g(a)f(stable)523
4799 y(state)27 b(is)f(completely)h(determined)f(b)n(y)h(the)g(input)g
(v)n(ector.)f(F)-7 b(or)26 b(a)g(no)r(de)h FC(v)j FK(\(wire)c
FC(w)r FK(\))i(and)523 4898 y(an)j(input)i(v)n(ector)d
FC(x)p FK(,)j(let)f FC(v)s FK(\()p FC(x)p FK(\))h(\()p
FC(w)r FK(\()p FC(x)p FK(\)\))h(denote)d(the)i(v)-5 b(alue)31
b(of)h(the)g(no)r(de)g(\(wire\))f(in)h(the)523 4998 y(stable)27
b(state)h(induced)g(b)n(y)f(input)h FC(x)p FK(.)606 5098
y(The)33 b(switc)n(hing)g(energy)f(of)i(a)f(circuit)g
FC(C)40 b FK(is)33 b(de\014ned)g(as)g(a)g(distance)g(metric)g(on)g(the)
523 5197 y(set)c(of)h(2)793 5167 y FA(n)867 5197 y FK(p)r(ossible)f
(inputs)h(v)n(ector.)e(W)-7 b(e)30 b(are)e(in)n(terested)h(in)h(what)f
(happ)r(ens)h(when)f(one)523 5297 y(input)g(v)n(ector)d
FC(x)i FK(to)g FC(C)34 b FK(is)27 b(replaced)g(b)n(y)h(another)e(input)
j(v)n(ector)d FC(y)s FK(.)i(A)g(wire)f FC(w)k FK(for)c(whic)n(h)p
eop end
%%Page: 40 50
TeXDict begin 40 49 bop 2572 299 a FF(4.1.)53 b(The)26
b(setting)154 b(40)523 506 y FC(w)r FK(\()p FC(x)p FK(\))34
b FD(6)p FK(=)e FC(w)r FK(\()p FC(y)s FK(\))i(is)f(said)g(to)g
FG(switch)p FK(.)h(Let)f FC(h)1883 518 y FA(w)r(;x;y)2083
506 y FK(b)r(e)g(the)h(Hamming)f(distance)g(b)r(et)n(w)n(een)523
606 y FC(w)r FK(\()p FC(x)p FK(\))d(and)e FC(w)r FK(\()p
FC(y)s FK(\).)h(A)g(wire)e FC(w)k FK(of)d(length)h FC(l)1844
618 y FA(w)1925 606 y FK(that)g(switc)n(hes,)e(in)n(v)n(olv)n(es)g(a)h
(switc)n(hing)f(area)523 706 y(of)j FC(h)668 718 y FA(w)r(;x;y)834
706 y FC(A)896 718 y FA(w)950 706 y FK(,)g(where)f FC(A)1307
718 y FA(w)1388 706 y FK(=)d FC(l)1504 718 y FA(w)1558
706 y FC(\025)k FK(is)g(the)g(area)e(of)i(one)g(of)f(the)i(constituen)n
(t)e(mono)n(wires)523 805 y(of)40 b FC(w)r FK(.)g(Letting)g
FC(p)f FK(b)r(e)h(the)g(area)e(accoun)n(ting)h(for)g(1)g(unit)h(of)g
FG(switching)h(ener)l(gy)p FK(,)f(this)523 905 y(accoun)n(ts)26
b(for)h FC(A=p)f FK(switc)n(hing)h(energy)-7 b(.)26 b(Similarly)-7
b(,)27 b(a)f(no)r(de)h FC(v)k FK(of)c(area)e FC(A)j FK(that)f(switc)n
(hes)523 1005 y(\()p FC(v)s FK(\()p FC(x)p FK(\))i FD(6)p
FK(=)f FC(v)s FK(\()p FC(y)s FK(\)\),)j(accoun)n(ts)e(for)h
FC(A=p)g FK(switc)n(hing)g(energy)f(\(for)h(con)n(v)n(enience)f(w)n(e)h
(assume)523 1104 y(that)e(all)f(of)h(the)g(area)e(of)h(the)h(no)r(de)g
(is)g(in)n(v)n(olv)n(ed)e(in)i(the)g(switc)n(hing\).)606
1204 y(The)g(total)g(switc)n(hing)g(energy)f(consumed)h(b)n(y)g
FC(C)35 b FK(when)28 b(switc)n(hing)g(from)g(input)h
FC(x)g FK(to)523 1303 y(input)f FC(y)j FK(is)1037 1482
y FC(E)5 b FK(\()p FC(C)q(;)14 b(x;)g(y)s FK(\))25 b(=)d(\()1559
1403 y Fn(X)1536 1581 y FA(w)r Fw(2)p FA(W)1716 1482
y FC(h)1764 1494 y FA(w)r(;x;y)1930 1482 y FC(A)1992
1494 y FA(w)2065 1482 y FK(+)2309 1403 y Fn(X)2148 1585
y FA(v)r Fw(2)p FA(V)t(;v)r FB(\()p FA(x)p FB(\))p Fw(6)p
FB(=)p FA(v)r FB(\()p FA(y)r FB(\))2603 1482 y FC(A)2665
1494 y FA(v)2705 1482 y FK(\))p FC(=p:)523 1745 y FK(Ob)n(viously)-7
b(,)17 b FC(E)5 b FK(\()p FC(C)q(;)14 b(x;)g(x)p FK(\))25
b(=)e(0,)18 b FC(E)5 b FK(\()p FC(C)q(;)14 b(x;)g(y)s
FK(\))24 b(=)f FC(E)5 b FK(\()p FC(C)q(;)14 b(y)s(;)g(x)p
FK(\),)20 b(and)e FC(E)5 b FK(\()p FC(C)q(;)14 b(x;)g(z)t
FK(\))24 b FD(\024)e FC(E)5 b FK(\()p FC(C)q(;)14 b(x;)g(y)s
FK(\)+)523 1844 y FC(E)5 b FK(\()p FC(C)q(;)14 b(y)s(;)g(z)t
FK(\),)25 b(so)f(that)h FC(E)5 b FK(\()p FC(C)h FK(\))26
b(is)e(indeed)i(a)e(distance)g(metric.)h(Also,)g FC(E)5
b FK(\()p FC(C)h FK(\))25 b(is)g(b)r(ounded)g(b)n(y)523
1944 y FC(A)585 1956 y FA(C)641 1944 y FC(=p)p FK(,)i(with)h
FC(A)1026 1956 y FA(C)1110 1944 y FK(the)g(total)f(area)f(of)i
FC(C)6 b FK(.)606 2044 y(The)28 b FG(worst)i(c)l(ase)g(uniswitch)g
(ener)l(gy)e FK(of)f(a)h(circuit)f FC(C)34 b FK(is)28
b(de\014ned)g(as)1324 2210 y FC(E)1385 2232 y FG(worst)1591
2210 y FK(\()p FC(C)6 b FK(\))24 b(=)112 b(max)1832 2267
y FA(x;y)r Fw(2f)p FB(0)p FA(;)p FB(1)p Fw(g)2125 2251
y Fx(n)2178 2210 y FC(E)5 b FK(\()p FC(C)q(;)14 b(x;)g(y)s
FK(\))p FC(;)523 2423 y FK(while)28 b(the)g FG(aver)l(age)j(c)l(ase)f
(uniswitch)g(ener)l(gy)e FK(is)g(de\014ned)g(as)1281
2601 y FC(E)1342 2613 y FG(avg)1472 2601 y FK(\()p FC(C)6
b FK(\))24 b(=)1819 2522 y Fn(X)1713 2704 y FA(x;y)r
Fw(2f)p FB(0)p FA(;)p FB(1)p Fw(g)2006 2687 y Fx(n)2059
2601 y FC(E)5 b FK(\()p FC(C)q(;)14 b(x;)g(y)s FK(\))p
FC(=)p FK(2)2498 2567 y FB(2)p FA(n)2577 2601 y FC(;)523
2869 y FK(where)k(2)796 2839 y FB(2)p FA(n)892 2869 y
FK(is)g(the)h(n)n(um)n(b)r(er)f(of)h(p)r(ossible)f(input)h(pairs)e(\()p
FC(x;)d(y)s FK(\).)19 b(This)g(de\014nition)f(of)h FC(E)3109
2881 y FG(avg)3239 2869 y FK(\()p FC(C)6 b FK(\))523
2969 y(assumes)27 b(that)h(the)g(input)g(v)n(ector)e(is)i(distributed)g
(uniformly)f(o)n(v)n(er)f FD(f)p FK(0)p FC(;)14 b FK(1)p
FD(g)2906 2939 y FA(n)2949 2969 y FK(.)606 3069 y(A)33
b(function)g FC(f)40 b FK(:)31 b FD(f)p FK(0)p FC(;)14
b FK(1)p FD(g)1371 3038 y Fw(\003)1439 3069 y FD(!)31
b(f)p FK(0)p FC(;)14 b FK(1)p FD(g)1758 3038 y Fw(\003)1827
3069 y FK(is)32 b FG(ener)l(gy)j(e\016cient)e FK(i\013)g(there)f
(exists)h(a)f(family)523 3168 y FC(C)h FK(=)27 b(\()p
FC(C)798 3180 y FA(n)843 3168 y FK(\))875 3183 y FB(\()p
FA(n)p Fw(2)p Fi(N)p FB(\))1106 3168 y FK(of)j(circuits)f(with)h
FC(C)1746 3180 y FA(n)1822 3168 y FK(realizing)f FC(f)9
b FD(jf)p FK(0)p FC(;)14 b FK(1)p FD(g)2435 3138 y FA(n)2507
3168 y FK(in)30 b FC(O)r FK(\(log)15 b FC(n)p FK(\))31
b(depth,)f(suc)n(h)523 3268 y(that)e FC(E)764 3290 y
FG(worst)970 3268 y FK(\()p FC(C)1061 3280 y FA(n)1107
3268 y FK(\))c(=)e FC(O)r FK(\()p FC(n)p FK(\).)606 3367
y(Throughout)27 b(this)g(c)n(hapter,)g(log)14 b FC(n)28
b FK(means)f(log)2110 3388 y FB(2)2161 3367 y FC(n)p
FK(.)523 3620 y Fl(4.1.1)99 b(Mo)-5 b(del)34 b(Motivation)523
3769 y FK(The)22 b(in)n(ten)n(t)g(of)f(this)h(section)f(is)h(to)f
(motiv)-5 b(ate)22 b(the)g FG(Uniswitch)j(Mo)l(del)e
FK(in)f(ligh)n(t)f(of)h(ph)n(ysical)523 3869 y(considerations.)34
b(USM)j(is)f(a)f(go)r(o)r(d)g(mo)r(del)i(for)e(obtaining)g(lo)n(w)n(er)
g(b)r(ounds)h(b)r(ecause)f(it)523 3969 y(conserv)-5 b(ativ)n(ely)29
b(estimates)h(a)g(circuit's)g(switc)n(hing)h(b)r(eha)n(viour.)e(Th)n
(us,)h(a)g(lo)n(w)n(er)f(b)r(ound)523 4068 y(in)f(USM)g(is)g(an)f
(equally)g(v)-5 b(alid)27 b(lo)n(w)n(er)f(b)r(ound)i(on)g
FG(multiswitch)g FK(energy)-7 b(.)606 4168 y(USM)24 b(tak)n(es)e(no)h
(notice)g(of)g(ho)n(w)g(a)g(circuit)g(arriv)n(es)e(at)i(a)g(particular)
f(state.)h(This)g(is)g(the)523 4267 y(domain)i(of)h(the)g
FG(Multiswitch)j(Mo)l(dels)p FK(,)e(whic)n(h)f(are)f(discussed)g(in)h
([7])g(and)f([6].)h(Ho)n(w)n(ev)n(er,)523 4367 y(in)35
b(order)e(to)i(discuss)f(the)h(relev)-5 b(ance)34 b(of)g(using)h(USM)g
(to)f(obtain)h(upp)r(er)g(b)r(ounds,)f(the)523 4467 y(follo)n(wing)27
b(m)n(ultiswitc)n(h)g(notions)g(are)g(in)n(tro)r(duced.)606
4566 y(The)34 b(switc)n(hing)f(b)r(eha)n(viour)f(of)i(ph)n(ysical)e
(circuits)i(is)f(in\015uenced)h(b)n(y)f(v)-5 b(arious)33
b(dela)n(y)523 4666 y(functions,)d(suc)n(h)f(as)h(gate)f(dela)n(y)f
FC(\016)s FK(,)i(wire)g(dela)n(y)e(\001)i(and)g(input)g(dela)n(y)f
FC(I)7 b FK(.)30 b FC(\016)j FK(determines)523 4766 y(the)c(switc)n
(hing)f(sp)r(eed)h(of)g(a)f(gate.)g(\001)h(determines)g(the)g(time)g
(to)g(transmit)g(bits)g(along)e(a)523 4865 y(wire.)g
FC(I)35 b FK(determines)27 b(when)h(an)f(input)i(v)-5
b(alue)27 b(arriv)n(es)f(at)h(an)g(input)i(p)r(ort.)606
4965 y(Let)34 b(\()p FC(C)852 4977 y FA(n)898 4965 y
FC(;)14 b(\016)o(;)g FK(\001)p FC(;)g(I)7 b FK(\))34
b(denote)g(a)g FG(cir)l(cuit)i(scheme)p FK(,)f(where)f
FC(C)2461 4977 y FA(n)2541 4965 y FK(is)g(a)g(VLSI)g(circuit)g(with)523
5064 y(gate)f(dela)n(y)g FC(\016)s FK(,)g(wire)g(dela)n(y)g(\001,)h
(and)f(input)i(dela)n(y)d FC(I)7 b FK(.)34 b(A)g(circuit)f(sc)n(heme)g
(\()p FC(C)3021 5076 y FA(n)3067 5064 y FC(;)14 b(\016)o(;)g
FK(\001)p FC(;)g(I)7 b FK(\))523 5164 y(exhibits)22 b(the)h
FG(uniswitch)i(pr)l(op)l(erty)e FK(if)g(eac)n(h)e(no)r(de)h(or)f(wire)h
(of)g FC(C)2514 5176 y FA(n)2582 5164 y FK(switc)n(hes)f(at)h(most)g
(once)523 5264 y(when)29 b FC(C)800 5276 y FA(n)874 5264
y FK(c)n(hanges)f(from)g(one)h(input)g(setting)g(to)f(another,)g
(according)f(to)i FC(\016)s FK(,)g(\001)g(and)g FC(I)7
b FK(.)523 5363 y(Otherwise,)27 b(\()p FC(C)1030 5375
y FA(n)1076 5363 y FC(;)14 b(\016)o(;)g FK(\001)p FC(;)g(I)7
b FK(\))27 b(exhibits)h(the)g(m)n(ultiswitc)n(h)g(prop)r(ert)n(y)-7
b(.)p eop end
%%Page: 41 51
TeXDict begin 41 50 bop 2572 299 a FF(4.1.)53 b(The)26
b(setting)154 b(41)523 1323 y @beginspecial 0 @llx 0
@lly 468 @urx 148.500000 @ury 3401 @rwi @setspecial
%%BeginDocument: combcirc.ps
%!
%%Creator: tromp@wolf.cwi.nl (John Tromp,M231,4078,02207-16322)
%%Title: combcirc (xpic)
%%CreationDate: Sat Sep 18 12:44:12 1993
%%Pages: 1
%%BoundingBox: 0 0 468 148.5
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%%EndComments
% Prolog for xpic to PostScript converter
% Author: Mark Moraes
% $Header: x2ps.pro,v 1.2 88/03/19 16:50:09 moraes Exp 
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%	%d S - change size (font size in points)
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%	%d %d m - moveto
%	%d %d s - spline segment
%	x - flush line, spline
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%	<wid> <ht> <x> <y> e - ellipse
%	%d ss - setscale
%	%d W - change linewidth
%	getpagesize - gets the values of PAGEHEIGHT and PAGEWIDTH
%	%d %d flip - translate by %d, PAGEHEIGHT - %d (this 
%		transforms to X windows coordinates)
save 50 dict begin /xpic exch def
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% Set defaults
/fontname /Times-Roman def
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% halign has the values for MIDLINE, TOPLINE, BOTLINE
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/s {rcurveto} def
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/b {
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} def
/mtrx matrix def
/e {
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% The next three take the text string, and moveto the right horiz. position
% leaving the string on the stack.
/lj {} def
/rj {dup stringwidth pop neg 0 rmoveto} def
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% And this is invoked after one of the three above, and 
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% in curscale for font scaling
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StartXpic
%%EndProlog
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 @endspecial 822 1497 a(FIGURE)25 b(4.1.)j(Preferred)e(Sequen)n(tial)f
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y FK(Using)k(USM)g(to)g(obtain)g(upp)r(er)g(b)r(ounds)g(is)g
(justi\014ed)h(for)f(circuit)g(sc)n(hemes)f(that)h(ex-)523
1860 y(hibit)e(the)f(uniswitc)n(h)g(prop)r(ert)n(y)-7
b(.)26 b(F)-7 b(or)26 b(example,)g(if)i(eac)n(h)e(no)r(de)g(of)h(a)g
(circuit)f(receiv)n(es)g(its)523 1960 y(inputs)i(at)g(the)f(same)g
(time,)i(race)d(conditions)h(cannot)g(arise.)g(The)g(uniswitc)n(h)h
(prop)r(ert)n(y)523 2060 y(is)g(th)n(us)f(ensured.)h(Some)f(real)g
(circuits)g(ha)n(v)n(e)g(this)h(timing)g(prop)r(ert)n(y)-7
b(.)26 b(Where)i(race)e(con-)523 2159 y(ditions)h(deriv)n(e)e(solely)h
(from)h(a)f(circuit's)g(async)n(hron)n(y)e(\(i.e..)k(the)f(paths)f(to)h
(a)f(no)r(de)h(v)-5 b(ary)523 2259 y(in)34 b(length\),)g(a)f(circuit)h
(sc)n(heme)f(can)g(acquire)g(the)g(uniswitc)n(h)h(prop)r(ert)n(y)f(if)h
(the)g(circuit)523 2359 y(can)i(b)r(e)g(made)g(sync)n(hronous.)d(A)k
("bad")d(input)j(sc)n(hedule)f(can)f(b)r(e)h(o\013set)g(b)n(y)g(v)-5
b(arying)523 2458 y(gate)36 b(dela)n(ys.)f(These)i(approac)n(hes)d(to)i
(designing)g(circuit)h(sc)n(hemes)e(that)i(ac)n(hiev)n(e)e(the)523
2558 y(uniswitc)n(h)28 b(prop)r(ert)n(y)e(are)h(discussed)g(in)h([7])f
(and)h([6].)f(F)-7 b(urther,)28 b(according)e(to)h(C.)h(Mead)523
2657 y([15)o(],)k(man)n(y)g(CMOS)g(designs)f(are)h(sync)n(hronized)e
(to)i(ensure)g(that)g(the)h(corresp)r(onding)523 2757
y(circuit)28 b(sc)n(hemes)e(ha)n(v)n(e)h(the)h(uniswitc)n(h)g(prop)r
(ert)n(y)-7 b(.)606 2857 y(USM)41 b(is)g(the)g(\014rst)f(step)h(in)g
(the)g(systematic)f(asymptotic)g(analysis)g(of)g(switc)n(hing)523
2956 y(energy)33 b(consumption)h(in)g(VLSI)h(circuits.)f(As)g(suc)n(h,)
g(USM)g(is)g(justi\014ed)h(as)f(an)g(upp)r(er)523 3056
y(b)r(ound)29 b(mo)r(del.)f(In)h(addition,)f(USM)h(is)f(motiv)-5
b(ated)28 b(b)n(y)g(designers')f(practical)g(e\013orts)h(to)523
3156 y(prev)n(en)n(t)21 b(hazards)g(and)h(th)n(us)h(ensure)e(the)i
(uniswitc)n(h)f(prop)r(ert)n(y)-7 b(.)22 b(USM)g(is)h(used)f(for)g(upp)
r(er)523 3255 y(b)r(ound)28 b(analysis)e(in)i([18)o(])g(and)f([1].)606
3355 y(USM)20 b(is)g(de\014ned)g(for)g(acyclic)f(circuits.)g(The)h
(study)g(of)g(com)n(binational)f(circuits)g(\(with-)523
3454 y(out)34 b(lo)r(ops\))g(has)g(a)g(long)g(and)g(distinguished)h
(history)-7 b(.)33 b(Krohn)g(and)i(Rho)r(des')f([10)o(,)h(11)o(])523
3554 y(seminal)g(w)n(ork)f(in)i(this)f(area)f(sho)n(w)n(ed)g(that)i
(eac)n(h)e(sequen)n(tial)h(mac)n(hine)g(\(with)h(lo)r(ops\))523
3654 y(can)h(b)r(e)h(decomp)r(osed)e(in)n(to)h(structures)g(consisting)
f(of)h(only)g(com)n(binational)f(circuits)523 3753 y(and)27
b(\015ip-\015ops.)606 3853 y(A)j(recommended)g(arc)n(hitecture)e(for)i
(sequen)n(tial)f(mac)n(hines)g(is)h(the)g(\014nite)h(state)e(ma-)523
3953 y(c)n(hine)k(in)h(whic)n(h)f(the)h(com)n(binational)e(logic)g(is)h
(isolated)g(from)g(the)h(lo)r(oping)e(structure)523 4052
y([14)o(].)g(See)f(Figure)g(4.1.)f(This)h(arc)n(hitecture)g(lends)g
(itself)h(to)f(analysis)f(of)h(the)h(com)n(bina-)523
4152 y(tional)27 b(logic)g(distinct)h(from)g(the)f(lo)r(oping)g
(bu\013ers.)523 4459 y FL(4.2)116 b(W)-10 b(orst)38 b(case)h(upp)s(er)g
(b)s(ounds)523 4675 y Fl(4.2.1)99 b(Ener)-5 b(gy-E\016cient)34
b Fh(k)s Fl(-Thr)-5 b(eshold)34 b(Cir)-5 b(cuit)523 4824
y FK(In)21 b(this)g(section,)g(a)f(no)n(v)n(el)g(energy-e\016cien)n(t)f
(threshold)i(circuit)f(is)h(describ)r(ed,)g(whic)n(h)f(gen-)523
4924 y(eralizes)28 b(the)h(tec)n(hniques)g(of)g(the)h(energy-e\016cien)
n(t)d FG(OR)32 b FK(and)d FG(AND)36 b FK(circuits)29
b(describ)r(ed)523 5024 y(in)g([8)o(])g(,[5)o(].)g(The)f
(energy-e\016cien)n(t)f FG(OR)k FK(circuit)d(used)g(the)h(observ)-5
b(ation)27 b(that)h(it)h(is)f(su\016-)523 5123 y(cien)n(t)d(to)g(turn)g
(on)g(one)g FG(OR)i FK(input)f(to)f(turn)g(on)g(the)g(output.)h(Th)n
(us,)f(when)g(man)n(y)f(inputs)523 5223 y(to)f FG(SOR)j
FK(\(Smart)e FG(OR)i FK(circuit\))d(are)g("1",)f(all)h(but)h(one)f(of)h
(these)f("1")g(signals)f(are)g(`killed'.)523 5322 y(In)32
b(a)f(completely)g(analogous)f(manner,)h(it)h(is)f(su\016cien)n(t)h(to)
f(turn)h(o\013)f(one)g FG(AND)39 b FK(input)p eop end
%%Page: 42 52
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b(orst)25 b(case)i(upp)r(er)e(b)r(ounds)153 b(42)523
506 y FK(in)31 b(order)f(to)h(turn)g(o\013)g(the)g(output.)g(Th)n(us,)g
(when)g(man)n(y)g(inputs)g(are)f(turned)h(o\013,)g(only)523
606 y(one)i(0)f(signal)g(m)n(ust)h(propagate)e(all)i(the)g(w)n(a)n(y)f
(to)h(the)g(output.)g(The)g(principle)g(idea)g(is)523
706 y(that)d(the)h(input)g(ma)n(y)e(pro)n(vide)g(more)g(information)g
(than)h(the)h(function)f(requires,)f(and)523 805 y(suppressing)h
(unneeded)h(input)h(bits)g(results)e(in)i(energy)e(sa)n(vings.)f(This)i
(idea)g(w)n(as)f(also)523 905 y(used)e(in)f([8])h(to)f(design)g
(energy-e\016cien)n(t)g(comparator)e(circuits.)606 1005
y(A)31 b(threshold)g(function,)h FC(T)1469 1017 y FA(k)1509
1005 y FK(,)f(is)g(de\014ned)h(on)e(a)h(b)r(o)r(olean)f(v)n(ector)g
FC(x)f FK(=)g FC(x)2913 1017 y FB(1)2950 1005 y FC(x)2997
1017 y FB(2)3049 1005 y FC(:)14 b(:)g(:)g(x)3207 1017
y FA(n)3283 1005 y FK(as)523 1104 y(follo)n(ws:)1440
1233 y FC(T)1489 1245 y FA(k)1529 1233 y FK(\()p FC(x)p
FK(\))24 b(=)1752 1116 y Fn(\032)1856 1182 y FK(1)82
b(if)2056 1120 y Fn(P)2158 1182 y FC(x)2205 1194 y FA(i)2256
1182 y FD(\025)23 b FC(k)1856 1282 y FK(0)82 b(otherwise)523
1419 y(Hence,)42 b(at)f(most)g FC(k)j FK(input)e(bits)g(that)f(are)g
("1")f(m)n(ust)h(reac)n(h)f(the)i(output;)g(the)f(rest)523
1519 y(can)29 b(b)r(e)h("killed".)e(The)h(energy-e\016cien)n(t)f
(threshold)h(circuit,)g FG(Thr)2665 1531 y FA(k)2706
1519 y FK(,)g(describ)r(ed)g(b)r(elo)n(w,)523 1618 y(e\013ectiv)n(ely)
24 b(uses)f(only)g(necessary)f(information,)h(killing)h(o\013)g(the)g
(rest,)f(resulting)g(in)h(only)523 1718 y(linear)j(w)n(orst)f(case)h
(switc)n(hing)g(energy)g(consumption.)606 1817 y(Let's)h(de\014ne)f(a)h
(sum-to-)p FC(k)h FK(function)g(on)e FC(x)c FK(=)g FC(x)2116
1829 y FB(1)2154 1817 y FC(x)2201 1829 y FB(2)2252 1817
y FC(:)14 b(:)g(:)g(x)2410 1829 y FA(n)2483 1817 y FK(as:)1524
1994 y FC(S)1575 2006 y FA(k)1616 1994 y FK(\()p FC(x)p
FK(\))24 b(=)f(min\()p FC(k)s(;)2092 1915 y Fn(X)2226
1994 y FC(x)2273 2006 y FA(i)2301 1994 y FK(\))p FC(:)523
2174 y FK(Circuit)33 b FG(Thr)956 2186 y FA(k)1030 2174
y FK(consist)g(of)h(a)e(comparator)f(b)r(olted)j(on)n(to)f(a)g(circuit)
g FG(Cnt)2869 2186 y FA(k)2909 2174 y FK(,)h(whic)n(h)f(uses)523
2274 y(\()p FC(k)17 b FK(+)c(1\)-ary)23 b(logic)h(to)h(compute)h
FC(S)1613 2286 y FA(k)1653 2274 y FK(.)g(The)f(comparator)e(compares)g
(the)j(output)f(of)g FG(Cnt)3317 2286 y FA(k)523 2374
y FK(with)g(the)g(constan)n(t)f FC(k)s FK(.)g(Circuit)h
FG(Cnt)1695 2386 y FA(k)1760 2374 y FK(con)n(tains)f(2)g(t)n(yp)r(es)g
(of)h(no)r(des:)f(+-no)r(des)g(that)h(sum)523 2473 y(the)j(inputs,)g
(and)g(min-no)r(des)f(that)h(`kill')g(inputs)g(exceeding)f
FC(k)s FK(.)606 2573 y(Lik)n(e)g(the)g FG(SOR)i FK(circuit,)e(the)h
FG(Cnt)1695 2585 y FA(k)1763 2573 y FK(circuit)f(is)g(laid)g(out)g(so)f
(that,)i(for)e(an)n(y)g(input,)i(the)523 2673 y(area)e(of)i(non-zero)e
(wires)g(is)i(at)f(most)h(linear)f(in)h(the)g(input)g(size.)606
2772 y(The)22 b(follo)n(wing)e(recurrence)g(describ)r(es)g(the)i(b)r(o)
r(olean)f(sum-to-)p FC(k)j FK(function)e FC(S)3013 2784
y FA(k)3053 2772 y FD(jf)p FK(0)p FC(;)14 b FK(1)p FD(g)3281
2742 y FB(2)p FA(n)523 2872 y FK(in)28 b(terms)f(of)h
FC(S)997 2884 y FA(k)1038 2872 y FD(jf)p FK(0)p FC(;)14
b FK(1)p FD(g)1266 2842 y FA(n)1309 2872 y FK(,)27 b(addition,)h(and)f
(minim)n(um.)639 3140 y FC(S)690 3152 y FA(k)731 3140
y FK(\()p FC(x)810 3152 y FB(1)848 3140 y FC(;)14 b(:)g(:)g(:)f(;)h(x)
1079 3152 y FB(2)p FA(n)1158 3140 y FK(\))801 3264 y(=)82
b(min)q(\()p FC(k)s(;)14 b(S)1253 3276 y FA(k)1294 3264
y FK(\()p FC(x)1373 3276 y FB(1)1411 3264 y FC(;)g(:)g(:)g(:)f(;)h(x)
1642 3276 y FA(n)1688 3264 y FK(\))k(+)g FC(S)1872 3276
y FA(k)1913 3264 y FK(\()p FC(x)1992 3276 y FA(n)p FB(+1)2122
3264 y FC(;)c(:)g(:)g(:)g(;)g(x)2354 3276 y FB(2)p FA(n)2432
3264 y FK(\)\))801 3389 y(=)82 b FC(S)999 3401 y FA(k)1040
3389 y FK(\()p FC(x)1119 3401 y FB(1)1157 3389 y FC(;)14
b(:)g(:)g(:)g(;)g(x)1389 3401 y FA(n)1434 3389 y FK(\))19
b(+)f(min\()p FC(k)k FD(\000)c FC(S)1937 3401 y FA(k)1978
3389 y FK(\()p FC(x)2057 3401 y FB(1)2095 3389 y FC(;)c(:)g(:)g(:)f(;)h
(x)2326 3401 y FA(n)2372 3389 y FK(\))p FC(;)g(S)2492
3401 y FA(k)2533 3389 y FK(\()p FC(x)2612 3401 y FA(n)p
FB(+1)2742 3389 y FC(;)g(:)g(:)g(:)f(;)h(x)2973 3401
y FB(2)p FA(n)3052 3389 y FK(\)\))801 3513 y(=)82 b(min)q(\()p
FC(S)1170 3525 y FA(k)1211 3513 y FK(\()p FC(x)1290 3525
y FB(1)1328 3513 y FC(;)14 b(:)g(:)g(:)f(;)h(x)1559 3525
y FA(n)1605 3513 y FK(\))p FC(;)g(k)21 b FD(\000)d FC(S)1872
3525 y FA(k)1913 3513 y FK(\()p FC(x)1992 3525 y FA(n)p
FB(+1)2122 3513 y FC(;)c(:)g(:)g(:)g(;)g(x)2354 3525
y FB(2)p FA(n)2432 3513 y FK(\)\))19 b(+)f FC(S)2649
3525 y FA(k)2690 3513 y FK(\()p FC(x)2769 3525 y FA(n)p
FB(+1)2899 3513 y FC(;)c(:)g(:)g(:)g(;)g(x)3131 3525
y FB(2)p FA(n)3209 3513 y FK(\))606 3682 y(The)23 b(structure)f(of)h
(the)g FG(Cnt)1492 3694 y FA(k)1556 3682 y FK(circuit)g(directly)f
(re\015ects)g(the)i(ab)r(o)n(v)n(e)d(recurrence.)g(W)-7
b(e'll)523 3781 y(next)26 b(describ)r(e)g(it's)g(la)n(y)n(out)f(for)h
(the)g(case)g(that)g FC(n)g FK(is)g(a)g(p)r(o)n(w)n(er)f(of)h(2.)g(The)
g(generalization)523 3881 y(to)e(other)g(v)-5 b(alues)24
b(of)g FC(n)g FK(can)g(b)r(e)h(obtained)f(in)g(a)g(straigh)n(tforw)n
(ard)d(manner)j(b)n(y)g(taking)f(the)523 3981 y(circuit)18
b(for)g(the)h(next)g(p)r(o)n(w)n(er)e(of)i(2,)f(hardwiring)f(the)i
(excess)f(inputs)h(to)f(0,)g(and)h(simplifying)523 4080
y(the)25 b(result.)g(F)-7 b(or)24 b(notational)f(con)n(v)n(enience,)h
(n)n(um)n(b)r(er)g(the)h FC(n)e FK(=)g(2)2571 4050 y
FA(l)2621 4080 y FK(inputs)i(0)p FC(;)14 b FK(1)p FC(;)g(:)g(:)g(:)f(;)
h(n)f FD(\000)g FK(1)523 4180 y(and)33 b(iden)n(tify)h(them)g(with)g
(their)f(length)g FC(l)i FK(binary)d(expansion.)h(A)g(binary)g(string)g
FC(s)g FK(of)523 4279 y(length)d(less)f(than)h FC(l)h
FK(then)f(naturally)e(corresp)r(ond)g(to)i(the)g(blo)r(c)n(k)f(of)g(2)
2769 4249 y FA(l)p Fw(\000j)p FA(s)p Fw(j)2947 4279 y
FK(consecutiv)n(e)523 4379 y(inputs)40 b(ha)n(ving)e
FC(s)h FK(as)g(a)f(pre\014x.)h(In)h(particular,)e(the)h(empt)n(y)h
(string)e FC(\017)h FK(iden)n(ti\014es)g(the)523 4479
y(sequence)27 b(of)h(all)f(inputs.)h(The)g(ab)r(o)n(v)n(e)e(recurrence)
g(can)h(no)n(w)g(b)r(e)h(rewritten)f(as)1128 4647 y FC(S)1179
4659 y FA(k)1220 4647 y FK(\()p FC(s)p FK(\))83 b(=)g(min\()p
FC(k)s(;)14 b(S)1858 4659 y FA(k)1899 4647 y FK(\()p
FC(s)p FK(0\))k(+)g FC(S)2196 4659 y FA(k)2237 4647 y
FK(\()p FC(s)p FK(1\)\))1406 4772 y(=)83 b FC(S)1605
4784 y FA(k)1646 4772 y FK(\()p FC(s)p FK(0\))18 b(+)g(min\()p
FC(k)k FD(\000)c FC(S)2261 4784 y FA(k)2302 4772 y FK(\()p
FC(s)p FK(0\))p FC(;)c(S)2535 4784 y FA(k)2575 4772 y
FK(\()p FC(s)p FK(1\)\))1406 4896 y(=)83 b(min\()p FC(S)1775
4908 y FA(k)1816 4896 y FK(\()p FC(s)p FK(0\))p FC(;)14
b(k)21 b FD(\000)d FC(S)2196 4908 y FA(k)2237 4896 y
FK(\()p FC(s)p FK(1\)\))h(+)f FC(S)2567 4908 y FA(k)2608
4896 y FK(\()p FC(s)p FK(1\))606 5064 y(Figures)i(4.2)h(and)g(4.3)f
(sho)n(w)h(the)g(la)n(y)n(out)f(of)i(circuit)f FG(Cnt)2367
5076 y FA(k)2430 5064 y FK(on)g(input)h(blo)r(c)n(k)e
FC(s)p FK(.)i(The)f(\014rst)523 5164 y(one)29 b(sho)n(ws)f(the)h(case)f
(where)h FD(j)p FC(s)p FD(j)c FK(=)g FC(l)c FD(\000)e
FK(1,)29 b(i.e.)g(part)g(of)g(the)g(b)r(ottom)h(la)n(y)n(er)d(of)i
FG(Cnt)3196 5176 y FA(k)3236 5164 y FK(\()p FC(\017)p
FK(\).)523 5264 y(There)d(are)g(2)942 5234 y Fw(j)p FA(s)p
Fw(j)1039 5264 y FK(=)d FC(n=)p FK(2)i(suc)n(h)i(parts)f(in)h(the)g(b)r
(ottom)f(la)n(y)n(er.)f(The)i(second)f(sho)n(ws)g(part)g(of)523
5363 y(the)i(middle/upp)r(er)g(la)n(y)n(ers)d(of)j FG(Cnt)1659
5375 y FA(k)1700 5363 y FK(\()p FC(\017)p FK(\),)g(where)f
FD(j)p FC(s)p FD(j)c FC(<)g(l)d FD(\000)e FK(1.)p eop
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	/yc exch def /xc exch def /yrad exch def /xrad exch def
	xc xrad add yc moveto
	/savematrix mtrx currentmatrix def
	xc yc translate
	xrad yrad scale
	0 0 1 0 360 arc
	savematrix setmatrix stroke
} def
% The next three take the text string, and moveto the right horiz. position
% leaving the string on the stack.
/lj {} def
/rj {dup stringwidth pop neg 0 rmoveto} def
/ce {dup stringwidth pop 2 div neg 0 rmoveto} def
% And this is invoked after one of the three above, and 
% computes the vert. pos, and then displays the string.
/t {halign exch get 0 exch rmoveto show newpath} def
% Store an array of patterns in /styles - a pattern is an array consisting 
% of an array and an offset. Corresp to xpic patterns
% solid, dotted, short-dashed, long-dashed, dot-dashed
/styles [ [] 0 ] [ [1 3] 0 ] [ [4 4] 0 ] [ [8 4] 0 ] [ [1 4 4 4] 0 ]
	5 array astore def
% change style to arg.
/D {stroke styles exch get aload pop setdash newpath} def
/W {stroke 0.5 mul setlinewidth newpath} def
% fontbox takes a fontname off the stack, and returns an array
% containing the values of the bottom line of the bounding box, the
% mid line of the bounding box, and the top line of the bounding box
% of that font, taken from the baseline, scaled to a font of size 1
/fontbox {
	findfont dup /FontMatrix get /fm exch def /FontBBox get aload pop
	/ytop exch def pop /ybot exch def pop
	/ymid ytop ybot sub 2 div def
	0 ybot fm dtransform exch pop % botline
	dup neg exch % midline - this works better than (ytop-ybot)/2!
	0 ytop fm dtransform exch pop exch %topline
	% now in the order midline, topline, botline.
	3 array astore
} def
% select font
/F {
	dup /fontname exch def fontbox
	/thisfontbox exch def SF 
} def
% set point size
/S {/ptsize exch def SF} def
% actually set font
/SF {
	fontname findfont ptsize curscale div scalefont setfont
	thisfontbox aload pop
	1 1 3 {
		pop ptsize mul curscale div neg 3 1 roll
	} for
	halign astore pop
} def
% sets the scale to 72 / n, where n is on the stack, and stores the value
% in curscale for font scaling
/curscale 1 def
/getpagesize{newpath clippath pathbbox /pageheight exch def 
	/pagewidth exch def pop pop newpath} def
/flip{pageheight exch sub translate} def
/ss {/curscale exch 72 exch div dup dup scale def} def
/land {90 rotate} def
StartXpic
%%EndProlog
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/Times-Roman F
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152 56 m
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32 56 m
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32 132 m
x
24 24 36 156 e
136 56 176 16 b
16 56 56 16 b
%%Trailer
showpage
% Trailer for xpic to PostScript converter
% $Header: x2ps.tra,v 1.2 89/07/02 15:59:53 moraes Exp $
xpic end restore

%%EndDocument
 @endspecial 898 2508 a(FIGURE)25 b(4.2.)i(Bottom)f(La)n(y)n(er)g(of)g
(an)g(Em)n(b)r(edding)f(of)i(Circuit)f Fg(Cnt)2945 2517
y Ff(k)523 4721 y @beginspecial 0 @llx 0 @lly 496.799988
@urx 208.800003 @ury 3401 @rwi @setspecial
%%BeginDocument: midlay.ps
%!
%%Creator: tromp@wolf.cwi.nl (John Tromp,M231,4078,02207-16322)
%%Title: midlay (xpic)
%%CreationDate: Sat Sep 18 16:15:37 1993
%%Pages: 1
%%BoundingBox: 0 0 496.8 208.8
%	(in inches) at 0 0, width 6.9, height 2.9
%%EndComments
% Prolog for xpic to PostScript converter
% Author: Mark Moraes
% $Header: x2ps.pro,v 1.2 88/03/19 16:50:09 moraes Exp 
%	%d D - change style SOLID, DOTTED, SHORT-DASH, LONG-DASH, DOT-DASH
%	%s F - change font to fontname
%	%d S - change size (font size in points)
%	(%s) rj %d t - text right just. (%d is TOPLINE, MIDLINE, BOTLINE)
%	(%s) lj %d t - text left just. (%d is TOPLINE, MIDLINE, BOTLINE)
%	(%s) ce %d t - text centered (%d is TOPLINE, MIDLINE, BOTLINE)
%	%d %d l - lineto
%	%d %d m - moveto
%	%d %d s - spline segment
%	x - flush line, spline
%	<wid> <ht> <x> <y> b - box
%	<wid> <ht> <x> <y> e - ellipse
%	%d ss - setscale
%	%d W - change linewidth
%	getpagesize - gets the values of PAGEHEIGHT and PAGEWIDTH
%	%d %d flip - translate by %d, PAGEHEIGHT - %d (this 
%		transforms to X windows coordinates)
save 50 dict begin /xpic exch def
/StartXpic {newpath 0 0 moveto [] 0 setdash 0 setgray 1 setlinecap} def
% Set defaults
/fontname /Times-Roman def
/ptsize 12 def
% halign has the values for MIDLINE, TOPLINE, BOTLINE
/halign 3 array def
/s {rcurveto} def
/x {stroke} def
/l {lineto} def
/m {moveto} def
/b {
	/ury exch def /urx exch def /lly exch def /llx exch def 
	llx lly moveto urx lly lineto urx ury lineto 
	llx ury lineto llx lly lineto stroke
} def
/mtrx matrix def
/e {
	/yc exch def /xc exch def /yrad exch def /xrad exch def
	xc xrad add yc moveto
	/savematrix mtrx currentmatrix def
	xc yc translate
	xrad yrad scale
	0 0 1 0 360 arc
	savematrix setmatrix stroke
} def
% The next three take the text string, and moveto the right horiz. position
% leaving the string on the stack.
/lj {} def
/rj {dup stringwidth pop neg 0 rmoveto} def
/ce {dup stringwidth pop 2 div neg 0 rmoveto} def
% And this is invoked after one of the three above, and 
% computes the vert. pos, and then displays the string.
/t {halign exch get 0 exch rmoveto show newpath} def
% Store an array of patterns in /styles - a pattern is an array consisting 
% of an array and an offset. Corresp to xpic patterns
% solid, dotted, short-dashed, long-dashed, dot-dashed
/styles [ [] 0 ] [ [1 3] 0 ] [ [4 4] 0 ] [ [8 4] 0 ] [ [1 4 4 4] 0 ]
	5 array astore def
% change style to arg.
/D {stroke styles exch get aload pop setdash newpath} def
/W {stroke 0.5 mul setlinewidth newpath} def
% fontbox takes a fontname off the stack, and returns an array
% containing the values of the bottom line of the bounding box, the
% mid line of the bounding box, and the top line of the bounding box
% of that font, taken from the baseline, scaled to a font of size 1
/fontbox {
	findfont dup /FontMatrix get /fm exch def /FontBBox get aload pop
	/ytop exch def pop /ybot exch def pop
	/ymid ytop ybot sub 2 div def
	0 ybot fm dtransform exch pop % botline
	dup neg exch % midline - this works better than (ytop-ybot)/2!
	0 ytop fm dtransform exch pop exch %topline
	% now in the order midline, topline, botline.
	3 array astore
} def
% select font
/F {
	dup /fontname exch def fontbox
	/thisfontbox exch def SF 
} def
% set point size
/S {/ptsize exch def SF} def
% actually set font
/SF {
	fontname findfont ptsize curscale div scalefont setfont
	thisfontbox aload pop
	1 1 3 {
		pop ptsize mul curscale div neg 3 1 roll
	} for
	halign astore pop
} def
% sets the scale to 72 / n, where n is on the stack, and stores the value
% in curscale for font scaling
/curscale 1 def
/getpagesize{newpath clippath pathbbox /pageheight exch def 
	/pagewidth exch def pop pop newpath} def
/flip{pageheight exch sub translate} def
/ss {/curscale exch 72 exch div dup dup scale def} def
/land {90 rotate} def
StartXpic
%%EndProlog
80 ss
0.5 W
/Times-Roman F
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146 56 m
(k) ce 0 t
426 56 m
(k) ce 0 t
0 D
24 24 516 196 e
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516 196 m
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36 196 m
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24 24 236 116 e
24 24 36 116 e
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288 144 544 8 b
24 24 316 116 e
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320 140 m
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232 140 m
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488 196 m
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200 196 m
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516 140 m
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516 172 m
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36 140 m
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248 136 m
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x
304 136 m
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248 176 l
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0.5 W
16 S
36 116 m
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236 116 m
(l) ce 0 t
316 116 m
(r) ce 0 t
516 116 m
(r) ce 0 t
12 S
320 196 m
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18 S
420 64 m
(r=Cnt  \(s1\)) ce 0 t
0 232 552 0 b
140 64 m
(l=Cnt  \(s0\)) ce 0 t
200 216 268 176 b
284 216 352 176 b
12 S
236 196 m
(min\(k-l,r\)) ce 0 t
%%Trailer
showpage
% Trailer for xpic to PostScript converter
% $Header: x2ps.tra,v 1.2 89/07/02 15:59:53 moraes Exp $
xpic end restore

%%EndDocument
 @endspecial 785 4895 a FF(FIGURE)f(4.3.)i(Middle/Upp)r(er)f(La)n(y)n
(er)g(of)g(an)g(Em)n(b)r(edding)f(of)h(Circuit)h Fg(Cnt)3057
4904 y Ff(k)p eop end
%%Page: 44 54
TeXDict begin 44 53 bop 2106 299 a FF(4.2.)53 b(W)-6
b(orst)25 b(case)i(upp)r(er)e(b)r(ounds)153 b(44)606
506 y FK(Eac)n(h)28 b(sub-circuit)g FG(Cnt)1367 518 y
FA(k)1408 506 y FK(\()p FC(s)p FK(\))i(o)r(ccupies)e(a)g(rectangular)f
(area)g(ab)r(o)n(v)n(e)h(the)h(input)g(blo)r(c)n(k)523
606 y FC(s)p FK(.)k(It)g(computes)f FC(S)1138 618 y FA(k)1179
606 y FK(\()p FC(s)p FK(\))h(whic)n(h)f(app)r(ears)g(at)g(b)r(oth)h
(the)g(top-left)g(and)f(top-righ)n(t)g(of)g(the)523 706
y(sub)r(circuit,)h(as)f(the)h(outputs)g(of)g(the)g(t)n(w)n(o)f
(addition)h(no)r(des)f(there.)h(Apart)f(from)h(these)523
805 y(addition)26 b(no)r(des,)g(it)g(adds)g(on)n(to)f(the)i(lo)n(w)n
(er)d(la)n(y)n(ers)h(6)g(short)h(\(constan)n(t)f(length\))i(wires,)e(2)
523 905 y(constan)n(t-area)20 b(subtract-and-minimize)i(no)r(des)g
(\(whic)n(h)h(ha)n(v)n(e)e(the)i(constan)n(t)f(threshold)523
1005 y FC(k)31 b FK(hardwired)26 b(in)n(to)h(them\),)i(and)e(2)h(long)e
(wires.)606 1104 y(All)36 b(wires)f(in)h(la)n(y)n(ers)e
FC(B)40 b FK(and)c(up)g(ha)n(v)n(e)e(constan)n(t)h(bandwidth)h
FC(B)41 b FK(=)36 b FD(d)p FK(log\()p FC(k)27 b FK(+)d(1\))p
FD(e)p FK(,)523 1204 y(su\016cien)n(t)32 b(to)f(carry)f(an)n(y)h(v)-5
b(alue)31 b(in)h(the)g(range)e FD(f)p FK(0)p FC(;)14
b FK(1)p FC(;)g(:)g(:)g(:)e(;)i(k)s FD(g)p FK(.)31 b(F)-7
b(or)31 b(the)h(b)r(ottom)g FC(B)25 b FD(\000)c FK(1)523
1303 y(la)n(y)n(ers,)f(where)h FD(j)p FC(s)p FD(j)i FC(>)f(l)8
b FD(\000)e FC(B)t FK(,)22 b FG(Cnt)1560 1315 y FA(k)1601
1303 y FK(\()p FC(s)p FK(\))g(sums)g(o)n(v)n(er)d(2)2142
1273 y FA(l)p Fw(\000j)p FA(s)p Fw(j)2312 1303 y FK(inputs,)j(hence)f
(its)h(wires)f(can)g(do)523 1403 y(with)27 b(bandwidth)g
FC(l)19 b FD(\000)d(j)p FC(s)p FD(j)h FK(+)f(1)22 b FD(\024)h
FC(B)t FK(.)k(This)g(sho)n(ws)e(that)i(the)g(input)h(no)r(de)f(densit)n
(y)f(is)h(not)523 1503 y(limited)j(so)e(m)n(uc)n(h)h(b)n(y)f(the)i
(wire-width)f(as)f(b)n(y)g(the)i(size)e(of)h(the)h(non-input)f(no)r
(des.)g(The)523 1602 y(width)35 b(a)n(v)-5 b(ailable)33
b(to)h(these)h(no)r(des)f(doubles)g(at)g(ev)n(ery)f(la)n(y)n(er,)g
(while)h(their)h(n)n(um)n(b)r(er)f(of)523 1702 y(input)28
b(bits)g(increases)e(b)n(y)h(only)g(2.)g(Hence,)g(the)h(inputs)g(can)f
(b)r(e)g(spaced)g FC(c\025)h FK(apart,)e(with)523 1802
y FC(c)g FK(a)f(small)g(constan)n(t)g(dep)r(ending)h(on)g(the)g
(implemen)n(tation)f(of)h(the)g(non-input)g(no)r(des)f(in)523
1901 y(the)j(b)r(ottom)g(la)n(y)n(ers.)523 2060 y Fv(Theorem)k
FH(4.1)41 b FG(F)-6 b(or)30 b(c)l(onstant)f FC(k)s FG(,)h
FC(S)1730 2072 y FA(k)1800 2060 y FG(is)g(ener)l(gy)g(e\016cient.)523
2219 y Fv(Pr)n(oof.)19 b FK(T)-7 b(ak)n(e)18 b FG(Cnt)1162
2231 y FA(k)1222 2219 y FK(for)h(increasingly)f(larger)f
FC(n)i FK(as)f(a)h(circuit)g(family)-7 b(.)20 b(Since)f(the)g(b)r
(ottom)523 2319 y(la)n(y)n(er)j(has)h(depth)h(1)g(and)f(eac)n(h)g
(additional)g(la)n(y)n(er)f(adds)h(only)h(2)f(to)h(the)g(depth,)g(the)g
(depth)523 2418 y(of)31 b(the)h(en)n(tire)f(circuit)g(is)g(2)p
FC(l)22 b FD(\000)f FK(1)28 b(=)h FC(O)r FK(\(log)15
b FC(n)p FK(\))32 b(as)e(required.)h(It)g(remains)g(to)g(sho)n(w)f
(that)523 2518 y FC(E)584 2540 y FG(worst)790 2518 y
FK(\()p FG(Cnt)964 2530 y FA(k)1005 2518 y FK(\))24 b(=)e
FC(O)r FK(\()p FC(n)p FK(\).)27 b(The)f(no)r(des)g(ha)n(v)n(e)e(b)r
(ounded)j(area,)d(since)i(they)g(ha)n(v)n(e)e(at)i(most)523
2617 y(2)p FC(B)31 b FK(input)e(bits.)f(The)f(total)h(n)n(um)n(b)r(er)f
(of)h(no)r(des)f(in)h FG(Cnt)2303 2629 y FA(k)2344 2617
y FK(,)g(including)g(input)g(no)r(des,)g(is)1636 2715
y Fn(X)1624 2897 y Fw(j)p FA(s)p Fw(j)p FA(<l)1782 2794
y FK(4)23 b(=)f(4\()p FC(n)c FD(\000)g FK(1\))p FC(:)523
3051 y FK(It)37 b(follo)n(ws)f(that)h(the)g(no)r(de)g(energy)e(of)i
(the)g(circuit)g(is)f(only)h(linear.)f(The)g(n)n(um)n(b)r(er)h(of)523
3151 y(v)n(ertical)26 b(wires)h(is)1133 3237 y Fn(X)1079
3419 y Fw(j)p FA(s)p Fw(j)p FB(=)p FA(l)p Fw(\000)p FB(1)1321
3315 y FK(4)18 b(+)1518 3237 y Fn(X)1464 3419 y Fw(j)p
FA(s)p Fw(j)p FA(<l)p Fw(\000)p FB(1)1707 3315 y FK(6)k(=)h(2)p
FC(n)18 b FK(+)g(6\()p FC(n=)p FK(2)f FD(\000)h FK(1\))23
b(=)f(5)p FC(n)c FD(\000)g FK(6)p FC(:)523 3572 y FK(These)25
b(to)r(o)g(ha)n(v)n(e)e(only)i(a)g(linear)f(total)h(area)e(making)i
(the)g(`v)n(ertical)f(wire)g(energy')g(of)h(the)523 3672
y(circuit)j(linear.)606 3772 y(So)23 b(w)n(e)f(are)g(left)h(to)g
(consider)f(the)h(`long)f(wire)g(energy'.)g(By)h(the)g(triangle)f
(inequalit)n(y)-7 b(,)22 b(it)523 3871 y(su\016ces)k(to)f(sho)n(w)h
(that)g FC(E)5 b FK(\()p FG(Cnt)1526 3883 y FA(k)1567
3871 y FC(;)14 b(x;)g FK(0\))23 b(=)f FC(O)r FK(\()p
FC(n)p FK(\).)27 b(The)f(simple)h(but)f(crucial)f(observ)-5
b(ation)523 3971 y(is)24 b(that)h(if)f(the)h(long)f(wire)f(in)i
FG(Cnt)1581 3983 y FA(k)1621 3971 y FK(\()p FC(s)p FK(\))g(running)f
(directly)g(ab)r(o)n(v)n(e)f FG(Cnt)2728 3983 y FA(k)2768
3971 y FK(\()p FC(s)p FK(0\))i(\(i.e.)f(the)h(left)523
4071 y(long)h(wire)g(sho)n(wn)g(in)h(Figure)f(4.3\))g(carries)f(a)h
(non-zero)f(v)-5 b(alue,)26 b(then)i FC(S)2797 4083 y
FA(K)2861 4071 y FK(\()p FC(s)p FK(\))23 b FC(>)g(S)3126
4083 y FA(K)3190 4071 y FK(\()p FC(s)p FK(0\).)523 4170
y(Similarly)36 b(for)f(the)i(long)e(wire)h(ab)r(o)n(v)n(e)f
FG(Cnt)1934 4182 y FA(k)1975 4170 y FK(\()p FC(s)p FK(1\).)i(More)e
(generally)-7 b(,)35 b(if)h(the)h(long)e(wire)523 4270
y(running)k(directly)h(ab)r(o)n(v)n(e)e FG(Cnt)1548 4282
y FA(k)1589 4270 y FK(\()p FC(s)p FK(\))i(is)f(non-zero,)g(then)h
FC(S)2452 4282 y FA(K)2516 4270 y FK(\()p FC(t)p FK(\))k
FC(>)e(S)2812 4282 y FA(K)2876 4270 y FK(\()p FC(s)p
FK(\))f(for)e(an)n(y)g FC(t)523 4369 y FK(whic)n(h)e(is)g(a)f(prop)r
(er)g(pre\014x)g(of)h FC(s)p FK(.)g(No)n(w)f(imagine)g(an)n(y)g(v)n
(ertical)g(cross-section)f(of)h(the)523 4469 y(circuit.)26
b(The)h(n)n(um)n(b)r(er)f(of)g(non-zero)f(long)g(wires)h(though)g(this)
h(cross-section)d(is)i(then)h(at)523 4569 y(most)i FC(S)779
4581 y FA(K)843 4569 y FK(\()p FC(\017)p FK(\))h(whic)n(h)f(clearly)f
(cannot)h(exceed)g FC(k)s FK(.)h(Th)n(us,)f(the)h(total)f(length)g(of)g
(non-zero)523 4668 y(long)c(wires)h(is)g(b)r(ounded)g(b)n(y)g
FC(k)j FK(times)d(the)h(width)f(of)h(the)f(circuit)g
FG(Cnt)2747 4680 y FA(k)2787 4668 y FK(,)h(whic)n(h)f(is)g
FC(k)s(nc\025)p FK(,)523 4768 y(and)32 b(the)h(area)e(in)n(v)n(olv)n
(ed)g(is)h(no)g(more)g(than)g FC(B)t(\025)h FK(times)g(that.)f(With)i
FC(k)s FK(,)p FC(c)p FK(,)p FC(B)t FK(,)e(and)g FC(\025)h
FK(all)523 4868 y(b)r(eing)21 b(constan)n(t,)f(this)h(pro)n(v)n(es)d
(that)j FC(E)5 b FK(\()p FG(Cnt)1908 4880 y FA(k)1949
4868 y FC(;)14 b(x;)g FK(0\))23 b(=)g FC(O)r FK(\()p
FC(n)p FK(\))f(and)e(thereb)n(y)g(the)h(theorem.)523
4967 y Fu(2)606 5109 y FK(Since)27 b(adding)f(a)g(comparator)e(to)i
FG(Cnt)1845 5121 y FA(k)1912 5109 y FK(adds)g(only)g(constan)n(t)g
(area)f(and)h(1)g(to)g(depth,)523 5209 y(w)n(e)h(ha)n(v)n(e)523
5368 y Fv(Cor)n(ollar)-6 b(y)31 b FH(4.2)41 b FK(F)-7
b(or)27 b(constan)n(t)g FC(k)s FK(,)g FC(T)1798 5380
y FA(k)1866 5368 y FK(is)h(energy)e(e\016cien)n(t.)p
eop end
%%Page: 45 55
TeXDict begin 45 54 bop 2106 299 a FF(4.2.)53 b(W)-6
b(orst)25 b(case)i(upp)r(er)e(b)r(ounds)153 b(45)523
506 y FL(4.3)116 b(Conclusions)523 706 y FK(W)-7 b(e)30
b(ha)n(v)n(e)e(describ)r(ed)g(a)h(construction)g(that)g(yields)g(fast,)
g(minim)n(um)h(energy)e(VLSI)i(cir-)523 805 y(cuits)k(to)g(compute)g
(threshold)g(functions)g(and)g(to)g(coun)n(t)g(up)g(to)g(a)g(constan)n
(t.)f(Energy)523 905 y(b)r(ounds)28 b(for)f(general)f(coun)n(ting)h
(functions,)h(including)g(ma)5 b(jorit)n(y)-7 b(,)26
b(remain)h(op)r(en.)p eop end
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2516 y FK([1])41 b(Aggarw)n(al,)33 b(A.,)i(A.)h(Chandra,)e(P)-7
b(.)35 b(Ragha)n(v)-5 b(an,)34 b FG(Ener)l(gy)j(Consumption)g(in)g
(VLSI)694 2615 y(Cir)l(cuits)p FK(,)28 b(Pro)r(ceedings)e(of)h(20th)g
(A)n(CM)h(STOC,)f(Ma)n(y)g(1988,)f(pp.)i(205-216.)565
2771 y([2])41 b(Boro)r(din,)24 b(A.,)i FG(On)i(R)l(elating)f(Time)i
(and)g(Sp)l(ac)l(e)f(to)g(Size)g(and)g(Depth)p FK(,)e(SIAM)g(Jour-)694
2871 y(nal)h(of)h(Computing,)f(V)-7 b(ol.)28 b(6,)f(No.)h(4,)f(Decem)n
(b)r(er)g(1977,)f(pp.)i(733-744.)565 3026 y([3])41 b(Bren)n(t,)h(R.P)-7
b(.,)43 b(H.T.)h(Kung,)e FG(A)i(R)l(e)l(gular)g(L)l(ayout)g(for)h(Par)l
(al)t(lel)h(A)l(dders)p FK(,)d(IEEE)694 3126 y(T)-7 b(ransactions)25
b(on)j(Computers,)f(V)-7 b(ol.)28 b(C-31,)e(No.)h(3,)h(Marc)n(h)e
(1982,)g(pp.)i(260-264.)565 3282 y([4])41 b(Bren)n(t,)26
b(R.P)-7 b(.,)27 b(H.T.)g(Kung,)f("The)h(Area-Time)e(Complexit)n(y)i
(of)f(Binary)g(Multipli-)694 3381 y(cation",)g(JA)n(CM,)i(V)-7
b(ol.)27 b(28,)g(No.)h(3,)f(July)g(1981,)f(pp.)i(521-534.)565
3537 y([5])41 b(Kissin,)19 b(G.,)h FG(Upp)l(er)j(and)g(L)l(ower)g
(Bounds)g(on)f(Switching)h(Ener)l(gy)g(in)g(VLSI)p FK(,)c(JA)n(CM,)694
3637 y(V)-7 b(ol.)27 b(38,)g(No.)h(1,)f(Jan)n(uary)f(1991.)565
3792 y([6])41 b(Kissin,)31 b(G.,)h FG(Mo)l(dels)k(of)e(Multiswitch)h
(Ener)l(gy)p FK(,)e(CWI)f(Quarterly)-7 b(,)31 b(V)-7
b(ol.)32 b(3,)f(No.)h(1,)694 3892 y(Marc)n(h)26 b(1990,)g(pp.)i(45-66.)
565 4048 y([7])41 b(Kissin,)27 b(G.,)h FG(Mo)l(deling)j(Ener)l(gy)f
(Consumption)g(in)g(VLSI)f(Cir)l(cuits)p FK(,)f(PhD)g(Thesis,)694
4147 y(Departmen)n(t)f(of)h(Computer)f(Science,)h(Univ)n(ersit)n(y)e
(of)i(T)-7 b(oron)n(to,)26 b(1987.)565 4303 y([8])41
b(Kissin,)30 b(G.,)g FG(F)-6 b(unctional)33 b(Bounds)f(on)h(Switching)g
(Ener)l(gy)p FK(,)e(Pro)r(ceedings)d(of)j(1985)694 4403
y(Chap)r(el)j(Hill)h(Conference)f(on)g(V)-7 b(ery)34
b(Large)e(Scale)i(In)n(tegration,)f(Ma)n(y)h(1985,)e(pp.)694
4502 y(181-196.)565 4658 y([9])41 b(Kissin,)30 b(G.,)h
FG(Me)l(asuring)i(Ener)l(gy)g(Consumption)g(in)f(VLSI)g(Cir)l(cuits:)i
(a)f(F)-6 b(ounda-)694 4758 y(tion)p FK(,)28 b(Pro)r(ceedings)d(of)j
(14th)f(A)n(CM)h(STOC,)f(Ma)n(y)g(1982,)f(pp.)i(99-104.)523
4913 y([10])41 b(Krohn,)31 b(K.,)h(J.)g(Rho)r(des,)g("Algebraic)f
(Theory)g(of)i(Mac)n(hines.)e(I.)i(Prime)e(Decom-)694
5013 y(p)r(osition)h(Theorem)f(for)h(Finite)h(Semigroups)e(and)h(Mac)n
(hines",)f(T)-7 b(ransactions)31 b(of)694 5113 y(American)c
(Mathematical)g(So)r(ciet)n(y)-7 b(,)27 b(V)-7 b(ol.)28
b(116,)e(1965,)g(pp.)i(450-464.)523 5268 y([11])41 b(Krohn,)31
b(K.,)i(J.)f(Rho)r(des,)h FG(R)l(esults)g(on)i(Finite)g(Semigr)l(oups)p
FK(,)e(Pro)r(ceedings)e(of)h(the)694 5368 y(National)27
b(Academ)n(y)g(of)g(Science,)h(USA,)g(V)-7 b(ol.)28 b(53,)f(1965,)f
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694 606 y(The)27 b(Netherlands,)h(F)-7 b(ebruary)26 b(1984.)523
772 y([13])41 b(Lengauer,)35 b(T.,)h(K.)g(Mehlhorn,)g
FG(On)h(the)h(Complexity)h(of)g(VLSI)e(Computations)p
FK(,)694 872 y(Pro)r(ceedings)22 b(of)j(CMU)g(Conference)e(on)h(VLSI,)h
(Computer)f(Science)h(Press,)e(Octo-)694 971 y(b)r(er)k(1981,)f(pp.)i
(89-99.)523 1137 y([14])41 b(Mead,)f(C.,)h(L.)g(Con)n(w)n(a)n(y)-7
b(,)40 b(In)n(tro)r(duction)g(to)h(VLSI)g(Systems,)g(Addison-W)-7
b(esley)g(,)694 1237 y(1980.)523 1403 y([15])41 b(Mead,)27
b(C.,)h(priv)-5 b(ate)27 b(comm)n(unication)523 1569
y([16])41 b(Sn)n(yder,)g(L.,)h(A.)g(T)n(y)n(agi,)e FG(The)j(Ener)l(gy)g
(Complexity)h(of)g(T)-6 b(r)l(ansitive)43 b(F)-6 b(unctions)p
FK(,)694 1669 y(Pro)r(ceedings)24 b(of)j(24th)f(Allerton)g(Conference)g
(on)g(Comm)n(unication,)g(Con)n(trol)f(and)694 1768 y(Computing,)i
(Octob)r(er)g(1986,)f(pp.)i(562-572.)523 1934 y([17])41
b(Thompson,)19 b(C.,)g FG(A)j(Complexity)i(The)l(ory)g(for)f(VLSI)p
FK(,)c(PhD)h(Thesis,)f(Dept.)i(of)e(Com-)694 2034 y(puter)27
b(Science,)h(Carnegie-Mellon)e(Univ)n(ersit)n(y)-7 b(,)26
b(1980.)523 2200 y([18])41 b(T)n(y)n(agi,)35 b(A.,)j
FG(Ener)l(gy)h(Complexity)h(of)f(L)l(o)l(gic)h(L)l(evel)f(Structur)l
(es,)c FK(Pro)r(ceedings)h(of)694 2300 y(Stanford)27
b(Conference)g(on)g(Adv)-5 b(anced)28 b(Researc)n(h)e(in)i(VLSI,")f
(MIT)h(Press,)e(1987.)523 2466 y([19])41 b(V)-7 b(uillemin,)36
b(J.,)g FG(A)h(Combinatorial)j(Limit)d(to)h(the)f(Computing)h(Power)g
(of)g(VLSI)694 2565 y(Cir)l(cuits)p FK(,)e(IEEE)e(T)-7
b(ransactions)33 b(on)i(Computers,)g(V)-7 b(ol.)36 b(C-30,)e(No.)i(2,)f
(1983,)e(pp.)694 2665 y(135-140.)p eop end
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b(Storage)g(Mo)t(di\014cation)523 972 y(Mac)l(hines)523
2549 y FL(5.1)116 b(In)m(tro)s(duction)523 2748 y FK(The)30
b(Storage)d(Mo)r(di\014cation)j(Mac)n(hine)f(\()p FG(SMM)14
b FK(\))30 b(is)f(a)g(mac)n(hine)g(mo)r(del)h(in)n(tro)r(duced)f(b)n(y)
523 2848 y(Sc)n(h\177)-42 b(onhage)38 b(in)i(1977)d([16)o(].)j(The)f
(mo)r(del)h(has)e(its)i(predecessor)d(in)j(the)g(Kolmogoro)n(v-)523
2947 y(Usp)r(enskii)28 b(mac)n(hine)f(\()p FG(KUM)14
b FK(\))28 b([10)o(].)g(Sc)n(h\177)-42 b(onhage)27 b(adv)n(o)r(cates)f
(his)i(mo)r(del)g(as)f FG(a)j(mo)l(del)h(of)523 3047
y(extr)l(eme)e(\015exibility)7 b FK(.)606 3147 y(The)35
b(mo)r(del)f(resem)n(bles)g(the)h(Random)f(Access)g(Mac)n(hine)g(\()p
FG(RAM)13 b FK(\))35 b([1])g(as)e(far)h(as)g(it)523 3246
y(has)39 b(a)h(stored)e(program)g(and)i(a)f(p)r(oten)n(tially)h
(in\014nite)g(memory)f(structure)g(where)g(it)523 3346
y(stores)20 b(its)i(data.)f(Whereas)g(the)h FG(RAM)35
b FK(uses)21 b(an)g(in\014nite)h(sequence)f(of)h(storage)d(registers,)
523 3445 y(eac)n(h)35 b(capable)f(of)h(storing)g(an)g(arbitrarily)e
(large)h(in)n(teger,)h(the)g FG(SMM)50 b FK(op)r(erates)34
b(on)h(a)523 3545 y(directed)18 b(graph)g(b)n(y)g(creating)f(no)r(des)h
(and)g(\(re\)directing)g(p)r(oin)n(ters.)g(The)g(main)h(di\013erence)
523 3645 y(b)r(et)n(w)n(een)37 b(the)g FG(SMM)51 b FK(and)37
b(the)h FG(KUM)50 b FK(is)37 b(that)g(the)h FG(KUM)50
b FK(op)r(erates)36 b(on)h(undirected)523 3744 y(instead)27
b(of)h(directed)g(graphs.)606 3844 y(W)-7 b(e)21 b(can)g(appro)n
(ximately)e(mo)r(del)i(an)g FG(SMM)35 b FK(b)n(y)21 b(a)g(P)n(ascal)e
(program)g(that)i(uses)f(records)523 3944 y(of)28 b(p)r(oin)n(ters)f
(to)g(records)f(to)h(describ)r(e)h(the)g(directed)f(graph)2418
3913 y FB(1)2454 3944 y FK(:)523 4101 y FH(t)m(yp)s(e)32
b FK(p)r(oin)n(ter)27 b(=)h(^no)r(de;)739 4200 y(no)r(de)113
b(=)28 b FH(record)g FK(a,b:)f(p)r(oin)n(ter)h FH(end)p
FK(;)523 4300 y FH(v)-5 b(ar)86 b FK(head)115 b(:)70
b(p)r(oin)n(ter;)606 4457 y(In)39 b(con)n(trast)f(with)h(P)n(ascal,)f
(p)r(oin)n(ters)g(are)g(not)h(allo)n(w)n(ed)f(to)g(b)r(e)i
FG(nil)48 b FK(or)38 b(unde\014ned;)523 4556 y(they)26
b(m)n(ust)h(alw)n(a)n(ys)d(p)r(oin)n(t)i(to)g(some)g(no)r(de.)g(The)g
(\(\014nite\))h(set)f(of)h(p)r(oin)n(ter)e(names,)h(in)g(the)523
4656 y(example)36 b FD(f)p FC(a;)14 b(b)p FD(g)p FK(,)34
b(is)i(called)f(the)i FG(alphab)l(et)g FK(of)f FG(dir)l(e)l(ctions)p
FK(,)h(denoted)f(\001.)g(The)g(p)r(oin)n(ters)523 4756
y(in)f(the)g(graph)f(are)g(lab)r(eled)g(with)i(the)f(elemen)n(ts)f(of)h
(\001)g(suc)n(h)g(that)g(eac)n(h)f(no)r(de)g(in)h(the)523
4855 y(digraph)i(has,)g(for)h(eac)n(h)f(direction)h FC(\016)43
b FD(2)e FK(\001,)d(exactly)f(one)h(outgoing)e FC(\016)s
FK(-p)r(oin)n(ter.)i(The)523 4955 y(graph)f(th)n(us)g(has)h(regular)e
(outdegree)g FD(j)p FK(\001)p FD(j)p FK(.)i(T)-7 b(o)38
b(complete)f(the)h(analogy)e(b)r(et)n(w)n(een)i(an)p
523 5079 499 5 v 627 5161 a Fm(1)661 5193 y FF(In)28
b(P)n(ascal)i Fe(^T)e FF(denotes)h(the)f(t)n(yp)r(e)f(`p)r(oin)n(ter)i
(to)f(T';)h(a)g(v)l(alue)f(of)h(this)f(t)n(yp)r(e)g(is)g(the)g(address)
523 5276 y(of)d(an)f(ob)t(ject)h(of)g(t)n(yp)r(e)e(T.)i(Indirection)f
(through)f(a)i(p)r(oin)n(ter)f(is)g(written)h(as)f Fe(p^)p
FF(,)h(whic)n(h)f(refers)h(to)523 5359 y(the)g(ob)t(ject)i(at)f(whic)n
(h)f(p)h(p)r(oin)n(ts.)p eop end
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(latter)h(m)n(ust)f(b)r(e)h(restricted)f(to)g(the)h(use)f(of)g(only)g
(one)523 606 y(v)-5 b(ariable;)19 b(the)h(p)r(oin)n(ter)f
Fk(head)p FK(.)f(By)i(rep)r(eated)f(application)g(of)g(the)i(P)n(ascal)
c Fk(new)i FK(statemen)n(t,)523 706 y(the)33 b(program)d(can)i(create)f
(an)h(arbitrarily)e(large)h(data)h(structure.)g(This)g(is)g(addressed)
523 805 y(with)25 b(expressions)f(lik)n(e)g Fk(head^.b^.a^.b^.b^)o(.a)o
FK(.)19 b(Similarly)-7 b(,)25 b(the)g FG(SMM)39 b FK(addresses)24
b(its)523 905 y(storage)39 b(with)h(w)n(ords)g(\(strings\))g(o)n(v)n
(er)e(\001,)j(lik)n(e)f FC(babba)p FK(.)f(In)i(the)g
FG(SMM)54 b FK(mo)r(del,)41 b(there)523 1005 y(is)33
b(only)h(a)f(conceptual)g(head)g(p)r(oin)n(ter|at)g(an)n(y)g(time,)h
(one)f(no)r(de,)h(call)f(the)h FG(c)l(enter)p FK(,)f(is)523
1104 y(distinguished)27 b(as)g(the)g(one)g(from)g(where)f(addressing)g
(starts.)g(Th)n(us)h(the)h(cen)n(ter,)e(whose)523 1204
y(iden)n(tit)n(y)i(can)g(c)n(hange)e(dynamically)-7 b(,)28
b(is)f(addressed)g(b)n(y)h(the)g(empt)n(y)g(w)n(ord)f
FC(\017)p FK(,)h(and)f(other)523 1303 y(no)r(des)g(are)g(addressed)f(b)
n(y)i(follo)n(wing)e(p)r(oin)n(ters)h(starting)g(from)g(the)h(cen)n
(ter.)606 1403 y(It)d(has)f(b)r(een)h(established)f(that)h(from)g(the)g
(p)r(ersp)r(ectiv)n(e)f(of)h(computational)f(complex-)523
1503 y(it)n(y)33 b(theory)g(the)g FG(SMM)48 b FK(\(if)34
b(equipp)r(ed)f(with)h(the)f(correct)f(space)h(measure)f([12)o(,)h(21)o
(]\))h(is)523 1602 y(computationally)f(equiv)-5 b(alen)n(t)34
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523 1702 y(lik)n(e)c(the)h(T)-7 b(uring)30 b(mac)n(hine)g(and)h(the)g
FG(RAM)13 b FK(.)31 b(This)f(equiv)-5 b(alence)30 b(amoun)n(ts)g(to)h
(the)g(fact)523 1802 y(that)37 b(these)g(mo)r(dels)g(sim)n(ulate)g(eac)
n(h)f(other)g(with)h(p)r(olynomially)g(b)r(ounded)g(o)n(v)n(erhead)523
1901 y(in)i(time)g(and)f(constan)n(t)g(factor)f(o)n(v)n(erhead)g(in)h
(space,)g(th)n(us)h(satisfying)e(the)i(so-called)523
2001 y FG(invarianc)l(e)31 b(thesis)k FK([17)o(,)28 b(22)o(].)606
2100 y(F)-7 b(or)24 b(most)h(sequen)n(tial)f(mo)r(dels)g(there)h(ha)n
(v)n(e)e(b)r(een)i(prop)r(osed)f(parallel)g(mac)n(hine)g(mo)r(d-)523
2200 y(els)19 b(based)g(on)h(the)g(classical)e(sequen)n(tial)h(v)n
(ersion.)f(F)-7 b(or)19 b(the)h(T)-7 b(uring)19 b(mac)n(hine)g(Sa)n
(vitc)n(h)g([15)o(])523 2300 y(has)27 b(prop)r(osed)g(a)g(parallel)g(v)
n(ersion)f(based)h(on)g(parallel)g(recursiv)n(e)f(branc)n(hing;)g(a)i
(mo)r(del)523 2399 y(based)33 b(on)g(nondeterministic)g(forking)f(on)h
(a)g(shared)f(set)h(of)h(tap)r(es)f(w)n(as)f(describ)r(ed)h(b)n(y)523
2499 y(Wiedermann)d([24)o(,)g(25)o(],)g(but)h(this)f(mo)r(del)h(turns)e
(out)h(to)g(b)r(e)h(p)r(olynomially)e(equiv)-5 b(alen)n(t)523
2599 y(in)33 b(time)h(and)f(space)f(with)i(the)f(standard)g(sequen)n
(tial)f(devices.)h(The)g(ric)n(hness)f(of)h(par-)523
2698 y(allel)i(mo)r(dels)f(based)h(on)f(the)h FG(RAM)49
b FK(is)34 b(ev)n(en)h(m)n(uc)n(h)f(greater,)g(whic)n(h)g(mak)n(es)g
(it)h(hard,)523 2798 y(if)i(not)f(imp)r(ossible)h(to)f(refer)f(to)i(a)f
(small)g(set)g(of)g(represen)n(tativ)n(e)f(mo)r(dels.)h(There)g(are)523
2897 y(mo)r(dels)25 b(based)f(on)g(shared)f(memory)h(and)h(alternativ)n
(e)e(mo)r(dels)h(based)g(on)h(lo)r(cal)f(storage)523
2997 y(and)37 b(message)f(passing.)h(Hybrid)g(com)n(binations)f(o)r
(ccur)h(as)g(w)n(ell.)g(Within)h(eac)n(h)f(class)523
3097 y(there)c(exist)g(more)f(re\014ned)h(distinctions)g(lik)n(e)f(the)
i(resolution)d(strategy)h(for)g(resolving)523 3196 y(write)c
(con\015icts)g(in)g(shared)f(memory)g(mo)r(dels,)h(the)h(a)n(v)-5
b(ailable)26 b(arithmetic)i(instructions)523 3296 y(and)e(the)g(mec)n
(hanism)f(for)g(restricting)g(the)h(n)n(um)n(b)r(er)g(of)g(pro)r
(cessors)d(activ)-5 b(ated)26 b(during)f(a)523 3396 y(computation.)h
(Moreo)n(v)n(er,)d(there)j(exist)g(sequen)n(tial)f(mo)r(dels)h(whic)n
(h)g(b)r(ecome)f(computa-)523 3495 y(tionally)e(equiv)-5
b(alen)n(t)24 b(to)g(parallel)e(mo)r(dels)i(due)g(to)g(their)g(p)r(o)n
(w)n(er)f(to)g(create)g(and)h(manipu-)523 3595 y(late)j(exp)r(onen)n
(tially)g(large)f(v)-5 b(alues)27 b(in)g(a)g(linear)f(n)n(um)n(b)r(er)h
(of)h(steps)f(in)g(the)h(uniform)f(time)523 3694 y(measure.)36
b(Also,)g(b)n(y)g(exploiting)g(the)h(alternating)e(mo)r(de)i(of)f
(computation)g([5],)g(some)523 3794 y(standard)26 b(sequen)n(tial)h
(devices)f(b)r(ecome)h(computationally)g(equiv)-5 b(alen)n(t)26
b(to)h(the)h(parallel)523 3894 y(mac)n(hines.)606 3993
y(F)-7 b(or)33 b(a)h(more)f(detailed)h(surv)n(ey)e(of)i(parallel)f(mo)r
(dels)h(w)n(e)f(refer)g(to)h([20)o(,)g(22)o(].)g(F)-7
b(or)33 b(the)523 4093 y(purp)r(ose)j(of)h(the)g(presen)n(t)f(exp)r
(osition,)g(it)h(su\016ces)g(to)f(giv)n(e)g(some)g(impression)g(of)g
(the)523 4193 y(o)n(v)n(erall)25 b(landscap)r(e)i(of)h(parallel)e(mac)n
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(parallel)e(mo)r(dels)i(prop)r(osed)f(in)i(the)f(literature)f(b)r
(elong)h(to)523 4392 y(the)g(so-called)d FG(Se)l(c)l(ond)k(Machine)i
(Class)k FK(consisting)29 b(of)h(mac)n(hine)g(mo)r(dels)g(whic)n(h)g
(ob)r(ey)523 4491 y(the)k FG(Par)l(al)t(lel)j(Computation)f(Thesis)7
b FK(.)35 b(This)e(thesis)h(expresses)e(that)h(the)h(class)f(of)g(lan-)
523 4591 y(guages)d(recognized)g(in)i(nondeterministic)f(p)r(olynomial)
g(time)h(on)g(the)g(parallel)e(device)523 4691 y(is)f(equal)f(to)g(the)
h(class)f FG(PSP)-6 b(A)n(CE)39 b FK(of)28 b(languages)f(recognized)g
(in)i(p)r(olynomial)f(space)g(on)523 4790 y(a)d(sequen)n(tial)f
(device.)h(Con)n(v)n(ersely)e(all)i(languages)f(in)h
FG(PSP)-6 b(A)n(CE)36 b FK(are)24 b(recognized)g(in)h(de-)523
4890 y(terministic)d(p)r(olynomial)f(time)h(on)f(the)h(parallel)e(mac)n
(hine.)h(In)h(our)f(reading)f(the)i(P)n(arallel)523 4990
y(Computation)k(Thesis)f(en)n(tails)h(the)g(equiv)-5
b(alence)25 b(of)h(deterministic)g(and)g(nondetermin-)523
5089 y(istic)20 b(p)r(olynomial)f(time)h(on)f(the)h(parallel)f(mo)r
(del.)g(The)h(mo)r(dels)g(for)f(whic)n(h)g(the)h(thesis)g(w)n(as)523
5189 y(originally)h(form)n(ulated)i(ob)r(ey)g(this)g(more)f(restricted)
h(thesis)g(as)f(w)n(ell.)h(And)h(indeed)f(those)523 5289
y(mo)r(dels)k(for)g(whic)n(h)h(nondeterministic)f(p)r(olynomial)g(time)
h(seems)f(to)g(exceed)g FG(PSP)-6 b(A)n(CE)p eop end
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TeXDict begin 50 59 bop 2536 299 a FF(5.1.)53 b(In)n(tro)r(duction)153
b(50)523 506 y FK(no)n(w)n(ada)n(ys)25 b(are)i(held)h(to)f(b)r(e)h
(more)f(p)r(o)n(w)n(erful.)606 606 y(Not)37 b(all)g(parallel)e(mo)r
(dels)i(ob)r(ey)g(the)g(ab)r(o)n(v)n(e)f(parallel)f(computation)i
(thesis.)g(Some)523 706 y(w)n(eak)24 b(mo)r(dels)h(turn)g(out)g(to)g(b)
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(sequen)n(tial)e(mo)r(d-)523 805 y(els)40 b(\(the)i(parallel)d(T)-7
b(uring)40 b(mac)n(hine)g(prop)r(osed)g(b)n(y)g(Wiedermann,)g(and)h
(its)f(equiv)-5 b(a-)523 905 y(len)n(ts)34 b([24)o(,)f(25)o(])h(b)r
(eing)g(a)f(t)n(ypical)g(example\).)h(Other)f(mo)r(dels,)g(lik)n(e)h
(the)g FG(P)o FD(\000)-9 b FG(RAM)46 b FK(pre-)523 1005
y(sen)n(ted)36 b(b)n(y)f(F)-7 b(ortune)35 b(and)h(Wyllie)g([7)o(])g
(deviate)f(from)h(the)g(thesis)f(b)n(y)g(recognizing)f(ex-)523
1104 y(p)r(onen)n(tially)h(time)h(b)r(ounded)f(languages)f(in)h(p)r
(olynomial)g(nondeterministic)g(time)h(on)523 1204 y(the)26
b(parallel)f(device;)h(some)f(parallel)g(devices)h(ev)n(en)f(recognize)
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(The)f(second)g(mac)n(hine)g(class)f(therefore)h(represen)n(ts)f(a)h
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n(w)n(er)f(of)g(uniform)h(unrestricted)f(parallelism)g(rather)g(than)
523 1503 y(the)i(union)g(of)f(all)h(p)r(ossible)f(parallel)g(mac)n
(hine)g(mo)r(dels.)g(Second)h(mac)n(hine)f(class)g(mem-)523
1602 y(b)r(ers)c(can)h(b)r(e)g(c)n(haracterized)d(as)i(pro)n(viding)g
(the)h(righ)n(t)f(mixture)g(of)h(exp)r(onen)n(tial)f(gro)n(wth)523
1702 y(p)r(oten)n(tial)19 b(together)f(with)h(the)h(prop)r(er)e(degree)
g(of)h(uniformit)n(y)-7 b(.)19 b(The)g(exp)r(onen)n(tial)f(gro)n(wth)
523 1802 y(p)r(oten)n(tial)25 b(is)g(required)g(for)f(the)i(implemen)n
(tation)g(of)f(the)g(transitiv)n(e)g(closure)f(algorithm)523
1901 y(on)29 b(a)f(directed)h(graph)f(of)h(exp)r(onen)n(tial)f(size)h
(\(whic)n(h)g(mo)r(dels)g(the)g(computation)g(graph)523
2001 y(of)j(some)g FG(PSP)-6 b(A)n(CE)11 b FK(-b)r(ounded)32
b(mac)n(hine\),)h(or)e(the)i(direct)f(solution)g(of)h(the)g
FG(PSP)-6 b(A)n(CE)10 b FK(-)523 2100 y(complete)40 b(problem)f
FG(QBF)51 b FK(in)40 b(p)r(olynomial)f(time.)h(The)g(uniformit)n(y)f
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e(of)h(a)g(p)r(olynomial-time)f(computation)h(of)g(the)h(nondeter-)523
2300 y(ministic)e(v)n(ersion)e(of)h(the)h(parallel)e(mac)n(hine)h(in)h
(p)r(olynomial)f(space.)g(See)g([22)o(])h(for)f(more)523
2399 y(details)33 b(on)f(the)h(standard)f(strategies)f(for)h(pro)n
(ving)f(mem)n(b)r(ership)i(in)g(the)g(second)f(ma-)523
2499 y(c)n(hine)c(class.)606 2599 y(In)k(this)f(c)n(hapter)g(w)n(e)g
(prop)r(ose)f(\(as)h(far)g(as)g(w)n(e)g(kno)n(w)f(for)h(the)h(\014rst)f
(time\))h(a)f(parallel)523 2698 y(v)n(ersion)f(of)h(the)g(storage)f(mo)
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(second)e(ma-)523 2798 y(c)n(hine)k(class.)f(T)-7 b(o)33
b(our)g(kno)n(wledge)g(few)h(parallel)f(v)n(ersions)f(of)h(p)r(oin)n
(ter)h(mac)n(hines)f(ha)n(v)n(e)523 2897 y(b)r(een)38
b(in)n(v)n(estigated)f(in)g(the)h(complexit)n(y)f(theory)g(literature.)
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(enskii)c(mac)n(hine)523 3097 y(whic)n(h)k(w)n(as)f(prop)r(osed)f(b)n
(y)i(Barzdin)f([2,)g(3].)h(This)g(mac)n(hine)f(op)r(erates)g(lik)n(e)g
(an)h(irregular)523 3196 y(cellular)c(arra)n(y)e(of)i(\014nite)h(state)
f(automata)g(in)g(a)g(graph)g(whic)n(h)g(is)g(dynamically)g(c)n(hanged)
523 3296 y(b)n(y)33 b(the)i(individual)e(no)r(des)h(in)n(teracting)f
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3396 y(putation)26 b(step)h(resem)n(bles)e(a)h(parallel)f(rewrite)g
(step)i(in)f(a)g(graph)f(grammar)f(deriv)-5 b(ation.)523
3495 y(In)26 b(this)h(mo)r(del)f(all)g(no)r(des)f(are)g(activ)n(e)h(in)
g(ev)n(ery)f(computation)g(step;)i(if)f(their)g(neigh)n(b)r(or-)523
3595 y(ho)r(o)r(d)i(matc)n(hes)g(the)h(pattern)f(required)g(b)n(y)g
(the)h(instruction)f(the)h(no)r(de)f(will)h(transform)523
3694 y(its)j(en)n(vironmen)n(t.)f(The)h(Hardw)n(are)d(Mo)r
(di\014cation)j(Mac)n(hine)f(\()p FG(HMM)14 b FK(\))33
b(in)n(tro)r(duced)e(b)n(y)523 3794 y(Dymond)23 b(and)f(Co)r(ok)g([6])g
(b)r(eha)n(v)n(es)g(in)h(a)f(similar)g(w)n(a)n(y)-7 b(.)21
b(This)i(mo)r(del)g(indeed)g(has)f(b)r(een)h(in-)523
3894 y(v)n(estigated)e(for)g(its)h(complexit)n(y)f(b)r(eha)n(vior.)g(F)
-7 b(rom)21 b(Lam)g(and)h(Ruzzo)f([11])g(it)i(follo)n(ws)d(that)523
3993 y(the)k(mac)n(hine)f(is)h(equiv)-5 b(alen)n(t)23
b(with)h(constan)n(t)f(factor)g(time)h(o)n(v)n(erheads)d(with)j(a)f
(restricted)523 4093 y(v)n(ersion)e(of)h(the)g FG(P)o
FD(\000)-9 b FG(RAM)36 b FK(of)22 b(F)-7 b(ortune)22
b(and)g(Wyllie.)h(F)-7 b(rom)21 b(this)i(result)f(one)g(can)f(observ)n
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b(another)i(example)g(of)g(the)g(class)g(of)g(devices)g(whic)n(h)g(are)
523 4292 y(lo)r(cated)25 b(b)r(ey)n(ond)g(the)h(second)f(mac)n(hine)g
(class)f(-)h(its)h(nondeterministic)f(v)n(ersion)f(accepts)523
4392 y FG(NEXPTIME)39 b FK(in)28 b(p)r(olynomial)f(time.)606
4491 y(The)19 b(computational)g(p)r(o)n(w)n(er)f(of)h(our)f
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FG(r)l(everse)43 b FK(order.)35 b(By)h(using)h(rev)n(erse)d
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f(asso)r(ciated)f(with)523 4790 y FC(x)28 b FK(b)n(y)g(p)r(oin)n(ting)g
(to)f FC(x)i FK(\(hence)f(the)g(name)1819 4760 y FB(2)1856
4790 y FK(\).)g(More)f(than)h(one)g(no)r(de)g(can)f(b)r(e)h(reac)n(hed)
f(on)523 4890 y(a)j(path)g(b)n(y)g(tra)n(v)n(ersing)d(p)r(oin)n(ters)i
(in)i(the)f(rev)n(erse)e(direction.)i(Note)g(that)g(at)g(this)h(p)r
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b(than)h(the)h(older)523 5089 y FG(KUM)h FK(mo)r(del;)21
b(in)g(an)g(undirected)g(graph)e(tra)n(v)n(ersing)f(p)r(oin)n(ters)i
(in)h(the)g(rev)n(erse)e(direction)p 523 5209 499 5 v
627 5291 a Fm(2)661 5323 y FF(compare)27 b(with)f Fg(c)l(ontent-addr)l
(essable)32 b(asso)l(ciative)d(memory)p eop end
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TeXDict begin 51 60 bop 2536 299 a FF(5.1.)53 b(In)n(tro)r(duction)153
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(accesses)g(the)h(storage)e(struc-)523 706 y(ture)e(b)n(y)g(means)f(of)
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b(used)g(only)f(in)h(t)n(w)n(o)f(t)n(yp)r(es)h(of)g(instructions:)f
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b(The)523 905 y(\014rst)h(argumen)n(t)f(of)h(the)h(ab)r(o)n(v)n(e)d(t)n
(w)n(o)i(instructions)f(is)h(a)g(path)g(whic)n(h)h(no)n(w)e(ma)n(y)g
(con)n(tain)523 1005 y(rev)n(erse)e(p)r(oin)n(ters.)i(This)g(path)g
(therefore)f(no)h(longer)f(denotes)h(a)g(single)f(no)r(de)h(but)h(a)f
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(e)h(empt)n(y\).)g(The)g(action)f(describ)r(ed)g(b)n(y)h(the)g
(instruction)523 1204 y(no)n(w)f(will)h(b)r(e)g(p)r(erformed)g(for)f
(all)h(no)r(des)f(in)h(this)g(set)g(in)g(parallel.)f(The)h(second)f
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b(is)g(required)f(to)g(b)r(e)h(a)g(path)g(consisting)f(of)g(forw)n(ard)
f(p)r(oin)n(ters)523 1403 y(only;)25 b(it)g(therefore)f(alw)n(a)n(ys)f
(denotes)h(a)h(single)f(no)r(de.)h(Therefore)f(the)h(action)f(p)r
(erformed)523 1503 y(b)n(y)j(the)h(t)n(w)n(o)f(instructions)g(ab)r(o)n
(v)n(e)g(is)g(deterministic.)606 1602 y(Our)38 b(mo)r(del)h(ma)n(y)f(b)
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(of)g(sequen)n(tial)523 1702 y(mac)n(hines)23 b(whic)n(h)h(op)r(erate)f
(on)g(large)g(ob)5 b(jects)23 b(in)h(unit)g(time)g(and)g(obtain)g
(their)f(p)r(o)n(w)n(er)g(of)523 1802 y(parallelism)28
b(thereof.)h(Other)g(mo)r(dels)g(of)g(this)g(c)n(haracter)f(are)g(the)h
(v)n(ector)f(mac)n(hines)h(of)523 1901 y(Pratt)d(and)g(Sto)r(c)n(kmey)n
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b(b)n(y)g(Hartmanis)g(and)g(Simon)h([9])523 2001 y(and)40
b(simpli\014ed)h(b)n(y)g(Bertoni)f(et)g(al.)h([4)o(],)g(and)f(also)g
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2100 y(Stegw)n(ee)27 b(et)h(al.)f([18)o(,)h(22)o(].)606
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2300 y(designing)19 b(a)h(parallel)f(v)n(ersion)g(of)h(the)h
FG(SMM)34 b FK(mo)r(del.)21 b(In)f(the)h(conclusion)e(of)h(this)h(c)n
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(suggested)h(b)n(y)g(an)h(anon)n(ymous)e(referee.)606
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2599 y(b)n(y)i(the)h FG(ASMM)43 b FK(mo)r(del)28 b(b)n(y)h
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b(time)h(b)n(y)f(nondeterministic)g FG(ASMM)32 b FK(devices)18
b(is)h(denoted)f(b)n(y)g FG(ASMM)5 b FD(\000)-9 b FG(NPTIME)10
b FK(.)523 2798 y(The)33 b(class)e FG(PSP)-6 b(A)n(CE)43
b FK(as)32 b(indicated)h(ab)r(o)n(v)n(e,)e(denotes)i(the)g(class)e(of)i
(languages)e(recog-)523 2897 y(nized)c(in)g(p)r(olynomial)f(space)g(on)
g(a)g(T)-7 b(uring)26 b(mac)n(hine.)h(The)f(fact)h(that)g(the)g
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(equalit)n(y:)606 3097 y FG(ASMM)5 b FD(\000)-9 b FG(PTIME)34
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(equalit)n(y)f(w)n(e)g(use)h(the)g(w)n(ell)f(kno)n(wn)g
FG(PSP)-6 b(A)n(CE)10 b FK(-complete)25 b(prob-)523 3296
y(lem:)606 3396 y FG(QUANTIFIED)k(BOOLEAN)g(F)n(ORMULAS)e
FK(\()p FG(QBF)12 b FK(\))28 b([19)o(])f(:)523 3562 y
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FC(x)2517 3574 y FB(1)2555 3562 y FC(;)i(:)g(:)g(:)g(;)g(x)2787
3574 y FA(n)2832 3562 y FK(\)],)523 3661 y(where)27 b(eac)n(h)g
FC(Q)1016 3673 y FA(i)1071 3661 y FK(equals)g FD(8)g
FK(or)g FD(9)p FK(,)h(and)f(where)h FC(P)12 b FK(\()p
FC(x)2141 3673 y FB(1)2178 3661 y FC(;)i(:)g(:)g(:)g(;)g(x)2410
3673 y FA(n)2456 3661 y FK(\))523 3761 y(is)28 b(a)f(prop)r(ositional)f
(form)n(ula)g(in)i(the)g(b)r(o)r(olean)f(v)-5 b(ariables)27
b FC(x)2422 3773 y FB(1)2459 3761 y FC(;)14 b(:)g(:)g(:)g(;)g(x)2691
3773 y FA(n)2736 3761 y FK(.)523 3960 y FG(QUESTION)p
FK(:)27 b(do)r(es)g(this)h(form)n(ula)f(ev)-5 b(aluate)27
b(to)g FG(true)6 b FK(?)523 4367 y FL(5.2)116 b(The)38
b Fd(SMM)58 b FL(and)38 b(the)h Fd(ASMM)57 b FL(mo)s(dels)523
4566 y FK(Our)21 b FG(ASMM)36 b FK(mo)r(del)23 b(is)f(based)f(on)h(the)
h(Storage)d(Mo)r(di\014cation)i(Mac)n(hine)g(as)f(in)n(tro)r(duced)523
4666 y(b)n(y)26 b(Sc)n(h\177)-42 b(onhage)25 b(in)i(1970)d([16].)i(The)
h FG(SMM)40 b FK(mo)r(del)27 b(resem)n(bles)e(the)i FG(RAM)40
b FK(mo)r(del)27 b(as)e(far)523 4765 y(as)h(it)g(has)g(a)g(stored)g
(program)e(and)i(a)g(similar)g(\015o)n(w)f(of)h(con)n(trol.)g(It)g(has)
g(a)g(single)g(storage)523 4865 y(structure,)19 b(called)g(a)h(\001)p
FG(-structur)l(e)6 b FK(.)18 b(Here)h(\001)h(denotes)f(a)h(\014nite)g
(alphab)r(et)g(consisting)e(of)i(at)523 4965 y(least)f(t)n(w)n(o)f(sym)
n(b)r(ols.)g(W)-7 b(e)19 b(denote)g(the)g(rev)n(erse)e(of)i(a)g
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b FC(a)p FK(.)19 b(F)-7 b(urthermore,)537 5043 y(\026)523
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FD(j)p FC(\013)23 b FD(2)h FK(\001)p FD(g)j FK(is)h(the)g(set)f(of)h
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y(~)2625 5064 y(\001)c(=)g(\001)18 b FD([)2980 5043 y
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FC(X)47 b FK(is)40 b(a)g(\014nite)h(directed)f(graph)f(eac)n(h)h(no)r
(de)g(of)g(whic)n(h)g(has)g FC(k)47 b FK(=)523 5264 y
FD(j)p FK(\001)p FD(j)31 b FK(outgoing)f(edges)g(whic)n(h)h(are)f(lab)r
(eled)h(b)n(y)f(the)i FC(k)h FK(elemen)n(ts)e(of)g(\001.)g(In)g(Sc)n
(h\177)-42 b(onhage's)523 5363 y(formalization,)24 b(a)h
(\001-structure)g(is)g(a)g(triple)g(\()p FC(X)r(;)14
b(c;)g(p)p FK(\),)25 b(where)g FC(X)31 b FK(denotes)25
b(the)h(\014nite)g(set)p eop end
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523 1104 y(b)n(y)j(tracing)f(a)g(w)n(ord)g FC(w)k FK(in)d(\001)1458
1074 y Fw(\003)1526 1104 y FK(starting)f(from)h(the)g(cen)n(ter)f
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1403 y(no)d(longer)g(can)g(b)r(e)h(disregarded)e(as)g(b)r(eing)i
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(hanging)f(\001-structure,)523 1602 y(whic)n(h)28 b(initially)f
(consists)g(of)h(a)f(single)g(no)r(de,)h(the)g(cen)n(ter.)f(The)h
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523 2100 y(instructions.)22 b(Consequen)n(tly)f(w)n(e)h(only)g
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2300 y(a)g(nondeterministic)h(manner,)g(but)g(suc)n(h)f(instructions)h
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(separate)e(lab)r(els)i(and)f(instructions)h(b)n(y)f(a)h(colon,)f
(whereas)523 2599 y(instructions)27 b(are)g(ended)h(b)n(y)f
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FK(0)p FC(;)14 b FK(1)p FD(g)p FK(\))523 2947 y FG(input)54
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2959 y FB(1)972 2947 y FK(;)523 3047 y FG(output)53 b
FC(\014)t FK(;)523 3147 y FG(goto)f FC(\025)p FK(;)523
3246 y FG(halt)9 b FK(;)523 3495 y(The)35 b FG(input)42
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FK(and)34 b(transfers)f(con)n(trol)h(to)g FC(\025)3112
3507 y FA(\014)3157 3495 y FK(.)h(The)523 3595 y(other)27
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3794 y(-)f(in)h(this)g(case)e(a)h(\001-structure)g FC(X)7
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FG(ASMM)47 b FK(the)33 b(single)f(argumen)n(t)g(of)h
FG(new)42 b FK(and)33 b(the)g(\014rst)523 3993 y(argumen)n(t)26
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3972 y(~)1830 3993 y(\001;)i(the)g(other)f(argumen)n(ts)f(\(second)h
(argumen)n(t)523 4093 y(of)35 b FG(set)79 b(to)40 b FK(and)35
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eop end
%%Page: 53 63
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b FF(\026)-42 b Fo(x)34 b FF(with)h Fo(v)j FF(a)d(v)l(ariable)523
5190 y(name,)27 b(it)f(is)h(desirable)h(for)f Fo(v)i
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(the)g FG(if)50 b FK(statemen)n(t.)32 b(Since)f(b)r(oth)h(paths)f(in)g
(this)523 1270 y(instruction)e(consist)g(of)g(forw)n(ard)e(p)r(oin)n
(ters)h(only)-7 b(,)29 b(the)h(meaning)e(of)i(this)f(instruction)g(is)
523 1370 y(equal)e(for)g(the)h FG(SMM)42 b FK(and)27
b(the)h FG(ASMM)14 b FK(.)606 1469 y(The)31 b FG(time)h(c)l(omplexity)
39 b FK(w)n(e)30 b(use)h(is)f(simply)h(the)g(n)n(um)n(b)r(er)f(of)h
(instructions)f(executed.)523 1569 y(W)-7 b(e)31 b(do)f(not)g(concern)f
(ourselv)n(es)g(with)h(the)h FG(sp)l(ac)l(e)i(c)l(omplexity)7
b FK(;)32 b(see)e([12)o(,)g(21)o(])g(for)g(a)g(dis-)523
1669 y(cussion)d(of)g(the)h(space)f(complexit)n(y)g(of)h(the)g
FG(SMM)14 b FK(.)523 1976 y FL(5.3)116 b(An)38 b(illustration)d(of)k
(the)f(p)s(o)m(w)m(er)g(of)g(asso)s(ciativit)m(y)523
2175 y FK(W)-7 b(e)29 b(demonstrate)g(the)g(p)r(o)n(w)n(er)f(of)h(the)g
FG(ASMM)43 b FK(mo)r(del)29 b(b)n(y)g(sho)n(wing)e(the)j(capabilit)n(y)
e(to)523 2275 y(manipulate)g(arbitrarily)d(large)h(sets)i(in)g(constan)
n(t)f(time.)606 2374 y(The)41 b(mo)r(del)f(allo)n(ws)f(the)i(follo)n
(wing)e(natural)h(represen)n(tation)f(of)h(sets.)h(If)f
FC(W)53 b FK(is)40 b(a)523 2474 y(w)n(ord)f(o)n(v)n(er)g(\001,)h(and)g
FC(\013)45 b FD(2)f FK(\001)d(a)e(direction,)h(then)h
FC(P)12 b FK(\()p FC(W)20 b FK(\026)-50 b FC(\013)p FK(\))41
b(is)f(the)h(set)f(of)g(all)g(no)r(des)523 2574 y(ha)n(ving)28
b(their)h FC(\013)p FK(-p)r(oin)n(ter)g(directed)g(to)g(the)g(no)r(de)g
FC(P)12 b FK(\()p FC(W)g FK(\).)30 b(Assume)f(that)g(our)g(alphab)r(et)
523 2673 y(is)d(\001)d(=)f FD(f)p FC(A;)14 b(B)t(;)g(C)q(;)g(\013;)g
(\014)t(;)g(\015)5 b FD(g)26 b FK(and)f(that)h(the)g
FC(A)p FK(,)g FC(B)t FK(,)g(and)g FC(C)6 b FK(-p)r(oin)n(ters)24
b(from)i(the)g(cen)n(ter)f(go)523 2773 y(to)f(three)f(di\013eren)n(t)h
(no)r(des)f FC(P)12 b FK(\()p FC(A)p FK(\),)24 b FC(P)12
b FK(\()p FC(B)t FK(\))25 b(and)e FC(P)12 b FK(\()p FC(C)6
b FK(\),)24 b(none)g(of)f(whic)n(h)h(is)f(the)h(cen)n(ter.)f(W)-7
b(e)523 2873 y(will)21 b(no)n(w)f(consider)g(the)h(sets)f
FC(P)12 b FK(\()p FC(A)c FK(\026)-50 b FC(\013)q FK(\),)21
b FC(P)12 b FK(\()p FC(B)1912 2851 y FK(\026)1900 2873
y FC(\014)5 b FK(\))21 b(and)f FC(P)12 b FK(\()p FC(C)q(;)18
b FK(\026)-46 b FC(\015)6 b FK(\))21 b(and)f(see)h(ho)n(w)f(the)h
(standard)523 2972 y(set)26 b(op)r(erators)e(can)i(b)r(e)g(applied)g
(to)g(them)g(b)n(y)g(using)g(appropriate)e FG(set)54
b(to)31 b FK(instructions.)523 3072 y(W)-7 b(e)38 b(ha)n(v)n(e)e(c)n
(hosen)h FC(A)p FK(,)h FC(B)k FK(and)37 b FC(C)44 b FK(to)38
b(b)r(e)g(directions)e(so)h(that)h(the)g(instructions)f(with)523
3171 y(whic)n(h)29 b(w)n(e)f(will)h(implemen)n(t)g(the)h(set)e(op)r
(erators)f(cannot)h(a\013ect)h(the)g(addressing)f(of)g(the)523
3271 y(no)r(des)33 b FC(P)12 b FK(\()p FC(A)p FK(\),)34
b FC(P)12 b FK(\()p FC(B)t FK(\))34 b(and)f FC(P)12 b
FK(\()p FC(C)6 b FK(\).)34 b(As)g(long)e(as)h(no)g(suc)n(h)g(in)n
(terference)f(exists,)h(w)n(e)g(can)523 3371 y(generalize)24
b(to)i(the)g(case)f(where)g FC(A)p FK(,)i FC(B)j FK(and)25
b FC(C)33 b FK(are)24 b(not)i(elemen)n(ts)g(of)g(\001)g(but)g(w)n(ords)
f(o)n(v)n(er)523 3470 y(\001.)606 3570 y(The)30 b(instruction)g
FG(set)63 b FC(A)8 b FK(\026)-50 b FC(\013)q(\014)59
b FG(to)h FC(B)t FK(;)31 b(has)e(the)i(e\013ect)g(of)f(adding)g(to)g
FC(P)12 b FK(\()p FC(B)3008 3548 y FK(\026)2996 3570
y FC(\014)5 b FK(\))30 b(the)h(set)523 3670 y FC(P)12
b FK(\()p FC(A)c FK(\026)-50 b FC(\013)q FK(\),)30 b(while)h(the)f
(instruction)g FG(set)63 b FC(A)8 b FK(\026)-50 b FC(\013\014)59
b FG(to)h FC(\017)p FK(;)30 b(remo)n(v)n(es)e(from)i
FC(P)12 b FK(\()p FC(B)2904 3648 y FK(\026)2892 3670
y FC(\014)5 b FK(\))30 b(the)h(no)r(des)523 3769 y(whic)n(h)d(are)e
(also)h(in)h FC(P)12 b FK(\()p FC(A)c FK(\026)-50 b FC(\013)p
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(ho)n(w)h(the)g(standard)f(set)h(op)r(erators,)e(sho)n(wn)h(as)h(as-)
523 3968 y(signmen)n(t)39 b(statemen)n(ts)f(in)i(the)f(b)r(o)n(xes,)f
(can)h(b)r(e)g(implemen)n(ted)h(in)f(terms)g(of)g FG(set)91
b(to)523 4068 y FK(instructions.)27 b(The)h(cen)n(ter)f
FC(\017)g FK(is)h(used)f(to)h(direct)f(p)r(oin)n(ters)g(a)n(w)n(a)n(y)f
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b FC(\017)1011 2106 y FE(\033)p 1011 2237 4 17 v 1011
2355 a(\032)1634 2106 y(\030)p 1634 2237 V 1634 2355
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a(\031)p 2374 2355 391 4 v 2374 2106 V 2281 2256 a FC(A)8
b FK(\026)-50 b FC(\013)24 b FK(:=)f FC(A)8 b FK(\026)-50
b FC(\013)19 b FD(\\)g FC(B)2818 2234 y FK(\026)2806
2256 y FC(\014)606 2582 y FK(The)h(follo)n(wing)e(program)g
(illustrates)g(ho)n(w)h(in)h(linear)f(time)h(a)f(set)g
FC(P)12 b FK(\()c(\026)-50 b FC(\013)q FK(\))20 b(of)f(exp)r(onen)n
(tial)523 2682 y(size)27 b(can)h(b)r(e)g(constructed)f(\(with)h(a)f
(singleton)g(alphab)r(et\):)645 2848 y FG(new)64 b FK(\026)-51
b FC(\013)q FK(;)645 2947 y FG(set)62 b FK(\026)-50 b
FC(\013)9 b FK(\026)-51 b FC(\013)47 b FG(to)k FC(\017)p
FK(;)645 3036 y(.)645 3069 y(.)645 3102 y(.)645 3202
y FG(new)64 b FK(\026)-51 b FC(\013)q FK(;)645 3301 y
FG(set)62 b FK(\026)-50 b FC(\013)9 b FK(\026)-51 b FC(\013)47
b FG(to)k FC(\017)p FK(;)606 3467 y(Initially)32 b(only)g(the)h(cen)n
(ter)f(exists,)g(so)g(all)g(no)r(des)g(p)r(oin)n(t)g(to)h(the)f(cen)n
(ter.)g(If)h(at)f(some)523 3567 y(p)r(oin)n(t)37 b(2)791
3537 y FA(k)867 3567 y FK(no)r(des)g(exist,)f(all)g(of)h(whic)n(h)f(p)r
(oin)n(t)g(to)h(the)g(cen)n(ter,)e(then)i(after)f(the)h
FG(new)523 3667 y FK(instruction,)22 b(eac)n(h)e(of)i(these)f(2)1478
3637 y FA(k)1540 3667 y FK(no)r(des)h(no)n(w)f(p)r(oin)n(ts)g(to)h(one)
f(of)h(2)2551 3637 y FA(k)2613 3667 y FK(newly)f(created)g(no)r(des,)
523 3766 y(whic)n(h)h(again)g(p)r(oin)n(t)g(to)g(the)h(cen)n(ter.)f
(Next)h(the)f FG(set)77 b FK(instruction)22 b(mak)n(es)f(all)h(2)3005
3736 y FA(k)q FB(+1)3152 3766 y FK(no)r(des)523 3866
y(p)r(oin)n(t)27 b(to)f(the)g(cen)n(ter.)g(Hence)h(after)f
FC(k)j FK(rep)r(etitions)d(of)h(these)f(t)n(w)n(o)f(instructions)h(the)
h(size)523 3966 y(of)h(the)g(set)f FC(P)12 b FK(\()c(\026)-50
b FC(\013)q FK(\))28 b(has)f(b)r(ecome)g(2)1588 3935
y FA(k)1629 3966 y FK(.)606 4065 y(In)h(the)g(next)f(section)g(w)n(e)h
(will)f(see)g(ho)n(w)g(these)h(and)f(similar)g(constructions)f(are)h
(used)523 4165 y(to)h(pro)r(cess)e(large)g(amoun)n(ts)h(of)h(data)f(in)
g(parallel.)523 4472 y FL(5.4)116 b Fd(PSP)-9 b(A)m(CE)48
b FL(=)33 b Fd(ASMM)5 b Fr(\000)-13 b FL(\()p Fd(N)19
b FL(\))p Fd(PTIME)523 4671 y FK(The)25 b(pro)r(of)f(of)h(mem)n(b)r
(ership)g(in)g(the)g(Second)g(Mac)n(hine)f(Class)g(is)h(usually)g
(split)g(in)n(to)g(t)n(w)n(o)523 4771 y(parts:)523 4937
y Fv(Lemma)32 b FH(5.1)40 b FG(PSP)-6 b(A)n(CE)34 b FD(\022)22
b FG(ASMM)5 b FD(\000)-9 b FG(PTIME)606 5103 y FK(W)i(e)31
b(pro)n(v)n(e)d(this)j(b)n(y)e(sk)n(etc)n(hing)h(an)f
FG(ASMM)45 b FK(whic)n(h)30 b(solv)n(es)e(the)j FG(PSP)-6
b(A)n(CE)10 b FK(-complete)523 5203 y(problem)27 b FG(QBF)39
b FK(in)28 b(p)r(olynomial)f(time.)523 5369 y Fv(Lemma)32
b FH(5.2)40 b FG(ASMM)5 b FD(\000)-9 b FG(NPTIME)34 b
FD(\022)22 b FG(PSP)-6 b(A)n(CE)p eop end
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b(A)n(CE)30 b FF(=)21 b Fg(ASMM)s Fp(\000)-9 b FF(\()p
Fg(N)12 b FF(\))p Fg(PTIME)162 b FF(56)523 1531 y @beginspecial
0 @llx 0 @lly 540 @urx 211 @ury 3401 @rwi @setspecial
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b(e)41 b(pro)n(v)n(e)d(this)j(b)n(y)f(sho)n(wing)f(ho)n(w)g(to)h(sim)n
(ulate)g FC(t)h FK(steps)f(of)g(a)g(nondeterministic)523
2069 y FG(ASMM)i FK(on)27 b(a)g(T)-7 b(uring)27 b(mac)n(hine)g(using)h
FC(O)r FK(\()p FC(t)1946 2038 y FB(2)1984 2069 y FK(\))g(space.)523
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b Fl(TIME)13 b Fb(\()p Fh(n)1916 2282 y Fa(2)1955 2318
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b(p)r(olynomial)f(time)i(pro)r(ceeds)523 2567 y(in)33
b(8)e(stages.)h(Let)g FC(X)37 b FK(=)31 b FD(f)p FC(x)1416
2579 y FB(0)1453 2567 y FC(;)14 b(:)g(:)g(:)f(;)h(x)1684
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FC(c)g FK(b)r(e)h(the)g(cen)n(ter.)e(Basically)-7 b(,)35
b(the)i(algorithm)523 2766 y(expands)29 b(the)i(form)n(ula)d(b)n(y)i
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b FD(\))24 b FC(F)12 b FK(\(0\))18 b FD(_)h FC(F)12 b
FK(\(1\))p FC(:)523 3264 y FK(The)32 b(resulting,)f(fully)h(expanded)f
(form)n(ula,)f(can)h(b)r(e)h(view)n(ed)f(as)g(a)g(tree.)g(It)h
(consists)f(of)523 3364 y(a)e(complete)g(binary)g(tree)g
FC(T)41 b FK(of)29 b(depth)h FC(k)s FK(,)f(with)h(an)f(instance)g(of)h
(the)f(form)n(ula)g(b)r(o)r(dy)h FC(B)523 3463 y FK(ro)r(oted)f(at)g
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b(assigned)f(to)h(them)h(along)d(the)j(path)f(from)g(the)h(ro)r(ot)e
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f(than)h(to)f(build)h(and)g(ev)-5 b(aluate)27 b(this)h(tree.)606
3762 y(Figure)e(5.1)g(depicts)g(the)h(structure)f(built)i(for)e(the)h
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FC(x)3248 3774 y FB(0)3302 3762 y FD(^)523 3862 y FC(x)570
3874 y FB(1)608 3862 y FK(.)k(The)f(part)h(on)f(the)h(righ)n(t)f
(represen)n(ts)g(the)h(expanded)f(form)n(ula)g(\(where)g(some)h(of)f
(the)523 3961 y FC(x)p FK(-p)r(oin)n(ters)33 b(ha)n(v)n(e)f(b)r(een)h
(omitted)h(for)f(clarit)n(y\).)f(W)-7 b(e)34 b(no)n(w)e(brie\015y)h
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FC(c;)14 b(x)1654 4256 y FB(0)1692 4244 y FC(;)g(x)1776
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b FK(link)n(ed)g(through)f(the)i(0-p)r(oin)n(ter.)731
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b FK(b)n(y)g(their)f(name,)h(and)g(use)523 1851 y(v)-5
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2051 y(linearly)21 b(sized)h(input)h(represen)n(tation)e(\(on)h(the)g
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(are)g(connected)h(only)g(b)n(y)f(the)523 2449 y FC(x)p
FK(-p)r(oin)n(ters)35 b(that)g(go)f(from)h(a)f(no)r(de)h(in)g(some)g
FC(C)6 b FK(\()p FC(v)s FK(\))36 b(to)f(either)g FC(v)j
FK(or)c(to)h(the)g(cen)n(ter)g FC(c)p FK(.)523 2549 y(Th)n(us)27
b(the)h(follo)n(wing)f(in)n(v)-5 b(arian)n(t)26 b(holds)i(throughout)e
(the)i(execution:)1483 2731 y FD(8)p FC(v)e FD(2)e FC(X)g
FD([)19 b FC(B)27 b FK(:)46 b FC(v)8 b FK(\026)-47 b
FC(x)24 b FD(\022)f FC(C)6 b FK(\()p FC(v)s FK(\))p FC(:)606
2914 y FK(F)-7 b(urthermore,)33 b(the)i FC(x)p FK(-p)r(oin)n(ter)f
(from)g FC(r)r FK(,)h(the)g(ro)r(ot)f(of)g FC(T)12 b
FK(,)34 b(remains)f(anc)n(hored)g(to)h FC(x)3320 2926
y FB(0)523 3014 y FK(un)n(til)28 b(ev)-5 b(aluation)27
b(is)h(completed.)606 3113 y(F)-7 b(or)27 b(a)g(clear)g(understanding)g
(of)g(the)h(construction,)f(it)h(is)g(imp)r(ortan)n(t)f(to)h
(distinguish)523 3213 y(b)r(et)n(w)n(een)43 b(truth-v)-5
b(alues)43 b(and)g(their)g(represen)n(tation.)f(Conceptually)-7
b(,)42 b(the)i(algorithm)523 3313 y(w)n(orks)36 b(with)h(truth)h(v)-5
b(alues,)37 b(1)f(\(true\))i(and)f(0)g(\(false\).)g(The)h(lea)n(v)n(es)
d(of)i(the)h(2)3063 3282 y FA(k)3140 3313 y FK(copies)523
3412 y(of)f FC(B)42 b FK(are)37 b(assigned)f(truth)i(v)-5
b(alues)37 b(in)h(the)g(ob)n(vious)e(w)n(a)n(y)g(according)g(to)h(whic)
n(h)h(v)-5 b(ari-)523 3512 y(able)30 b(they)g(represen)n(t.)f(The)h
(other)f(no)r(des)h(are)f(assigned)g(default)i(truth)f(v)-5
b(alues,)30 b(whic)n(h)523 3611 y(are)g(0)g(for)g(a)h(disjunction,)g
(and)g(1)f(for)g(a)g(conjunction)h(or)f(a)g(negation)g(\(recall)g(that)
h(the)523 3711 y(quan)n(ti\014ers)h(ha)n(v)n(e)f(b)r(een)j(transformed)
d(in)n(to)i(disjunction)g(and)g(conjunction\).)g(Next,)g(a)523
3811 y(b)r(ottom-up)c(pro)r(cess)e(rep)r(eatedly)h(c)n(hanges)e
(defaults,)j(that)g(are)e(in)h(disagreemen)n(t)f(with)523
3910 y(their)i(ev)-5 b(aluated)29 b(c)n(hildren,)f(in)n(to)h(the)g
(correct)f(ev)-5 b(aluation.)28 b(In)h(order)f(to)h(facilitate)f(this)
523 4010 y(pro)r(cess,)f(w)n(e)g(use)g(a)g(mixed)h(represen)n(tation)e
(of)i(truth)g(v)-5 b(alues,)27 b(as)g(follo)n(ws.)606
4110 y(Call)h(a)g(no)r(de)h FC(u)1102 4079 y Fw(0)1149
4110 y FD(2)c FC(C)6 b FK(\()p FC(u)p FK(\))29 b FG(active)g
FK(if)g(its)g FC(x)p FK(-p)r(oin)n(ter)f(is)g(directed)h(to)f
FC(u)g FK(or)g FG(p)l(assive)i FK(if)f(its)523 4209 y
FC(x)p FK(-p)r(oin)n(ter)j(is)g(directed)g(to)g FC(c)p
FK(.)g(A)h(truth)f(v)-5 b(alue)32 b(is)g(represen)n(ted)f(b)n(y)h(the)h
(activ)n(e)e(state)h(i\013)523 4309 y(that)d(truth)h(v)-5
b(alue)29 b(disagrees)e(with)i(the)h(default)f(of)g(its)g(paren)n(t,)f
(and)h(with)h(the)f(passiv)n(e)523 4408 y(state)e(otherwise.)g(T)-7
b(o)27 b(summarize:)p 1394 4487 1094 5 v 1392 4587 5
100 v 1443 4557 a(paren)n(t)g(t)n(yp)r(e)p 2003 4587
V 2020 4587 V 209 w FD(^)p 2175 4587 V 100 w(_)p 2330
4587 V 100 w(:)p 2485 4587 V 1394 4591 1094 5 v 1394
4608 V 1392 4707 5 100 v 1443 4677 a FK(paren)n(t)g(default)p
2003 4707 V 2020 4707 V 124 w(1)p 2175 4707 V 113 w(0)p
2330 4707 V 113 w(1)p 2485 4707 V 1394 4711 1094 5 v
1392 4811 5 100 v 1443 4781 a(activ)n(e)g(v)-5 b(alue)p
2003 4811 V 2020 4811 V 209 w(0)p 2175 4811 V 113 w(1)p
2330 4811 V 113 w(1)p 2485 4811 V 1394 4815 1094 5 v
606 4952 a(As)23 b(an)g(example,)f(supp)r(ose)h(a)f FD(:)i
FK(no)r(de)f FC(u)1873 4922 y Fw(0)1918 4952 y FD(2)h
FC(C)6 b FK(\()p FC(u)p FK(\))23 b(has)g(a)f(paren)n(t)g
FD(_)i FK(no)r(de.)f(The)g(paren)n(t)523 5052 y(gets)g(a)f(default)i(v)
-5 b(alue)23 b(of)g(0,)f(whic)n(h)h(is)g(to)g(b)r(e)g(c)n(hanged)f(in)n
(to)h(a)g(1)g(i\013)g(either)g(of)g(its)g(c)n(hildren)523
5152 y(ev)-5 b(aluates)28 b(to)g(1.)f(Th)n(us,)h(1)g(is)g(the)h(activ)n
(e)e(v)-5 b(alue)28 b(for)g FC(u)2239 5122 y Fw(0)2262
5152 y FK(.)g(The)g(v)-5 b(alue)28 b(0)g(is)g(passiv)n(e)f(for)h
FC(u)3312 5122 y Fw(0)3335 5152 y FK(,)523 5251 y(since)g(it)h(agrees)e
(with)i(the)g(default)g(0)f(v)-5 b(alue)28 b(of)h(its)g
FD(_)f FK(paren)n(t.)g(Since)h(a)f FD(:)h FK(no)r(de)g(is)f(1)g(b)n(y)
523 5351 y(default,)g FC(u)873 5321 y Fw(0)924 5351 y
FK(is)f(activ)n(e)g(b)n(y)g(default,)h(hence)g(its)g
FC(x)p FK(-p)r(oin)n(ter)f(is)h(initially)f(directed)h(to)f
FC(u)p FK(.)p eop end
%%Page: 58 68
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b(at)g(a)g(no)r(de)g FC(u)g FK(therefore)g(dep)r(ends)g(on)g(the)523
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(This)523 706 y(holds)27 b(also)f(for)h(the)h(no)r(des)f(in)g(the)h
(tree)f FC(T)38 b FK(whic)n(h)27 b(are)g(asso)r(ciated)f(to)h(the)h(v)
-5 b(ariables)26 b FC(x)3307 718 y FA(i)3335 706 y FK(.)523
805 y(In)h(this)g(tree)f(the)h(copies)f(of)h(the)g(v)-5
b(ariables)25 b(ha)n(v)n(e)h(b)r(een)h(treated)f(as)g(logical)g
(connectiv)n(es)523 905 y(according)39 b(to)h(the)h(t)n(yp)r(e)f(of)h
(the)g(quan)n(ti\014er)e(binding)i(this)f(v)-5 b(ariable.)40
b(Note)g(that)h(b)n(y)523 1005 y(k)n(eeping)27 b(the)h
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FC(x)1956 1017 y FB(0)1994 1005 y FK(,)h(it)g(is)f(made)g(activ)n(e)g
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f(during)523 1204 y(stage)i(4)h(the)h(truth)f(v)-5 b(alues)24
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(the)f(copies)g(of)523 1303 y FC(B)t FK(,)k(and)g(in)h(stages)e(5)g
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(default)f(v)-5 b(alues.)523 1403 y(During)27 b(the)h(\014nal)g(t)n(w)n
(o)f(stages)f(the)i(en)n(tire)f(tree)h(is)f(ev)-5 b(aluated.)606
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(linearly)f(sized)h(list)523 1702 y(and)c(tree)f(represen)n(ting)g(the)
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f(no)r(de)h FC(u)c FD(2)g FC(X)g FD([)18 b FC(B)523 1802
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523 1901 y(these)27 b(four)f(sym)n(b)r(ols)g(will)i(also)d(b)r(e)j
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523 2001 y(of)36 b FC(B)k FK(are)35 b(of)h(t)n(yp)r(e)g
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(the)g(appropriate)f FC(x)3307 2013 y FA(i)3335 2001
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FC(x)1466 2112 y FA(i)1534 2100 y FK(ha)n(v)n(e)f FC(ty)s(pe)p
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FC(x)1660 2270 y Fw(0)1660 2321 y FA(i)1688 2300 y FC(s)p
FK(,)g(the)g(algorithm)f(needs)h(to)g(b)r(e)g(able)g(to)g(detect)h(its)
523 2399 y(end.)h(Since)g(the)g(no)r(des)f(in)h FC(B)k
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(their)f(c)n(hildren)523 2499 y(\(or)23 b(single)g(c)n(hild)g(in)h
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FC(x)2317 2511 y FA(i)2368 2499 y FK(direct)g(their)f(1-p)r(oin)n(ter)f
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f FC(v)s(x)i FK(with)g FC(\017)f FK(can)g(tell)g(whether)h
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(build)h(an)e(exp)r(onen)n(tially)523 2897 y(large)k(tree)h(in)h
(linear)f(time.)h(This)g(is)f(ac)n(hiev)n(ed)g(b)n(y)g(the)h(piece)g
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3362 y FC(\025)24 b FK(:)j FG(new)55 b FC(v)8 b FK(\026)-47
b FC(x)q FK(0;)1212 b(0-c)n(hildren)27 b(for)g FC(C)6
b FK(\()p FC(x)2865 3374 y FA(i)2893 3362 y FK(\))645
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FC(C)6 b FK(\()p FC(x)2736 3474 y FA(i)p FB(+1)2849 3462
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FK(0;)955 b(classify)27 b(in)g FC(C)6 b FK(\()p FC(v)s
FK(0\))523 4799 y FC(\025)571 4811 y FB(1)609 4799 y
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FK(1;)955 b(classify)27 b(in)g FC(C)6 b FK(\()p FC(v)s
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eop end
%%Page: 61 71
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(ts)h(to)f FC(x)i FK(\(unless)f FC(W)35 b FK(=)22 b FC(\017)p
FK(,)h(in)g(whic)n(h)g(case)e FC(p)p FK(\()p FC(x;)14
b(\013;)g(i)p FK(\))24 b(=)523 1660 y FG(c)l(enter)9
b FK([)p FC(i)i FD(\000)g FK(1]\).)23 b(No)n(w)h FC(p)p
FK(\()p FC(x;)14 b(\013;)g(i)p FK(\))23 b(=)g FC(p)p
FK(\()p FC(y)s(;)14 b(\016)o(;)g(i)d FD(\000)g FK(1\),)22
b(where)i FC(\016)j FK(is)d(the)g(direction)g FC(W)36
b FK(ends)23 b(in.)523 1760 y(On)28 b(the)h(other)e(hand,)i(supp)r(ose)
f FC(x)c(<)g FG(no)l(des)7 b FK([)p FC(i)19 b FD(\000)f
FK(1].)28 b(If)h FC(W)36 b FK(=)24 b FC(U)17 b FK(~)-50
b FC(\013)28 b FK(\(i.e.)h FC(\013)p FK(-p)r(oin)n(ters)e(ma)n(y)523
1860 y(ha)n(v)n(e)h(c)n(hanged\))h(and)h FC(Q)p FK(\()p
FC(x;)14 b(W)n(;)g(i)20 b FD(\000)f FK(1\),)29 b(then)i
FC(x)f FK(has)f(generated)f FC(p)p FK(\()p FC(x;)14 b(\013;)g(i)p
FK(\))27 b(=)f FG(no)l(des)8 b FK([)p FC(i)19 b FD(\000)523
1959 y FK(1])f(+)g FD(jf)p FC(y)25 b(<)e(x)p FD(j)p FC(q)s
FK(\()p FC(y)s(;)14 b(W)n(;)g(i)19 b FD(\000)f FK(1\))p
FD(gj)p FK(.)27 b(Otherwise)f FC(p)p FK(\()p FC(x;)14
b(\013;)g(i)p FK(\))24 b(=)f FC(p)p FK(\()p FC(x;)14
b(\013;)g(i)k FD(\000)g FK(1\).)606 2059 y(Second)32
b(and)g(last,)h(consider)e(the)i(case)e FG(instr)9 b
FK([)p FC(i)p FK(])31 b(=)f FG(set)39 b FC(W)k FG(to)36
b FC(V)19 b FK(.)33 b(If)g FC(W)43 b FK(=)30 b FC(U)17
b FK(~)-50 b FC(\013)33 b FK(and)523 2159 y FC(Q)p FK(\()p
FC(x;)14 b(W)n(;)g(i)d FD(\000)g FK(1\),)22 b(then)j
FC(p)p FK(\()p FC(x;)14 b(\013;)g(i)p FK(\))24 b(is)f(the)i(unique)e
FC(y)k FK(satisfying)c FC(P)12 b FK(\()p FC(y)s(;)i(V)5
b(;)14 b(i)d FD(\000)g FK(1\).)22 b(Otherwise)523 2258
y FC(p)p FK(\()p FC(x;)14 b(\013;)g(i)p FK(\))24 b(=)e
FC(p)p FK(\()p FC(x;)14 b(\013;)g(i)19 b FD(\000)f FK(1\).)606
2358 y(These)36 b(functions)g(can)f(easily)g(b)r(e)h(co)r(ded)g(on)f(a)
g(T)-7 b(uring)36 b(Mac)n(hine)f(using)g(recursion)523
2457 y(\(stac)n(k)f(frames\).)g(The)g(recursion)f(depth)i(is)g(b)r
(ounded)f(b)n(y)g FC(ct)p FK(,)h(where)f FC(c)g FK(is)h(a)f(constan)n
(t)523 2557 y(dep)r(ending)j(only)f(on)h(the)g(maxim)n(um)f(path)h
(length)g(of)g(the)g(ASMM)g(program.)e(Eac)n(h)523 2657
y(stac)n(k)26 b(frame)h(holds)g(a)g(return)g(address)f(and)h(some)f(no)
r(de)i(n)n(um)n(b)r(ers)e(and)h(coun)n(ters)g(eac)n(h)523
2756 y(of)j(whic)n(h)f(\014ts)h(in)g FC(t)g FK(bits.)g(T)-7
b(ogether)29 b(with)h(the)g(three)g(arra)n(ys,)d(space)i
FC(O)r FK(\()p FC(t)2874 2726 y FB(2)2912 2756 y FK(\))h(su\016ces)f
(for)523 2856 y(the)f(sim)n(ulation)f(of)h FC(t)f FK(steps)h(of)f(the)h
(ASMM.)523 3163 y FL(5.5)116 b(Conclusion)523 3362 y
FK(Of)39 b(all)f(the)h(parallel)e(mo)r(dels)i(whic)n(h)f(ha)n(v)n(e)f
(b)r(een)i(sho)n(wn)f(to)h(b)r(elong)f(to)g(the)h(Second)523
3462 y(Mac)n(hine)24 b(Class,)f(the)i FG(ASMM)38 b FK(is)25
b(the)f(\014rst)g(to)h(obtain)f(its)g(p)r(o)n(w)n(er)f(from)h(the)h
(use)f(of)g(asso-)523 3562 y(ciativ)n(e)19 b(addressing,)g(th)n(us)h
(making)f(it)h(an)g(in)n(teresting)f(addition)h(to)g(the)g(realm)f(of)h
(Second)523 3661 y(Mac)n(hine)29 b(Class)f(devices.)g(It)h(pro)n(vides)
f(another)g(example)h(that)g(a)f(small)h(mo)r(di\014cation)523
3761 y(of)c(a)g(mac)n(hine)g(mo)r(del)g(can)g(enforce)f(a)h(substan)n
(tial)f(increase)g(in)i(computational)e(p)r(o)n(w)n(er.)523
3861 y(In)29 b([4])g(it)h(w)n(as)e(sho)n(wn)h(that)g(this)g(increase)f
(is)h(pro)n(v)n(ok)n(ed)e(b)n(y)i(adding)g(m)n(ultiplicativ)n(e)g(in-)
523 3960 y(structions)35 b(to)g(the)h(unit-time)g(standard)f
FG(RAM)49 b FK(mo)r(del.)35 b(Similarly)g(the)h FG(EDITRAM)523
4060 y FK(mo)r(del)31 b(obtains)f(its)g(p)r(o)n(w)n(er)f(from)i(in)n
(tro)r(ducing)f(a)g(few)g(edit)h(op)r(erators)e(that)h(are)g(a)n(v)-5
b(ail-)523 4159 y(able)32 b(on)f(most)h(real)f(life)h(text)g(editors)f
(an)n(yho)n(w.)g(In)h(the)g FG(ASMM)46 b FK(mo)r(del)32
b(it)g(turns)g(out)523 4259 y(that)j(tra)n(v)n(ersing)d(p)r(oin)n(ters)
j(in)g(the)g(rev)n(erse)e(direction)h(is)h(all)f(w)n(e)h(need)g(to)g
(obtain)f(full)523 4359 y(parallel)h(p)r(o)n(w)n(er.)g(A)n(t)h(the)g
(same)g(time,)g(the)g(fact)g(that)h(the)f(storage)e(structure)h(of)h
(the)523 4458 y FG(ASMM)j FK(is)24 b(manipulated)h(b)n(y)f(a)g
(\014nite)h(program)e(that)i(in)n(teracts)e(with)i(the)g
(\001-structure)523 4558 y(b)n(y)i(means)g(of)h(a)f(single)g(cen)n(ter)
g(seems)g(to)h(b)r(e)g(the)g(main)f(reason)f(wh)n(y)h(the)h(mac)n(hine)
f(has)523 4658 y(not)c(b)r(ecome)h(to)r(o)f(p)r(o)n(w)n(erful.)g(As)g
(sho)n(wn)g(b)n(y)g(Lam)g(and)g(Ruzzo)g([11)o(],)h(a)f(mo)r(del)g
(where)g(the)523 4757 y(no)r(des)34 b(b)r(ecome)h(indep)r(enden)n(tly)g
(activ)n(e)e(\014nite)j(automata)d(b)r(ecomes)h(equiv)-5
b(alen)n(t)34 b(with)523 4857 y(a)i(restricted)f(v)n(ersion)g(of)h(the)
h FG(P)o FD(\000)-9 b FG(RAM)49 b FK(of)37 b(F)-7 b(ortune)36
b(and)g(Wyllie.)g(This)g(su\016ces)g(for)523 4956 y(making)26
b(the)g(nondeterministic)g(v)n(ersion)f(more)g(p)r(o)n(w)n(erful)h
(than)g FG(PSP)-6 b(A)n(CE)37 b FK(\(except)26 b(for)523
5056 y(the)i(unlik)n(ely)g(case)e(that)i FG(PSP)-6 b(A)n(CE)34
b FK(=)22 b FG(NEXPTIME)12 b FK(\).)28 b(This)f(situation)h(resem)n
(bles)e(the)523 5156 y(relation)33 b(b)r(et)n(w)n(een)i(the)g
FG(SIMD)n(A)n(G)41 b FK(describ)r(ed)34 b(b)n(y)g(Goldsc)n(hlager)e
([8],)i(where)g(a)g(single)523 5255 y(pro)r(cessor)i(broadcasts)g(its)i
(instructions)f(to)h(a)g(collection)f(of)h(p)r(eripheral)f(pro)r
(cessors)523 5355 y(and)31 b(the)h FG(P)o FD(\000)-9
b FG(RAM)44 b FK(mo)r(del)31 b(of)g(F)-7 b(ortune)31
b(and)g(Wyllie)h([7)o(])g(where)e(the)i(lo)r(cal)e(pro)r(cessors)p
eop end
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TeXDict begin 62 71 bop 2589 299 a FF(5.5.)53 b(Conclusion)155
b(62)523 506 y FK(are)27 b(indep)r(enden)n(t.)606 606
y(Clearly)34 b(there)h(are)g(other)f(mo)r(dels)i(whic)n(h)f(could)g
(serv)n(e)f(as)g(a)h(parallelized)f(v)n(ersion)523 706
y(of)h(the)h FG(SMM)14 b FK(.)36 b(In)g(our)e(mo)r(del)i(the)g(set-to)f
(instruction)g(is)g(rather)g(limited.)h(Since)g(its)523
805 y(second)23 b(argumen)n(t)f(addresses)f(a)i(single)g(no)r(de,)g(it)
h(cannot)f(b)r(e)g(used)h(for)e(setting)i(di\013eren)n(t)523
905 y(p)r(oin)n(ters)j(to)h(di\013eren)n(t)h(destinations.)e(This)h
(sev)n(erely)f(limits)h(the)h(scop)r(e)e(of)h(pro)r(ofs)f(that)523
1005 y(our)34 b(mac)n(hine)g(is)g(indeed)h(so)f(p)r(o)n(w)n(erful.)g(A)
h(more)f(con)n(v)n(en)n(tional)e(approac)n(h,)h(based)h(on)523
1104 y(the)25 b(construction)f(of)h(the)g(transition)f(graph)f(of)i(a)f
(p)r(olynomial)g(space)g(b)r(ounded)h(T)-7 b(uring)523
1204 y(mac)n(hine,)33 b(and)g(the)h(computation)f(of)h(its)f(transitiv)
n(e)g(closure)f(b)n(y)h(p)r(oin)n(ter)g(jumping|)523
1303 y(as)40 b(suggested)f(b)n(y)h(the)h(referee|is)e(rendered)g
(infeasible)i(b)n(y)f(the)h(limitation)f(of)g(the)523
1403 y(set-to)26 b(instruction.)g(Ov)n(ercoming)e(this)j(limitation)f
(w)n(ould)g(require)f(a)h(di\013eren)n(t)h(\015a)n(v)n(our)523
1503 y(of)33 b(set-to)g(instruction.)h(A)f(natural)g(p)r(ossibilit)n(y)
g(is)g(to)h(allo)n(w)e(the)i(con)n(v)n(en)n(tional)d(set-to)523
1602 y(instruction)19 b(of)g(the)g FG(SMM)34 b FK(to)18
b(b)r(e)i(executed)f(in)g(parallel)f(with)i(resp)r(ect)e(to)h(man)n(y)g
(di\013eren)n(t)523 1702 y(`cen)n(ters',)25 b(the)i(latter)f(b)r(eing)g
(sp)r(eci\014ed)g(b)n(y)g(a)g(third)g(argumen)n(t)f(whic)n(h)h(is)g(a)g
(string)f(in)3279 1681 y(~)3265 1702 y(\001)q(.)523 1802
y(This)35 b(mo)r(del)h(has)f(some)g(dra)n(wbac)n(ks,)e(ho)n(w)n(ev)n
(er.)g(One)i(is)h(the)g(p)r(ossibilit)n(y)f(of)g(con\015icts)523
1901 y(arising)c(when)h(a)f(p)r(oin)n(ter)h(m)n(ust)g(b)r(e)g(set)g(to)
g(one)g(no)r(de)g(when)g(addressed)f(through)g(one)523
2001 y(cen)n(ter,)23 b(and)h(to)g(another)e(no)r(de)i(when)g(addressed)
f(through)g(another)g(cen)n(ter.)g(Resolving)523 2100
y(this)35 b(problem)e(w)n(ould)h(probably)f(detract)h(from)g(the)h
(elegance)e(of)h(the)g(mo)r(del,)h(one)f(of)523 2200
y(its)f(prime)g(features.)g(Another)g(problem)f(is)h(that)h(it)f(b)r
(ecomes)g(harder)f(to)h(manage)f(all)523 2300 y(the)f(p)r(oin)n(ters,)f
(since)g(there)g(is)g(no)g(simple)h(w)n(a)n(y)e(in)i(whic)n(h)f(to)g
(direct)g(a)g(bunc)n(h)h(of)f(them)523 2399 y(to)f(some)g(\014xed)g(no)
r(de)g(where)f(they)i(can)e(b)r(e)i(`out)f(of)g(the)h(w)n(a)n(y'.)e(Th)
n(us)h(it)g(is)g(not)g(a)g(strict)523 2499 y(generalization)23
b(of)h(our)g(mo)r(del,)h(although)f(it)h(should)g(b)r(e)g(p)r(ossible)f
(to)h(sim)n(ulate)f(our)g(set-)523 2599 y(to)e(instruction)g(with)g
(this)g(new)g(one)g(b)n(y)g(k)n(eeping)f(around)g(an)g(extra)g
(direction)h(to)g(alw)n(a)n(ys)523 2698 y(p)r(oin)n(t)28
b(to)f(the)h(real)f(cen)n(ter.)p eop end
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TeXDict begin 63 72 bop 3357 223 2 75 v 3283 225 150
2 v 3357 300 2 75 v 523 543 a FI(Bibliograph)l(y)565
2516 y FK([1])41 b(Aho,)d(A.V.,)i(Hop)r(croft,)e(J.E.)g(and)h(Ullman,)g
(J.D.,)g FG(The)i(Design)f(and)g(A)n(nalysis)694 2615
y(of)35 b(Computer)f(A)n(lgorithms)7 b FK(,)34 b(Addison-W)-7
b(esley)32 b(Publ.)g(Comp.,)h(Reading,)e(Mass.,)694 2715
y(1974.)565 2867 y([2])41 b(Barzdin',)35 b(Y)-7 b(a.)36
b(M.,)g FG(Universal)i(pulsing)g(elements)7 b FK(,)35
b(So)n(viet)h(Ph)n(ysics-Doklady)d(9)694 2967 y(\(1965\))26
b(523{525.)565 3120 y([3])41 b(Barzdin',)20 b(Y)-7 b(a.)20
b(M.,)h FG(Universality)k(pr)l(oblems)f(in)g(the)f(the)l(ory)h(of)h(gr)
l(owing)f(automata)6 b FK(,)694 3219 y(So)n(viet)27 b(Ph)n
(ysics-Doklady)e(9)i(\(1965\))f(535{537.)565 3372 y([4])41
b(Bertoni,)c(A.,)h(Mauri,)f(G.)h(and)f(Sabadini,)g(N.,)h
FG(Simulations)i(among)f(classes)h(of)694 3472 y(r)l(andom)35
b(ac)l(c)l(ess)g(machines)h(and)g(e)l(quivalenc)l(e)f(among)h(numb)l
(ers)e(suc)l(cinctly)h(r)l(ep-)694 3571 y(r)l(esente)l(d)9
b FK(,)27 b(Ann.)h(Discr.)f(Math.)h(25)f(\(1985\))f(65{90.)565
3724 y([5])41 b(Chandra,)25 b(A.K.,)h(Kozen,)f(D.C.)i(and)e(Sto)r(c)n
(kmey)n(er,)g(L.J.,)h FG(A)n(lternation)6 b FK(,)26 b(J.)g(Asso)r(c.)
694 3824 y(Comput.)i(Mac)n(h.)f(28)f(\(1981\))h(114{133.)565
3976 y([6])41 b(Dymond,)27 b(P)-7 b(.W.)27 b(and)g(Co)r(ok,)f(S.A.,)i
FG(Har)l(dwar)l(e)i(c)l(omplexity)g(and)g(p)l(ar)l(al)t(lel)h(c)l
(ompu-)694 4076 y(tation)6 b FK(,)35 b(Pro)r(c.)f(21st)g(Ann.)h(IEEE)f
(Symp.)h(F)-7 b(oundations)34 b(of)h(Computer)g(Science,)694
4176 y(1980,)25 b(pp.)j(360{372.)565 4328 y([7])41 b(F)-7
b(ortune,)27 b(S.)h(and)f(Wyllie,)h(J.,)f FG(Par)l(al)t(lelism)32
b(in)e(r)l(andom)g(ac)l(c)l(ess)g(machines)7 b FK(,)29
b(Pro)r(c.)694 4428 y(10th)e(Ann.)h(A)n(CM)g(Symp.)g(Theory)e(of)i
(Computing,)f(1978,)f(pp.)i(114{118.)565 4581 y([8])41
b(Goldsc)n(hlager,)27 b(L.M.,)j FG(A)g(universal)i(inter)l(c)l(onne)l
(ction)f(p)l(attern)g(for)i(p)l(ar)l(al)t(lel)g(c)l(om-)694
4680 y(puters)7 b FK(,)27 b(J.)g(Asso)r(c.)h(Comput.)f(Mac)n(h.)h(29)e
(\(1982\))h(1073{1086.)565 4833 y([9])41 b(Hartmanis,)23
b(J.)i(and)f(Simon,)g(J.,)h FG(On)h(the)h(structur)l(e)e(of)i(fe)l
(asible)i(c)l(omputations)7 b FK(,)24 b(in)694 4932 y(Rubino\013,)31
b(M.)g(and)f(Y)-7 b(o)n(vits,)30 b(M.C.)h(\(Eds.\),)g(Adv)-5
b(ances)30 b(in)h(Computers,)f(V)-7 b(ol.)31 b(14)f(,)694
5032 y(Acad.)d(Press,)f(New)i(Y)-7 b(ork,)27 b(1976,)f(pp.)i(1{43.)523
5185 y([10])41 b(Kolmogoro)n(v,)19 b(A.N.)24 b(and)f(Usp)r(enskii,)g
(V.A.,)h FG(On)h(the)h(de\014nition)g(of)g(an)g(algorithm)6
b FK(,)694 5284 y(Usp)r(ehi)29 b(Mat.)f(Nauk)g(13)g(\(1958\))f(3{28)f
(;)j(AMS)g(T)-7 b(ransl.)27 b(2nd)h(ser.)g(29)g(\(1963\))f(217{)694
5384 y(245.)p eop end
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b(64)523 506 y FK([11])41 b(Lam,)25 b(T.W.)h(and)f(Ruzzo,)g(W.L.,)h
FG(The)j(p)l(ower)g(of)f(p)l(ar)l(al)t(lel)i(p)l(ointer)e(manipulation)
6 b FK(,)694 606 y(Pro)r(c.)26 b(1st)i(Ann.)g(A)n(CM)g(Symp.)h(P)n
(arallel)c(Algorithms)j(and)f(Arc)n(hitectures,)h(1989,)694
706 y(pp.)g(92{102)523 872 y([12])41 b(Lugin)n(buhl,)46
b(D.R.)i(and)e(Loui,)h(M.C.,)g FG(Hier)l(ar)l(chies)i(and)f(sp)l(ac)l
(e)g(me)l(asur)l(es)f(for)694 971 y(p)l(ointer)29 b(machines)7
b FK(,)27 b(Inf.)g(and)f(Comput.,)g(1993,)e(to)i(app)r(ear;)g(also:)f
(Rep)r(ort)h(UILU-)694 1071 y(ENG-88-2245,)45 b(Departmen)n(t)50
b(of)f(Electr.)g(Engin.,)f(Univ)n(ersit)n(y)h(of)g(Illinois)g(at)694
1171 y(Urbana-Champaign,)25 b(1988.)523 1337 y([13])41
b(P)n(arb)r(erry)-7 b(,)22 b(I.,)j FG(Par)l(al)t(lel)k(sp)l(e)l(e)l
(dup)f(of)g(se)l(quential)f(machines:)i(a)f(defense)g(of)g(the)f(p)l
(ar-)694 1436 y(al)t(lel)k(c)l(omputation)f(thesis)7
b FK(,)28 b(SIGA)n(CT)g(News)g(18,)e(nr.)i(1,)f(1986,)f(pp.)i(54{67.)
523 1602 y([14])41 b(Pratt,)18 b(V.R.)i(and)f(Sto)r(c)n(kmey)n(er,)f
(L.J.,)h FG(A)j(char)l(acterization)i(of)f(the)f(p)l(ower)h(of)g(ve)l
(ctor)694 1702 y(machines)7 b FK(,)28 b(J.)g(Comput.)g(Syst.)g(Sci.)f
(12)g(\(1976\))f(198{221.)523 1868 y([15])41 b(Sa)n(vitc)n(h,)g(W.J.,)i
FG(R)l(e)l(cursive)g(T)-6 b(uring)43 b(machines)7 b FK(,)44
b(In)n(ter.)e(J.)g(Comput.)h(Math.)f(6)694 1968 y(\(1977\))26
b(3{31.)523 2134 y([16])41 b(Sc)n(h\177)-42 b(onhage,)21
b(A.,)i FG(Stor)l(age)j(mo)l(di\014c)l(ation)h(machines)7
b FK(,)24 b(SIAM)f(J.)g(Comput.)g(9)f(\(1980\))694 2233
y(490{508.)523 2399 y([17])41 b(Slot,)g(C.)g(and)g(v)-5
b(an)41 b(Emde)g(Boas,)f(P)-7 b(.,)41 b FG(The)i(pr)l(oblem)g(of)g(sp)l
(ac)l(e)g(invarianc)l(e)h(for)694 2499 y(se)l(quential)30
b(machines)7 b FK(,)28 b(Inf.)h(and)e(Comp.)h(77)e(\(1988\))h(93{122.)
523 2665 y([18])41 b(Stegw)n(ee,)28 b(R.A.,)i(T)-7 b(oren)n(vliet,)27
b(L.)i(and)g(v)-5 b(an)29 b(Emde)g(Boas,)e(P)-7 b(.,)29
b FG(The)j(p)l(ower)g(of)f(your)694 2765 y(e)l(ditor)9
b FK(,)27 b(Rep)r(ort)f(RJ)f(4711)g(\(50179\),)f(IBM)i(Researc)n(h)e
(Lab.,)i(San)g(Jose,)f(Ca.,)h(1985.)523 2931 y([19])41
b(Sto)r(c)n(kmey)n(er,)17 b(L.,)i FG(The)j(p)l(olynomial)i(time)e(hier)
l(ar)l(chy)7 b FK(,)21 b(Theor.)c(Comp.)i(Sci.)g(3)f(\(1977\))694
3030 y(1{22.)523 3196 y([20])41 b(v)-5 b(an)34 b(Emde)g(Boas,)g(P)-7
b(.,)34 b FG(The)j(se)l(c)l(ond)f(machine)i(class)f(2:)g(an)f(encyclop)
l(ae)l(dic)j(view)694 3296 y(on)28 b(the)h(Par)l(al)t(lel)h
(Computation)g(Thesis)7 b FK(,)27 b(in:)g(Rasio)n(w)n(a,)d(H.)j
(\(Ed.\),)f(Mathematical)694 3396 y(Problems)32 b(in)h(Computation)g
(Theory)-7 b(,)33 b(Banac)n(h)f(Cen)n(ter)h(Publications,)f(V)-7
b(ol.)34 b(21,)694 3495 y(W)-7 b(arsa)n(w,)26 b(1987,)g(pp.)i(235{256.)
523 3661 y([21])41 b(v)-5 b(an)21 b(Emde)g(Boas,)e(P)-7
b(.,)21 b FG(Sp)l(ac)l(e)k(me)l(asur)l(es)e(for)i(stor)l(age)f(mo)l
(di\014c)l(ation)h(machines)7 b FK(,)22 b(Inf.)694 3761
y(Pro)r(c.)k(Lett.)i(30)f(\(1989\))f(103{110.)523 3927
y([22])41 b(v)-5 b(an)26 b(Emde)g(Boas,)e(P)-7 b(.,)26
b FG(Machine)k(mo)l(dels)g(and)e(simulations)7 b FK(,)27
b(in:)g(v)-5 b(an)26 b(Leeu)n(w)n(en,)f(J.)694 4027 y(\(Ed.\),)33
b(Handb)r(o)r(ok)f(of)g(Theoretical)f(Computer)i(Science,)f
(North-Holland)g(Publ.)694 4126 y(Comp.)27 b(1990,)f(pp.)i(1{66.)523
4292 y([23])41 b(W)-7 b(agner,)30 b(K.)h(and)g(W)-7 b(ec)n(hsung,)30
b(G.,)i FG(Computational)i(Complexity)7 b FK(,)33 b(Mathematis-)694
4392 y(c)n(he)39 b(Monographien)f(V)-7 b(ol.)39 b(19,)g(VEB)g(Deutsc)n
(her)g(V)-7 b(erlag)39 b(der)g(Wissensc)n(haften,)694
4491 y(Berlin)27 b(\(DDR\),)i(1986,)d(also:)g(Reidel)i(Publ.)g(Comp.,)f
(Dordrec)n(h)n(t,)g(1986.)523 4658 y([24])41 b(Wiedermann,)29
b(J.,)g FG(Par)l(al)t(lel)k(T)-6 b(uring)31 b(machines)i
FK(,)c(T)-7 b(ec)n(hn.)30 b(Rep.)f(R)n(UU-CS-84-11,)694
4757 y(Dept.)f(of)g(Computer)f(Science,)h(Univ)n(ersit)n(y)e(of)i
(Utrec)n(h)n(t,)f(Utrec)n(h)n(t,)h(1984.)523 4923 y([25])41
b(Wiedermann,)22 b(J.,)g FG(We)l(ak)j(p)l(ar)l(al)t(lel)i(machines;)g
(a)e(new)g(class)g(of)h(physic)l(al)t(ly)i(fe)l(asible)694
5023 y(p)l(ar)l(al)t(lel)38 b(machine)g(mo)l(dels)7 b
FK(,)36 b(I.M.)g(Ha)n(v)n(el)e(&)h(V.)g(Koub)r(ek)g(\(Eds.\),)g(pro)r
(c.)g(Mathe-)694 5122 y(matical)25 b(F)-7 b(oundations)25
b(of)h(Computer)f(Science)h(1992,)d(Springer)i(Lecture)g(notes)g(in)694
5222 y(Computer)i(Science)h(629)e(\(1992\))g(pp.)i(95{111.)p
eop end
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b(to)g(Construct)f(an)h(A)l(tomic)g(V)-13 b(ariable)523
2549 y FL(6.1)116 b(In)m(tro)s(duction.)523 2748 y FK(Comm)n(unication)
27 b(pla)n(ys)f(a)h(vital)g(role)g(in)h(an)n(y)e(distributed)i(system,)
f(allo)n(wing)f(m)n(ultiple)523 2848 y(pro)r(cessors)21
b(to)h(share)g(and)h(exc)n(hange)e(information.)i(Con)n(v)n(en)n
(tionally)e(this)i(comm)n(unica-)523 2947 y(tion)i(is)f(based)h(on)f(m)
n(utually)h(exclusiv)n(e)e(access)h(to)g(a)h(shared)e
FG(variable)p FK(.)k(This)e(is)g(the)g(case)523 3047
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h(the)h(t)n(w)n(o)f(endp)r(oin)n(ts)g(of)h(a)f(link)523
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b(,)28 b(this)g(exclusiv)n(e)e(nature)h(of)h(access)e(ma)n(y)523
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523 3346 y(the)33 b(parallelism)f(inheren)n(t)g(in)i(distributed)f
(systems.)f(In)h(the)g(last)g(y)n(ears)e(in)n(terest)h(has)523
3445 y(fo)r(cussed)21 b(on)g FG(wait-fr)l(e)l(e)h FK(v)-5
b(ariables,)21 b(whic)n(h)g(can)g(b)r(e)h(accessed)e(concurren)n(tly)g
(without)i(an)n(y)523 3545 y(form)33 b(of)h(w)n(aiting.)f(The)g
(question)g(is)h(ho)n(w)f(to)g(construct)g(suc)n(h)g(v)-5
b(ariables)33 b(in)h(terms)f(of)523 3645 y(lo)n(w)n(er-lev)n(el)25
b(hardw)n(are,)h(lik)n(e)h(\015ip-\015ops.)606 3744 y(P)n(eterson)35
b(w)n(as)h(one)g(of)h(the)g(\014rst)f(to)h(in)n(v)n(estigate)e(this)i
(question)f(in)h([15)o(],)g(giving)f(a)523 3844 y(construction)f(for)h
(a)f(single-writer,)g(m)n(ulti-reader,)f(m)n(ulti-bit)j(atomic)e(v)-5
b(ariable)35 b(from)523 3944 y(single-writer,)25 b(m)n(ulti-reader)g
(atomic)h(bits.)h(Later,)e(Lamp)r(ort)52 b([10],)26 b(spark)n(ed)f
(o\013)h(in)n(ter-)523 4043 y(est)k(in)g(the)g(sub)5
b(ject)29 b(b)n(y)h(dev)n(eloping)e(a)h(precise)g(theory)g(and)h
(formalisms|apart)d(from)523 4143 y(presen)n(ting)e(some)h(solutions)f
(to)h(subproblems.)g(It)g(is)g(w)n(orth)f(noting)h(that)h(most)e(pap)r
(ers)523 4242 y(use)i(the)h(w)n(ord)f(\\register")e(instead)j(of)f(v)-5
b(ariable.)606 4342 y(The)35 b(ultimate)g(goal)f(is)g(to)h(build)g(a)f
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(capabilities|whic)n(h)g(can)g(hold)g(an)n(y)g(\014xed)h(n)n(um-)523
4541 y(b)r(er)j(of)f(v)-5 b(alues,)25 b(and)h(whose)f(accesses)f(b)r
(eha)n(v)n(e)h(atomically)-7 b(.)24 b(The)i(latter)f(means)g(that)h
(for)523 4641 y(an)n(y)32 b(sequence)f(of)h(op)r(erations)f(on)h(the)h
(v)-5 b(ariable,)31 b(the)i(partial)e(precedence)h(order)f(\(de-)523
4741 y(\014ned)g(later\))f(among)g(those)g(op)r(erations)f(m)n(ust)i
(ha)n(v)n(e)e(a)h(total)g(extension)g(\(for)h(external)523
4840 y(consistency\))k(suc)n(h)g(that)h(eac)n(h)f(read)f(op)r(eration)h
(returns)g(the)h(v)-5 b(alue)35 b(from)g(the)h(write)523
4940 y(op)r(eration)26 b(whic)n(h)h(is)g(the)h(last)f(to)g(precede)g
(it)g(in)h(the)g(total)e(order)g(\(for)h(in)n(ternal)g(consis-)523
5039 y(ten)n(tency\).)606 5139 y(The)35 b(construction)f(of)h(this)g
(\\ultimate")f(v)-5 b(ariable)34 b(is)g(not)h(done)f(directly)h(from)f
(the)523 5239 y(most)21 b(primitiv)n(e)g(kind)h(of)f(v)-5
b(ariable.)20 b(Rather,)h(this)h(task)f(is)g(more)f(con)n(v)n(enien)n
(tly)g(split)i(in)n(to)523 5338 y(t)n(w)n(o)j(subtasks:)f(the)i
(construction)f(of)g(atomic,)g(m)n(ulti-bit,)h(single-writer,)e
(single-reader)p eop end
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b(ABLE)26 b(6.1.)h(w)n(orst)f(case)h(n)n(um)n(b)r(er)d(of)j(bits)523
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(atomic,)e(m)n(ulti-bit,)i(m)n(ulti-user)f(v)-5 b(ariables)523
1444 y(from)27 b(the)h(former)f(t)n(yp)r(e.)606 1544
y(This)36 b(partition)f(can)g(b)r(e)h(justi\014ed)g(b)n(y)f(the)h
(nature)f(of)h(the)g(problems)e(in)n(v)n(olv)n(ed.)h(In)523
1643 y(the)26 b(\014rst)g(construction)f(\(as)g(in)h([10)o(]\),)g(the)g
(m)n(ulti-bit)h(v)-5 b(alue)25 b(is)h(to)g(b)r(e)g FG(distribute)l(d)g
FK(o)n(v)n(er)e(a)523 1743 y(m)n(ultiple)f(n)n(um)n(b)r(er)f(of)g
(bits.)h(In)g(the)f(second)g(construction)g(\(as)g(in)g([20)o(]\),)h
(the)g(v)-5 b(alue)22 b(of)g(the)523 1842 y(m)n(ulti-user)f(v)-5
b(ariable)20 b(is)h FG(r)l(eplic)l(ate)l(d)i FK(among)e(all)g(users)
2207 1812 y FB(1)2264 1842 y FK(along)g(with)g(con)n(trol)f
(information)523 1942 y(whic)n(h)28 b(allo)n(ws)f(eac)n(h)g(user)h(to)g
(iden)n(tify)g(the)h(most)f(recen)n(tly)f(written)h(v)-5
b(alue.)28 b(This)g(is)g(the)523 2042 y(more)39 b(complex)g(problem,)g
(as)g(witnessed)g(b)n(y)h(the)g(fact)f(that)h(man)n(y)f(prop)r(osed)g
(\(and)523 2141 y(often)25 b(pro)n(v)n(en\))e(solutions)h(w)n(ere)g
(later)g(found)g(to)h(b)r(e)g(erroneous.)d(The)j(in)n(terested)f
(reader)523 2241 y(is)k(referred)e(to)h([2,)h(8)o(,)g(14)o(,)g(16)o(,)f
(20)o(].)606 2341 y(In)32 b(this)h(c)n(hapter)e(w)n(e)g(attac)n(k)h
(the)g(\014rst)g(problem,)f(and)h(also)f(giv)n(e)g(sp)r(ecial)h(atten)n
(tion)523 2440 y(to)c(the)f(case)g(of)h(constructing)f(a)g(single-bit)g
(atomic)g(v)-5 b(ariable.)523 2751 y FL(6.2)116 b(Comparison)36
b(with)h(Related)h(W)-10 b(ork)523 2950 y FK(There)36
b(are)g(basically)f(t)n(w)n(o)h(approac)n(hes)e(that)j(can)g(b)r(e)g
(tak)n(en)f(in)h(order)e(to)h(construct)523 3050 y(a)f(m)n(ulti-bit)g
(v)-5 b(ariable)34 b(from)h(a)g(linear)f(n)n(um)n(b)r(er)h(of)g(single)
f(bits.)h(The)h(\014rst)e(w)n(as)g(tak)n(en)523 3150
y(b)n(y)d(G.L.)h(P)n(eterson)e(in)i([15)o(])f(and)h(in)n(v)n(olv)n(es)e
(k)n(eeping)g(3)h(copies)g(of)h(the)g(m)n(ulti-bit)g(v)-5
b(alue,)523 3249 y(called)23 b(the)i FG(tr)l(acks)f FK(\(in)g(the)g
(original)f(pap)r(er,)g(they)h(are)f(called)g FG(bu\013ers)p
FK(\).)h(Apart)g(from)f(the)523 3349 y(3)33 b(trac)n(ks,)f(there)h(are)
f(some)g FG(c)l(ontr)l(ol)i FK(bits)f(whic)n(h)g(w)n(e)g(collectiv)n
(ely)f(call)h(the)h FG(switch)p FK(.)g(In)523 3449 y(this)f(approac)n
(h)d(the)i(writer)g(writes)g(the)g(new)g(v)-5 b(alue)32
b(to)g(all)g(three)g(trac)n(ks.)f(The)h(reader)523 3548
y(reads)27 b(from)h(all)f(trac)n(ks,)g(but)i(in)f(a)f(di\013eren)n(t)i
(order.)d(The)i(switc)n(h)g(allo)n(ws)f(the)h(reader)f(to)523
3648 y(determine)h(whic)n(h)f(trac)n(k)g(w)n(as)f(read)h(without)h(in)n
(terference)f(from)g(the)h(writer.)606 3747 y(In)j(Kirousis)f(et.)h
(al.)g([9)o(],)h(the)f(second)g(approac)n(h)e(w)n(as)h(tak)n(en.)g(The)
h(idea)g(is)g(that)g(the)523 3847 y(writer)h(and)g(the)h(reader)e
(access)g(only)h(a)g(single)g(trac)n(k,)f(and)i(that)f(the)h(switc)n(h)
g(ensures)523 3947 y(that)27 b(they)g(nev)n(er)e(access)g(the)i(same)f
(trac)n(k)f(sim)n(ultaneously)-7 b(.)26 b(The)h(price)f(to)g(b)r(e)h
(paid)f(for)523 4046 y(the)i(reduced)f(n)n(um)n(b)r(er)h(of)f(trac)n
(k-accesses)e(is)i(the)h(necessit)n(y)f(of)g(using)h(four)f(trac)n(ks.)
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(cols)g(for)h(m)n(ultiple)h(readers)e(but)i(w)n(e'll)f(consider)523
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y(In)40 b(this)f(c)n(hapter)g(a)g(simpli\014cation)g(of)h(the)g(single)
f(reader)f(construction)g(of)i([9)o(])g(is)523 4445 y(presen)n(ted.)20
b(T)-7 b(able)21 b(6.2)e(giv)n(es)h(a)g(comparison)f(of)i(these)g
(constructions)e(for)h(a)h FC(b)p FK(-bit)f(atomic)523
4544 y(v)-5 b(ariable.)37 b(The)i(\\used")e(column)h(displa)n(ys)g(the)
g(total)g(n)n(um)n(b)r(er)g(of)h(safe)f(bits)g(used)g(in)523
4644 y(the)27 b(construction)f(\(\\space)g(complexit)n(y"\).)h(The)f
(\\write")g(\(\\read"\))g(column)h(giv)n(es)e(the)523
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n(ust)f(b)r(e)h(accessed)d(in)j(a)f(write)f(\(read\))h(action)523
4843 y(on)d(the)h(atomic)f(v)-5 b(ariable.)27 b(A)h(trade-o\013)f(b)r
(et)n(w)n(een)g(time)h(and)g(space)f(is)g(clearly)g(visible.)606
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(ecial)g(case)f(of)h(constructing)f(an)h(atomic)523 5043
y(bit)c(with)f(a)g(minimal)g(n)n(um)n(b)r(er)g(of)g(non-atomic)f(bits.)
i(This)f(problem)g(w)n(as)f(solv)n(ed)g(in-)p 523 5162
499 5 v 627 5244 a Fm(1)661 5276 y FF(to)d(ha)n(v)n(e)f(a)h(comm)n
(unication)g(path)f(b)r(et)n(w)n(een)g(an)n(y)g(pair)h(of)g(users,)h(w)
n(e)e(need)g(to)h(main)n(tain)523 5359 y(\012\()p Fo(n)654
5327 y Fm(2)689 5359 y FF(\))26 b(copies.)p eop end
%%Page: 67 77
TeXDict begin 67 76 bop 1890 299 a FF(6.2.)53 b(Comparison)27
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(],)i(using)g(a)f(v)n(ery)g(similar)523 606 y(construction.)c(The)h
(use)g(of)g(\014nite)g(state)g(mac)n(hines)f(for)g(\(automated\))h
(correctness)e(v)n(er-)523 706 y(i\014cation)k(is)f(related)h(to)f(the)
h(w)n(ork)f(b)n(y)g(Clark)n(e)g(and)h(Emerson)e(\(e.g.)i([4)o(]\))h
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(written)f(b)n(y)h(one)f(user,)g(called)523 1418 y(the)j
FG(writer)p FK(,)g(and)g(read)e(b)n(y)i(another,)f(the)h
FG(r)l(e)l(ader)p FK(.)g(Both)g(users)e(ma)n(y)h(b)r(e)h(accessing)e
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(relativ)n(e)f(sp)r(eed)i(of)f(the)g(users,)g(and)f(that)523
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(aim)g(to)g(construct)g(an)g(atomic,)g(m)n(ulti-bit,)h(single-)523
1916 y(writer,)26 b(single-reader)e(v)-5 b(ariable.)26
b(The)h(ob)5 b(jects)26 b(w)n(e)h(use)f(in)h(this)g(construction)f(are)
g FG(safe)p FK(,)523 2016 y(single-bit,)31 b(single-writer,)f
(single-reader)g(v)-5 b(ariables,)30 b(or)h(simply)g
FG(safe)k(bits)p FK(.)c(These)h(are)523 2115 y(the)27
b(mathematical)g(coun)n(terparts)e(of)i(\015ip-\015ops,)f(in)h(the)h
(sense)e(that)h(real-life)f(\015ip-\015ops)523 2215 y(can)h(b)r(e)i
(argued)d(to)i(satisfy)f(the)i FG(safety)f FK(prop)r(ert)n(y)-7
b(.)27 b(Before)g(giving)g(rigorous)f(de\014nitions)523
2315 y(for)h(the)h(notions)f(of)h(safe)f(and)g(atomic,)h(w)n(e)f
(\014rst)g(state)g(some)h(preliminary)e(de\014nitions.)606
2414 y(In)k(order)f(to)h(distinguish)g(the)g(accesses)f(to)h(the)g
(constructed)g(atomic)f(v)-5 b(ariable)29 b(\(the)523
2514 y FG(higher)39 b(level)p FK(\))e(from)f(the)g(accesses)e(to)i(the)
g(safe)g(bits)g(\(the)h FG(lower)h(level)p FK(\),)f(w)n(e)f(call)f(the)
523 2614 y(former)40 b FG(actions)p FK(,)i(and)f(the)h(latter)f
FG(sub)l(actions)p FK(.)g(As)h(w)n(e)f(will)g(see,)g(eac)n(h)f
(higher-lev)n(el)523 2713 y(action)31 b(is)h(comp)r(osed)f(of)h(a)f(n)n
(um)n(b)r(er)h(of)g(subactions|where)e(the)i FG(wait-fr)l(e)l(e)h
FK(condition)523 2813 y(requires)26 b(this)i(n)n(um)n(b)r(er)g(to)f(b)r
(e)h(b)r(ounded.)606 2912 y(Let)23 b FC(V)43 b FK(b)r(e)23
b(a)g(v)-5 b(ariable)22 b(and)h FD(A)h FK(\(the)f(set)h(of)f
(accesses\))f(b)r(e)h(the)h(union)f(of)g(a)g(set)g(of)g(writes)523
3012 y FD(W)36 b FK(to)29 b FC(V)48 b FK(and)28 b(a)h(set)g(of)g(reads)
f FD(R)h FK(from)g FC(V)19 b FK(.)29 b(The)g(result)f(of)h(a)g(read)f
(is)h(a)f(v)-5 b(alue)29 b(whic)n(h)g(is)523 3112 y(said)i(to)h(b)r(e)g
FG(r)l(eturne)l(d)e FK(b)n(y)i(that)g(read.)e(Eac)n(h)h(access)f
FC(a)g FD(2)g(A)i FK(o)r(ccupies)f(a)g FG(time)j(interval)523
3211 y FK(\()p FC(s)p FK(\()p FC(a)p FK(\))p FC(;)14
b(f)9 b FK(\()p FC(a)p FK(\)\),)39 b(where)e FC(s)p FK(\()p
FC(a)p FK(\))i(is)e(the)i(start)e(time)i(and)e FC(f)9
b FK(\()p FC(a)p FK(\))38 b(the)h(\014nish)f(time)g(of)g(access)523
3311 y FC(a)p FK(.)31 b(All)g(start)f(and)g(\014nish)h(times)g(are)f
(assumed)g(to)g(b)r(e)h(pairwise)f(distinct.)h(W)-7 b(e)31
b(de\014ne)g(a)523 3411 y(precedence)g(relation)h FD(!)g
FK(on)g FD(A)g FK(as)f(follo)n(ws:)h FC(a)e FD(!)h FC(b)g
FK(i\013)i FC(f)9 b FK(\()p FC(a)p FK(\))31 b FC(<)f(s)p
FK(\()p FC(b)p FK(\).)i(W)-7 b(e)33 b(sa)n(y)d(that)j
FC(a)523 3510 y FG(overlaps)21 b FC(b)p FK(,)d(or)g FC(a)h
FK(and)g FC(b)f FK(o)n(v)n(erlap,)f(if)i(they)g(are)f
FD(!)p FK(-incomparable,)f(that)i(is,)g FC(a)k FD(6!)g
FC(b)q FD(^)q FC(b)g FD(6!)g FC(a)p FK(.)523 3610 y FG(Complete)32
b(overlap)f FK(of)d FC(a)h FK(b)n(y)g FC(b)f FK(means)g(that)h
FC(s)p FK(\()p FC(b)p FK(\))c FC(<)g(s)p FK(\()p FC(a)p
FK(\))g FC(<)f(f)9 b FK(\()p FC(a)p FK(\))25 b FC(<)g(f)9
b FK(\()p FC(b)p FK(\).)28 b(W)-7 b(e)29 b(assume)523
3709 y(that)d(the)f(set)h FD(f)p FC(a)p FD(j)p FC(a)c
FD(!)h FC(b)p FD(g)i FK(is)g(\014nite)h(for)e(an)n(y)h(action)g
FC(b)g FK(\(\014nite)h(history)e(assumption\).)h(W)-7
b(e)523 3809 y(call)29 b(the)g(pair)g(\()p FD(A)p FC(;)14
b FD(!)p FK(\))30 b(a)f FG(run)p FK(.)f(The)i(writes)f(in)g
FD(W)36 b FK(are)28 b(totally)h(ordered)f(b)n(y)h FD(!)p
FK(,)g(and)g(so)523 3909 y(are)j(the)h(reads)e(in)i FD(R)p
FK(,)g(in)g(accordance)e(with)i(the)g(requiremen)n(t)f(that)h(a)f(user)
g(can)g(only)523 4008 y(p)r(erform)27 b(one)g(access)g(at)g(a)g(time.)
606 4108 y(W)-7 b(e)32 b(relate)g(the)g(reads)f(to)h(the)g(writes)g(in)
g(terms)g(of)g(a)g(reading)f(function.)h(A)h(partial)523
4208 y(function)d FC(\031)g FK(:)d FD(R)g(!)g(W)37 b
FK(is)30 b(a)f(reading)g(function)h(if)g(for)g(ev)n(ery)e(read)h
FC(r)h FD(2)d(R)j FK(on)g(whic)n(h)f FC(\031)523 4307
y FK(is)j(de\014ned,)g FC(\031)s FK(\()p FC(r)r FK(\))h(writes)e(to)g
FC(V)51 b FK(the)32 b(v)-5 b(alue)32 b(returned)f(b)n(y)g
FC(r)r FK(.)h(Unless)g(explicitly)g(stated,)523 4407
y(the)g(reading)d(function)j(will)f(b)r(e)h(total)f(\(non-total)f
(reading)g(functions)i(will)f(b)r(e)h(needed)523 4506
y(in)c(the)g(de\014nition)g(of)f(safet)n(y\).)h(W)-7
b(e)28 b(call)f(the)h(triple)g(\()p FD(A)p FC(;)14 b
FD(!)p FC(;)g(\031)s FK(\))28 b(a)f FG(system)j(exe)l(cution)p
FK(.)606 4606 y(W)-7 b(e)28 b(can)f(no)n(w)g(de\014ne)h(atomicit)n(y:)
523 4759 y Fv(Definition)k FH(6.1)41 b FK(A)24 b(system)g(execution)f
FC(\033)k FK(=)22 b(\()p FD(A)p FC(;)14 b FD(!)p FC(;)g(\031)s
FK(\))25 b(of)f(the)g(v)-5 b(ariable)23 b FC(V)43 b FK(is)23
b(atomic)523 4858 y(i\013)38 b(there)g(is)g(a)f(total)g(extension)h
FD(!)1697 4828 y Fw(0)1758 4858 y FK(of)g FD(!)f FK(consisten)n(t)h
(with)g FC(\031)s FK(,)g(i.e.)g(for)f(ev)n(ery)g(read)523
4958 y FC(r)26 b FD(2)d(R)p FK(,)28 b FC(\031)s FK(\()p
FC(r)r FK(\))i(is)d(the)h(last)f(write)h(preceding)f
FC(r)j FK(in)e(the)g(total)f(order)g FD(!)2744 4928 y
Fw(0)2767 4958 y FK(.)606 5111 y(In)g(the)f(case)g(of)g(a)g(single)g
(writer,)g(a)g(simpli\014cation)g(of)h(the)f(general)f(de\014nition)i
(ab)r(o)n(v)n(e)523 5211 y(can)g(b)r(e)h(giv)n(en)f(whic)n(h)h(a)n(v)n
(oids)d(the)j(use)g(of)f(a)h(total)f(ordering,)f([10)o(,)i(14)o(,)f
(1]:)523 5363 y Fv(Definition)32 b FH(6.2)41 b FK(A)19
b(system)g(execution)g FC(\033)26 b FK(=)d(\()p FD(A)p
FC(;)14 b FD(!)p FC(;)g(\031)s FK(\))20 b(of)f(the)g(single-writer)f(v)
-5 b(ariable)p eop end
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TeXDict begin 68 77 bop 2515 299 a FF(6.3.)53 b(Preliminaries)156
b(68)523 506 y FC(V)58 b FK(is)38 b(atomic)h(i\013)g(the)g(follo)n
(wing)f(three)g(prop)r(erties)g(hold)g(for)h(all)f FC(r)n(;)14
b(r)2827 518 y FB(1)2865 506 y FC(;)g(r)2939 518 y FB(2)3018
506 y FD(2)42 b(R)d FK(and)523 606 y FC(w)26 b FD(2)d(W)7
b FK(:)523 763 y FH(A0)41 b FK(not)28 b(\()p FC(r)e FD(!)d
FC(\031)s FK(\()p FC(r)r FK(\)\))523 925 y FH(A1)41 b
FK(if)29 b FC(r)798 937 y FB(1)858 925 y FD(!)23 b FC(r)1001
937 y FB(2)1039 925 y FK(,)28 b(then)g(not)g(\()p FC(\031)s
FK(\()p FC(r)1578 937 y FB(2)1616 925 y FK(\))23 b FD(!)h
FC(\031)s FK(\()p FC(r)1897 937 y FB(1)1935 925 y FK(\)\))523
1087 y FH(A2)41 b FK(not)28 b(\()p FC(\031)s FK(\()p
FC(r)r FK(\))d FD(!)e FC(w)j FD(!)d FC(r)r FK(\))606
1306 y(Equiv)-5 b(alence)27 b(of)g(de\014nitions)h(6.1)f(and)g(6.2)g
(is)h(sho)n(wn)e(b)n(y)i(pro)n(ving)e(\([10)o(]\))714
1477 y FD(9)p FK(consisten)n(t)i(total)f(extension)22
b FD(!)1790 1443 y Fw(0)2050 1477 y FD(,)236 b(!)23 b
FK(satis\014es)k FC(A)p FK(0)p FC(;)14 b(A)p FK(1)p FC(;)g(A)p
FK(2)523 1649 y(First)28 b(of)g(all,)f(if)i(an)n(y)e(of)g(the)i(three)e
(conditions)h(on)f FD(!)h FK(is)g(violated,)f(then)i(it)f(will)g
(clearly)523 1749 y(b)r(e)22 b(imp)r(ossible)g(to)g(\014nd)g(a)g
(consisten)n(t)f(extension)h FD(!)2174 1719 y Fw(0)2197
1749 y FK(,)g(th)n(us)g(pro)n(ving)e(the)i(\014rst)g(direction.)523
1848 y(T)-7 b(o)24 b(pro)n(v)n(e)f(the)i(con)n(v)n(erse,)d(w)n(e)i
(construct)g(the)h(follo)n(wing)e FD(!)2404 1818 y Fw(0)2427
1848 y FK(:)i(Merge)e(the)i(reads)e(in)n(to)h(the)523
1948 y(totally)g(ordered)f(set)h(of)g(writes,)g(suc)n(h)g(that)h(eac)n
(h)e(write)h FC(w)j FK(is)d(immediately)h(follo)n(w)n(ed)e(b)n(y)523
2048 y(the)29 b(reads)e FC(r)k FK(with)d FC(\031)s FK(\()p
FC(r)r FK(\))e(=)d FC(w)r FK(,)29 b(putting)g(the)g(reads)e(in)h(some)g
(total)g(order)e(that)j(extends)523 2147 y(their)21 b(partial)f
(precedence)h(order.)f(Naturally)-7 b(,)20 b(this)i(order)d(is)i
(consisten)n(t)g(with)g FC(\031)s FK(.)h(By)f(A1,)523
2247 y(the)g(merging)e(pro)r(cess)g(preserv)n(es)f(the)j(ordering)e
(among)g(reads.)g(A0)h(and)h(A2)f(ensure)f(that)523 2346
y(the)27 b(precedence)f(b)r(et)n(w)n(een)h(a)f(read)g(and)g(a)h(write)f
(is)h(also)e(extended.)i(If)g FC(w)f FD(!)d FC(r)r FK(,)28
b(then)f(b)n(y)523 2446 y(A2,)d FD(:)p FK(\()p FC(\031)s
FK(\()p FC(r)r FK(\))h FD(!)f FC(w)r FK(\),)h(so)e(b)n(y)h(the)h
(construction)e(of)h FD(!)2183 2416 y Fw(0)2231 2446
y FK(w)n(e)g(ha)n(v)n(e)f FC(w)i FD(!)2705 2416 y Fw(0)2752
2446 y FC(r)r FK(.)g(If)f FC(r)i FD(!)d FC(w)r FK(,)i(then)523
2546 y(b)n(y)i(A0,)h FC(\031)s FK(\()p FC(r)r FK(\))d
FD(!)e FC(w)r FK(,)28 b(so)f(again)f(b)n(y)i(the)g(construction)e(of)i
FD(!)2424 2516 y Fw(0)2475 2546 y FK(w)n(e)f(ha)n(v)n(e)g
FC(r)e FD(!)2934 2516 y Fw(0)2981 2546 y FC(w)r FK(.)j
Fu(2)606 2645 y FK(Th)n(us,)g(in)h(atomic)g(runs,)f(the)h(partially)f
(ordered)f(set)i(of)f(accesses)f(can)h(b)r(e)i(linearized)523
2745 y(while)f(resp)r(ecting)g(the)h(logical)d(read/write)h(order.)g
(In)h(addition)g(to)g(de\014nition)h(6.2,)e(w)n(e)523
2845 y(ha)n(v)n(e:)523 3006 y Fv(Definition)k FH(6.3)41
b FC(\033)30 b FK(is)d FG(r)l(e)l(gular)g FK(i\013)h(A0)f(and)g(A2)f
(hold.)h FC(\033)k FK(is)c FG(safe)h FK(i\013)f(\()p
FD(A)18 b(\000)f(R)3012 2976 y Fw(0)3035 3006 y FC(;)d
FD(!)p FC(;)g(\031)s FK(\))28 b(is)523 3106 y(atomic,)c(where)f
FD(R)1121 3076 y Fw(0)1168 3106 y FK(=)f FD(f)p FC(r)k
FD(2)d(Rj9)p FC(w)j FD(2)e(W)7 b FK(\()p FC(w)26 b FD(6!)d
FC(r)13 b FD(^)e FC(r)27 b FD(6!)c FC(w)r FK(\))p FD(g)h
FK(is)g(the)g(set)g(of)g(reads)f(whic)n(h)523 3205 y(o)n(v)n(erlap)j(a)
h(write)g(\(in)h(whic)n(h)g(case)f FC(\031)k FK(is)c(left)h
(unde\014ned\).)606 3367 y(Th)n(us,)g(in)h(a)f(safe)g(run,)h(a)f(read)g
(o)n(v)n(erlapping)e(a)i(write)g(ma)n(y)g(return)g(an)n(y)g(v)-5
b(alue)29 b(in)f(the)523 3467 y(domain)35 b(of)g(the)g(v)-5
b(ariable.)35 b(The)g(other)f(actions)h(will)g(then)h(b)r(e)f(totally)g
(ordered,)f(suc)n(h)523 3566 y(that)26 b(eac)n(h)f(non-o)n(v)n
(erlapping)e(read)i(returns)g(the)h(v)-5 b(alue)26 b(written)g(b)n(y)f
(the)h(last)g(preceding)523 3666 y(write.)606 3766 y(In)33
b(a)g(regular)e(run,)i(a)g(read)f(ma)n(y)g(return)h(the)h(v)-5
b(alue)33 b(of)g(either)f(the)i(last)f(completed)523
3865 y(write)c(or)g(of)h(an)n(y)e(of)i(the)g(o)n(v)n(erlapping)d
(writes.)i(Th)n(us,)h(during)f(a)g(long)g(write,)g(one)h(read)523
3965 y(ma)n(y)h(obtain)f(the)i(new)f(v)-5 b(alue,)31
b(and)g(the)h(next)f(read)g(the)g(old)g(v)-5 b(alue.)31
b(This)h(violation)e(of)523 4065 y(A1)e(is)h(called)f(a)g
FG(new-old)k(inversion)p FK(,)d(and)g(is)f(what)h(distinguishes)f(a)g
(regular)f(run)h(from)523 4164 y(an)f(atomic)g(one.)523
4326 y Fv(Definition)32 b FH(6.4)41 b FK(V)-7 b(ariable)40
b FC(V)60 b FK(is)41 b(atomic)g(\(regular,safe\))f(i\013)i(for)e(eac)n
(h)h(of)g(its)g(runs)523 4425 y(\()p FD(A)p FC(;)14 b
FD(!)p FK(\),)22 b(there)g(exists)f(a)g(reading)f(function)i
FC(\031)s FK(,)f(suc)n(h)h(that)f(the)h(system)f(execution)g(\()p
FD(A)p FC(;)14 b FD(!)523 4525 y FC(;)g(\031)s FK(\))28
b(is)g(atomic)f(\(regular,safe\).)606 4687 y(It)g(is)g(clear)e(from)i
(these)f(de\014nitions)h(that)g(an)g(atomic)f(v)-5 b(ariable)25
b(is)i(regular,)e(and)h(that)523 4786 y(a)32 b(regular)e(v)-5
b(ariable)31 b(is)h(safe.)g(Whereas)f(an)h(atomic)g(bit)h(app)r(ears)e
(to)h(c)n(hange)f(its)h(v)-5 b(alue)523 4886 y(in)29
b(a)g(single,)f(indivisible)h(time)h(instan)n(t,)f(the)g(v)-5
b(alue)29 b(of)g(a)f(c)n(hanged)g(regular)f(bit)j FG(\015ickers)523
4986 y FK(during)j(the)g(c)n(hange,)f(and)h(only)f(stabilizes)g(to)h
(the)h(complemen)n(tary)d(v)-5 b(alue)33 b(when)g(the)523
5085 y(c)n(hanging)26 b(write)i(\014nishes.)606 5185
y(W)-7 b(e)25 b(call)e(a)h(reading)f(function)i FG(normal)f
FK(if)h(it)g(satis\014es)e(A0,)h(i.e.)g(it)h(do)r(esn't)f(map)g(a)g
(read)523 5284 y(to)h(a)g(write)h(whic)n(h)f(starts)f(after)i(the)f
(read)g(\014nishes.)g(In)h(practice,)f(only)g(normal)f(reading)523
5384 y(functions)k(are)f(considered.)p eop end
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TeXDict begin 69 78 bop 2515 299 a FF(6.3.)53 b(Preliminaries)156
b(69)523 506 y Fl(6.3.1)99 b(Making)34 b(Safe)g(Bits)h(R)-5
b(e)g(gular)523 656 y FK(A)23 b(w)n(ell)g(kno)n(wn)f(tec)n(hnique)g(to)
h(mak)n(e)f(a)g(safe)h(bit)g(regular)e(\([10)o(]\),)j(is)e(to)h(a)n(v)n
(oid)e(writing)h(the)523 756 y(old)27 b(v)-5 b(alue)28
b(to)f(the)h(bit,)g(i.e.)f(only)g FG(changing)i FK(it.)f(Then)f(for)g
(a)g(read)g(o)n(v)n(erlapping)e(a)i(write,)523 855 y(the)h(v)-5
b(alue)27 b(it)g(returns,)g(b)r(e)g(it)h(a)e(0)h(or)f(a)h(1,)g(will)g
(alw)n(a)n(ys)e(b)r(e)j(either)f(the)g(v)-5 b(alue)27
b(of)h(the)f(last)523 955 y(completed)d(write,)g(or)f(the)h(v)-5
b(alue)24 b(of)g(an)g(o)n(v)n(erlapping)e(write.)h(The)h(safe)g(bits)g
(used)g(in)h(our)523 1054 y(constructions)i(are)g(only)h(c)n(hanged,)f
(not)i(o)n(v)n(erwritten)d(with)j(the)f(old)g(v)-5 b(alue,)29
b(and)f(so)f(w)n(e)523 1154 y(can)33 b(assume)f(for)h(them)g(the)h
(existence)f(of)g(a)f(total)h(reading)f(function,)i(as)e(for)h(regular)
523 1254 y(bits.)523 1561 y FL(6.4)116 b(Problem)37 b(Statemen)m(t)523
1760 y FK(It)32 b(remains)e(to)i(de\014ne)g(what)f(it)h(means)f(to)g
(construct)g(an)h(atomic)f(v)-5 b(ariable)30 b(from)h(safe)523
1860 y(bits.)i(\(Recall)g(that)g(in)g(this)h(c)n(hapter)e(all)g(v)-5
b(ariables)32 b(are)g(single-writer,)f(single-reader,)523
1959 y(hence)e(t)n(w)n(o)f(users\).)g(Suc)n(h)h(a)f(construction)g(is)h
(de\014ned)g(b)n(y)g(an)f FG(ar)l(chite)l(ctur)l(e)h
FK(and)g(a)f(pair)523 2059 y(of)35 b FG(pr)l(oto)l(c)l(ols)p
FK(,)g(one)f(for)g(eac)n(h)g(user.)g(The)h(arc)n(hitecture)e(sp)r
(eci\014es)h(the)h(n)n(um)n(b)r(er)g(of)f(safe)523 2159
y(bits,)29 b(their)f(names)f(and)i(ho)n(w)e(they)h(are)g(connected)g
(among)f(the)h(t)n(w)n(o)g(users.)f(Eac)n(h)g(safe)523
2258 y(bit)37 b(can)e(b)r(e)i(connected)e(to)h(the)h(reader)d(and)i
(the)g(writer)g(in)g(one)f(of)h(only)g(t)n(w)n(o)f(w)n(a)n(ys:)523
2358 y(c)n(hanged)c(b)n(y)h(the)g(writer)f(and)h(read)f(b)n(y)h(the)g
(reader,)f(or)g(c)n(hanged)f(b)n(y)i(the)g(reader)f(and)523
2457 y(read)c(b)n(y)g(the)h(writer.)f(The)h(user)f(that)h(can)f(c)n
(hange)f(a)h(bit)i(is)e(said)g(to)h(b)r(e)g(the)g FG(owner)p
FK(.)606 2557 y(A)35 b(proto)r(col)d(sp)r(eci\014es)i(ho)n(w)g(the)g
(writer)g(\(reader\))f(can)h(c)n(hange)f(\(read\))h(the)g(atomic)523
2657 y(v)-5 b(ariable)31 b(in)g(terms)h(of)f(c)n(hanges)f(to)h(and)h
(reads)e(from)h(the)h(safe)f(bits.)h(In)g(addition,)f(the)523
2756 y(proto)r(cols)21 b(ma)n(y)i(mak)n(e)f(use)g(of)h
FG(lo)l(c)l(al)k(variables)p FK(,)e(whic)n(h)d(can)h(b)r(e)g(view)n(ed)
g(as)f(safe)g(bits)h(that)523 2856 y(are)h(c)n(hanged)f(and)h(read)g(b)
n(y)g(the)h(same)f(user.)g(These)g(are)g(ho)n(w)n(ev)n(er)e(not)j
(considered)e(part)523 2956 y(of)28 b(the)g(arc)n(hitecture,)e(whic)n
(h)i(sp)r(eci\014es)f(only)g FG(shar)l(e)l(d)i FK(bits.)606
3055 y(W)-7 b(e)29 b(consider)e(only)i FG(wait-fr)l(e)l(e)g
FK(proto)r(cols,)e(i.e.,)i(the)g(n)n(um)n(b)r(er)f(of)g(safe)g(bit)i
(accesses)c(in)523 3155 y(a)f(single)f(proto)r(col)g(execution)g(m)n
(ust)h(b)r(e)h(b)r(ounded)f(b)n(y)g(a)f(\014xed)h(constan)n(t.)f(This)h
(require-)523 3254 y(men)n(t)38 b(forbids)f(solutions)f(in)i(whic)n(h)f
(a)g(user)g(migh)n(t)g(ha)n(v)n(e)g(to)g(w)n(ait)g(for)g(a)g(safe)g
(bit)h(to)523 3354 y(c)n(hange)26 b(v)-5 b(alue.)606
3454 y(A)26 b(read)f(or)g(write)h(action)f(on)g(the)i(atomic)e(v)-5
b(ariable)25 b(consists)g(of)h(an)f(execution)h(of)f(the)523
3553 y(corresp)r(onding)31 b(proto)r(col.)g(W)-7 b(e)34
b(use)e(the)h(terms)g(\\action")e(and)i(\\proto)r(col)e(execution")523
3653 y(in)n(terc)n(hangeably)-7 b(.)30 b(A)i(construction)g(is)g
(initialized)g(b)n(y)g(an)f(initial)h(write)g(that)h(sets)e(the)523
3753 y(atomic)c(v)-5 b(ariable)26 b(to)h(the)g(v)-5 b(alue)27
b(0.)g(This)g(allo)n(ws)f(the)i(de\014nition)f(of)g(a)g(reading)f
(function)523 3852 y(on)20 b(ev)n(ery)f(read)g(action.)h(All)g(other)g
(shared)f(bits)h(and)g(lo)r(cal)g(v)-5 b(ariables)18
b(are)i(also)f(initialized)523 3952 y(to)26 b(0.)g(Finally)-7
b(,)27 b(eac)n(h)e(run)h(of)h(the)f(construction)g(m)n(ust)g(satisfy)g
(the)h(atomicit)n(y)f(criterion.)606 4051 y(In)34 b(the)g(next)f
(section)g(w)n(e)g(consider)g(the)h(sp)r(ecial)f(case)g(of)g(a)g(2-v)-5
b(alued)33 b(atomic)g(v)-5 b(ari-)523 4151 y(able.)29
b(After)h(pro)n(ving)e(that)i(3)f(safe)h(bits)g(are)e(needed)i(to)f
(construct)g(an)h(atomic)f(bit,)h(w)n(e)523 4251 y(dev)n(elop)g(a)h
(construction)f(that)i(ac)n(hiev)n(es)d(this)j(lo)n(w)n(er)d(b)r(ound,)
j(follo)n(w)n(ed)e(b)n(y)h(a)f(pro)r(of)h(of)523 4350
y(correctness.)20 b(The)i(general)f(case)g(of)h(a)g FC(b)p
FK(-bit)g(\(2)1996 4320 y FA(b)2029 4350 y FK(-v)-5 b(alued\))22
b(atomic)f(v)-5 b(ariable)21 b(is)h(dealt)g(with)523
4450 y(in)28 b(Section)f(6.6.)523 4757 y FL(6.5)116 b(Optimal)36
b(Construction)g(of)j(A)m(tomic)d(Bits)523 4973 y Fl(6.5.1)99
b(A)35 b(L)-5 b(ower)35 b(Bound)f(on)g(the)h(Numb)-5
b(er)35 b(of)g(Safe)f(Bits)h(ne)-5 b(e)g(de)g(d)33 b(to)832
5089 y(Construct)i(an)f(A)n(tomic)h(Bit)523 5239 y FK(W)-7
b(e)34 b(demonstrate)f(that)h(3)g(safe)f(bits)h(are)f(required)g(in)h
(a)f(construction)g(of)h(an)f(atomic)523 5338 y(bit,)28
b(in)g(particular,)e(2)i(bits)g(o)n(wned)f(b)n(y)g(the)h(writer,)f(and)
g(1)g(o)n(wned)g(b)n(y)h(the)g(reader.)p eop end
%%Page: 70 80
TeXDict begin 70 79 bop 1700 299 a FF(6.5.)53 b(Optimal)26
b(Construction)g(of)h(A)n(tomic)e(Bits)155 b(70)606 506
y FK(The)28 b(reason)e(that)i(a)f(single)h(bit)g(\(call)g(it)g
FC(V)19 b FK(\),)28 b(o)n(wned)f(b)n(y)h(the)g(writer)f(do)r(esn't)h
(su\016ce,)523 606 y(is)k(exempli\014ed)h(b)n(y)f(a)f(run)i(\()p
FD(A)p FC(;)14 b FD(!)p FC(;)g(\031)s FK(\),)33 b(where)f
FD(A)f FK(=)f FD(f)p FC(w)r(;)14 b(r)2410 618 y FB(1)2448
606 y FC(;)g(r)2522 618 y FB(2)2559 606 y FD(g)p FK(,)32
b(and)g FD(!)p FK(=)f FD(f)p FK(\()p FC(r)3112 618 y
FB(1)3149 606 y FC(;)14 b(r)3223 618 y FB(2)3261 606
y FK(\))p FD(g)p FK(,)523 706 y(i.e.)23 b(write)f FC(w)j
FK(\(whic)n(h)e(c)n(hanges)e(the)i(atomic)f(bit)h(from)f(its)g(initial)
h(v)-5 b(alue)22 b(0)g(to)h(1\),)f(o)n(v)n(erlaps)523
805 y(t)n(w)n(o)k(reads)f FC(r)932 817 y FB(1)996 805
y FK(and)i FC(r)1194 817 y FB(2)1231 805 y FK(.)g(Without)g(loss)f(of)g
(generalit)n(y)-7 b(,)25 b(w)n(e)h(can)g(assume)g(that)h(the)f(writer)
523 905 y(c)n(hanges)38 b FC(V)58 b FK(at)39 b(least)g(once)f(during)h
(the)h(execution)e(of)i(its)f(proto)r(col.)f(W)-7 b(e)39
b(ma)n(y)g(also)523 1005 y(assume)25 b(that,)g(dep)r(ending)h(on)f(the)
h(v)-5 b(alues)25 b(obtained)g(b)n(y)g(reading)f FC(V)19
b FK(,)26 b(but)g(indep)r(enden)n(t)523 1104 y(of)31
b(the)h(v)-5 b(alues)30 b(of)h(lo)r(cal)g(v)-5 b(ariables)30
b(at)h(the)g(start)g(of)g(a)g(read,)f(the)i(reader)d(proto)r(col)h(can)
523 1204 y(return)d(b)r(oth)h(a)e(0)h(and)g(a)g(1.)g(Consider)g(no)n(w)
f(the)i(case)e(when)i(b)r(oth)f(reads)g(o)r(ccur)f(during)523
1303 y(the)i(\014rst)g(time)g(that)g FC(w)j FK(c)n(hanges)26
b FC(V)19 b FK(.)28 b(Then)g(due)g(to)f(the)h(\015ic)n(k)n(ering)f(of)g
FC(V)19 b FK(,)28 b(it)g(is)g(p)r(ossible)523 1403 y(for)21
b FC(r)681 1415 y FB(1)740 1403 y FK(to)h(return)f(1)g(while)h
FC(r)1395 1415 y FB(2)1454 1403 y FK(returns)f(0.)g(But)h(no)n(w)e
(there)i(is)f(no)g(reading)g(function)h(whic)n(h)523
1503 y(satis\014es)27 b(all)g(three)h(atomicit)n(y)f(conditions.)606
1602 y(The)36 b(example)f(sho)n(ws)f(that)i(a)f(comm)n(unication)f(c)n
(hannel)h(from)g(the)h(writer)f(to)g(the)523 1702 y(reader)26
b(of)i(only)f(one)g(safe)g(bit)h(is)g(to)r(o)f(narro)n(w|at)f(least)h
(2)g(safe)g(bits)h(are)f(necessary)-7 b(.)606 1802 y(In)22
b([10)o(],)h(Lamp)r(ort)e(has)h(sho)n(wn)f(the)i(necessit)n(y)e(of)h(t)
n(w)n(o-w)n(a)n(y)e(comm)n(unication,)h(i.e.,)h(the)523
1901 y(reader)30 b(m)n(ust)h(o)n(wn)g(at)g(least)g(1)f(bit.)i(The)f
(sequence)g(of)g(safe)g(bits)g(c)n(hanged)g(during)f(the)523
2001 y(proto)r(col)25 b(execution)i(of)f(the)h(writer,)f(m)n(ust)h
(therefore)f(dep)r(end)h(on)f(information)g(that)h(it)523
2100 y(receiv)n(es)f(from)h(the)h(reader.)523 2265 y
Fv(Lemma)k FH(6.5)40 b FG(A)22 b(c)l(onstruction)g(of)h(an)f(atomic)h
(bit)f(fr)l(om)h(non-atomic,)g(safe)g(bits)f(r)l(e)l(quir)l(es)523
2365 y(at)30 b(le)l(ast)g FK(2)f FG(bits)h(owne)l(d)g(by)g(the)g
(writer,)h(and)f(at)g(le)l(ast)g FK(1)f FG(bit)h(owne)l(d)g(by)g(the)g
(r)l(e)l(ader.)606 2529 y FK(The)25 b(next)g(few)h(sections)e(deal)h
(with)g(the)g(dev)n(elopmen)n(t)g(of)g(a)f(solution)h(meeting)g(these)
523 2629 y(b)r(ounds.)j(Let)g(us)f(\014rst)g(describ)r(e)h(the)g(arc)n
(hitecture)e(of)i(the)g(construction.)523 2878 y Fl(6.5.2)99
b(The)34 b(A)n(r)-5 b(chite)g(ctur)g(e)523 3028 y FK(W)e(e)26
b(aim)f(to)g(attain)h(the)f(optimal)h(n)n(um)n(b)r(er)f(of)g(shared)g
(safe)g(bits,)g(whic)n(h)h(is)f(3.)g(As)h(sho)n(wn)523
3128 y(ab)r(o)n(v)n(e,)c(the)h(reader)f(will)h(b)r(e)h(the)g(o)n(wner)e
(of)h(one)f(of)h(these,)h(whic)n(h)f(w)n(e'll)g(call)g(\\R.")f(One)h
(of)523 3227 y(the)g(t)n(w)n(o)f(bits)i(o)n(wned)e(b)n(y)h(the)g
(writer)f(will)h(b)r(e)g(used)g(to)g(hold)g(the)g(v)-5
b(alue)23 b(of)g(the)g(sim)n(ulated)523 3327 y(atomic)g(bit)h(and)g(is)
f(called)g(\\V.")g(The)h(other)f(bit)h(o)n(wned)e(b)n(y)i(the)g(writer)
e(is)i(named)f(\\W".)523 3426 y(T)-7 b(o)27 b(sum)h(up,)g(w)n(e)f(ha)n
(v)n(e)g(the)h(follo)n(wing)e(3)h(safe)h(bits:)523 3599
y FH(W)-8 b(riter)32 b FD(!)p 915 3517 115 4 v 915 3624
4 107 v 21 w FH(V)p 1026 3624 V 915 3627 115 4 v 21 w
FD(!)g FH(Reader)857 b FK(v)-5 b(alue)27 b(of)h(sim)n(ulated)f(atomic)g
(bit)523 3786 y FH(W)-8 b(riter)32 b FD(!)p 915 3704
V 915 3811 4 107 v 9 w FH(W)p 1026 3811 V 915 3814 115
4 v 6 w FD(!)g FH(Reader)1415 b FK(\015ag)26 b(for)h(writer)523
3972 y FH(W)-8 b(riter)32 b FD( )p 915 3890 V 915 3997
4 107 v 21 w FH(R)p 1026 3997 V 915 4000 115 4 v 21 w
FD( )g FH(Reader)1405 b FK(\015ag)27 b(for)g(reader)523
4222 y Fl(6.5.3)99 b(The)34 b(Pr)-5 b(oto)g(c)g(ols)523
4372 y FK(In)26 b(the)h(proto)r(cols,)e(w)n(e)h(mak)n(e)f(use)i(of)f
(the)g(follo)n(wing)g(statemen)n(ts.)g(The)g(o)n(wner)f(of)h(a)g(safe)
523 4471 y(bit)33 b FC(B)k FK(can)32 b(execute)h(the)g(statemen)n(t)f
(\\)p Fk(change)41 b(B)p FK(")32 b(to)g(c)n(hange)g(its)g(v)-5
b(alue.)33 b(Remem)n(b)r(er)523 4571 y(that)25 b(during)g(this)g(c)n
(hange,)f(the)i(v)-5 b(alue)25 b(ma)n(y)f(\015ic)n(k)n(er)g(b)r(et)n(w)
n(een)h(0)g(and)g(1.)g(Lo)r(cal)f(v)-5 b(ariables)523
4671 y(ha)n(v)n(e)37 b(lo)n(w)n(er)g(case)g(names)h(to)h(distinguish)f
(them)h(from)f(the)h(shared)e(bits.)i(In)f(all)g(the)523
4770 y(proto)r(cols)30 b(presen)n(ted)g(here,)h(the)g(lo)r(cal)g(v)-5
b(ariables)29 b(are)i(2-v)-5 b(alued)30 b(\(bits\),)i(and)f(are)f(used)
523 4870 y(to)g(hold)f(a)h(cop)n(y)e(of)i FC(V)19 b FK(.)30
b(F)-7 b(or)29 b(this)h(purp)r(ose)f(there)g(is)h(a)f(statemen)n(t)h
(\\)p Fk(read)41 b(loc)i(:=)g(V)p FK(,")523 4969 y(whose)21
b(e\013ect)i(is)f(to)g(read)f FC(V)41 b FK(and)22 b(store)f(the)h
(result)g(in)g(the)g(lo)r(cal)g(v)-5 b(ariable)21 b FC(l)r(oc)p
FK(.)h(Giv)n(en)f(the)523 5069 y(regularit)n(y)30 b(of)h(a)g(c)n
(hanged-only)f(safe)h(bit,)h(w)n(e)f(kno)n(w)g(that)h(the)g(c)n(hanges)
e(to)i(and)f(reads)523 5169 y(from)h(it)h(ob)r(ey)g(the)g(conditions)f
(A0)g(and)h(A2)f(\(see)h(de\014nition)g(6.4\),)f(for)g(some)g(reading)
523 5268 y(function.)c(There)e(is)h(the)g(conditional)f(\\)p
Fk(if)g FK(test)h Fk(then)f FK(statemen)n(t.",)g(with)i(the)f(ob)n
(vious)523 5368 y(seman)n(tics.)f(The)g(test)h(is)f(either)h(")p
Fk(W==R)p FK(")d(or)h(\\)p Fk(W<>R)p FK(.")f(P)n(erforming)h(suc)n(h)h
(a)g(test)h(is)f(done)p eop end
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TeXDict begin 71 80 bop 1700 299 a FF(6.5.)53 b(Optimal)26
b(Construction)g(of)h(A)n(tomic)e(Bits)155 b(71)523 506
y FK(b)n(y)34 b(\014rst)f(reading)g(the)h(\015ag)g(o)n(wned)f(b)n(y)h
(the)g(other)f(user)h(\(e.g.)f(the)i(writer)e(reads)g
FC(R)h FK(in)523 606 y(its)h(test\).)f(This)h(read)e(is)h(implicit)i
(in)e(the)h(test,)f(and)h(is)f(not)g(stated)g(explicitly)h(in)f(the)523
706 y(proto)r(col)i(as)g(a)g(separate)g Fk(read)f FK(statemen)n(t.)i
(In)g(order)f(to)g(b)r(e)i(able)e(to)h(compare)e(this)523
805 y(v)-5 b(alue)31 b(against)f(the)i(v)-5 b(alue)31
b(of)g(one's)f(o)n(wn)h(\015ag,)f(w)n(e)h(assume)g(that)g(the)h(o)n
(wner)e(of)h(a)g(\015ag)523 905 y(k)n(eeps)26 b(trac)n(k)f(of)h(its)g
(v)-5 b(alue.)27 b(This)f(abbreviated)f(notation)h(for)g(tests)g(will)h
(pro)n(v)n(e)d(to)j(mak)n(e)523 1005 y(the)g(proto)r(cols)e(more)h
(concise)f(and)i(readable.)e(The)i(\014nal)f(statemen)n(t)h(in)f(our)g
(rep)r(ertoire)523 1104 y(is)f(\\)p Fk(return)41 b(loc)p
FK(,")24 b(with)i FC(l)r(oc)f FK(again)g(a)g(lo)r(cal)g(bit.)h(It)g(is)
f(used)h(b)n(y)f(the)h(reader)e(to)i(exit)f(the)523 1204
y(execution)i(of)h(its)g(proto)r(col)e(and)h(to)h(sp)r(ecify)g(the)g
(return)f(v)-5 b(alue.)606 1303 y(W)e(e)28 b(can)f(no)n(w)g(state)h
(the)g(proto)r(col:)523 1469 y Fk(WRITER)41 b(PROTOCOL)781
b(READER)41 b(PROTOCOL)523 1669 y(change)g(V)1089 b(1.)43
b(if)g(W==R)f(then)g(return)f(v)523 1768 y(if)i(W==R)f(then)g(change)f
(W)522 b(2.)43 b(read)f(x)h(:=)g(V)1961 1868 y(3.)g(if)g(W<>R)f(then)g
(change)f(R)1961 1968 y(4.)i(read)f(v)h(:=)g(V)1961 2067
y(5.)g(if)g(W==R)f(then)g(return)f(v)1961 2167 y(6.)i(read)f(v)h(:=)g
(V)1961 2267 y(7.)g(return)e(x)523 2516 y Fl(6.5.4)99
b(Handshaking)523 2665 y FK(A)27 b(handshaking)e(mec)n(hanism)h(is)g
(emplo)n(y)n(ed)f(to)i(let)f(the)h(reader)e(detect)i(when)f(a)g(c)n
(hange)523 2765 y(of)h FC(V)45 b FK(is)27 b(\014nalised.)f(After)h(c)n
(hanging)f FC(V)19 b FK(,)26 b(the)h(writer)f(p)r(erforms)g(a)h
(handshak)n(e)e(b)n(y)h(trying)523 2864 y(to)38 b(mak)n(e)f
FC(W)52 b FD(6)p FK(=)39 b FC(R)q FK(.)f(The)g(basic)f(plan)h(for)f
(the)h(reader)f(is)h(to)f(p)r(erform)h(a)f(handshak)n(e)523
2964 y(\(trying)27 b(to)f(mak)n(e)g FC(W)35 b FK(=)23
b FC(R)q FK(\))k(in)g(b)r(et)n(w)n(een)g(t)n(w)n(o)f(reads)g(of)g
FC(V)19 b FK(.)27 b(If)h(afterw)n(ards,)d FC(W)39 b FK(is)26
b(found)523 3064 y(to)i(still)g(equal)f FC(R)h FK(in)g(v)-5
b(alue,)28 b(then)g(the)g(read)f(returns)g(the)h(second)f(v)-5
b(alue,)28 b(and)f(un)n(til)h(the)523 3163 y(next)35
b(writer's)e(handshak)n(e)g(is)i(detected,)f(future)h(reads)e(can)h
(return)g(this)h(same)f(v)-5 b(alue)523 3263 y(with)32
b(no)e(risk)g(of)h(new-old)g(in)n(v)n(ersions.)e(If,)i(on)g(the)g
(other)g(hand,)g FC(W)43 b FK(is)31 b(found)g(to)g(ha)n(v)n(e)523
3362 y(b)r(een)j(c)n(hanged)e(b)n(y)g(the)i(writer,)e(then)i(the)g
(\014rst)e(v)-5 b(alue,)34 b FC(x)f FK(is)g(returned.)g(In)g(that)h
(case,)523 3462 y FC(v)s FK(,)g(whic)n(h)f(migh)n(t)g(b)r(e)h(returned)
e(in)i(line)f(1)g(of)g(the)h(next)f(read)g(action,)f(is)h(assigned)f
(the)523 3562 y(curren)n(t)27 b(v)-5 b(alue)27 b(of)h
FC(V)19 b FK(,)27 b(since)h(that)g(cannot)f(form)g(a)g(new-old)g(in)n
(v)n(ersion)f(with)i FC(x)p FK(.)523 3811 y Fl(6.5.5)99
b(Pr)-5 b(o)g(of)34 b(of)h(Corr)-5 b(e)g(ctness)523 3960
y FK(Let)29 b(\()p FD(A)p FC(;)14 b FD(!)p FK(\))30 b(b)r(e)f(a)g(run)f
(of)h(the)g(atomic)g(bit)g(construction.)f(Then)i(w)n(e)e(can)h(\014nd)
g(a)f(lo)n(w)n(er)523 4060 y(lev)n(el)e(regular)e(run)j(for)e(eac)n(h)h
(of)g(the)h(three)f(safe)g(bits,)h(consisting)e(of)h(all)g(the)h(c)n
(hanges)e(to)523 4159 y(and)h(reads)f(of)i(that)f(safe)g(bit,)h(and)f
(the)h(precedence)f(relation)f(de\014ned)i(from)f(their)g(start)523
4259 y(and)e(\014nish)g(times.)g(Let)f FC(\031)1334 4229
y Fw(0)1382 4259 y FK(b)r(e)h(a)f(reading)g(function)h(that)g(mak)n(es)
f(the)h(run)f(on)h FC(V)42 b FK(regular.)523 4359 y(Let)33
b FD(W)41 b FK(b)r(e)33 b(the)h(set)f(of)g(write)g(actions)g(in)g
FD(A)p FK(,)h(and)f FD(R)h FK(the)g(set)f(of)g(read)f(actions)h(in)g
FD(A)p FK(.)523 4458 y(W)-7 b(e)24 b(m)n(ust)h(pro)n(v)n(e)d(the)i
(atomicit)n(y)g(of)g FC(\033)i FK(=)d(\()p FD(A)p FC(;)14
b FD(!)p FC(;)g(\031)s FK(\))25 b(for)e(some)h(reading)e(function)j
FC(\031)s FK(.)f(W)-7 b(e)523 4558 y(de\014ne)29 b FC(\031)j
FK(in)d(a)f(natural)f(w)n(a)n(y)h(as)g(follo)n(ws.)f(Let)i
FC(r)e FD(2)e(R)k FK(b)r(e)g(an)n(y)f(read)g(action)g(and)g(let)h
FC(l)r(oc)523 4658 y FK(b)r(e)f(the)g(lo)r(cal)g(v)-5
b(ariable)26 b(returned)i(b)n(y)f FC(r)r FK(.)i(W)-7
b(e)28 b(can)f(de\014ne)h FC(\032)p FK(,)g(the)g(subread)f(of)h
FC(r)r FK('s)g(return)523 4757 y(v)-5 b(alue,)27 b(as)g(the)h(last)f
(subread)g(from)g FC(V)46 b FK(in)n(to)27 b FC(l)r(oc)h
FK(b)r(efore)f FC(r)j FK(returns.)d(E.g.)g(if)g FC(r)k
FK(returns)26 b(in)523 4857 y(line)j(7)g(then)h FC(\032)f
FK(is)g(the)g(read)g(in)g(line)g(2)g(of)g FC(r)r FK(,)h(and)f(if)h
FC(r)i FK(returns)c(in)h(line)h(1,)f(then)g FC(\032)g
FK(is)g(the)523 4956 y(read)h(in)h(line)h(4)e(or)g(line)i(6)e(of)h
(some)g(earlier)e(read)h(action.)h(Let)g FC(w)j FK(b)r(e)d(the)g(write)
g(action)523 5056 y(whic)n(h)c(executed)g FC(\031)1153
5026 y Fw(0)1176 5056 y FK(\()p FC(\032)p FK(\))g(in)g(the)h(\014rst)e
(line)h(of)g(its)g(proto)r(col.)e(Then)i(w)n(e)g(de\014ne)g
FC(\031)s FK(\()p FC(r)r FK(\))d(=)f FC(w)r FK(.)606
5156 y FH(Pro)s(of)j(of)h(A0)c FK(In)n(tuitiv)n(ely)-7
b(,)23 b(since)f(the)h(underlying)g(bits)g(are)f(safe,)g(a)h(read)f
(action)g(can)523 5255 y(only)33 b(return)g(the)h(v)-5
b(alue)34 b(of)f(a)h(past)f(or)g(concurren)n(t)f(\(o)n(v)n(erlapping\))
g(write)h(action.)g(W)-7 b(e)523 5355 y(formally)31 b(pro)n(v)n(e)g(A0)
h(b)n(y)g(con)n(tradiction:)f(Assume)h(that)g(for)g(some)g
FC(r)h FD(2)e(R)p FK(,)i FC(r)g FD(!)e FC(\031)s FK(\()p
FC(r)r FK(\).)p eop end
%%Page: 72 82
TeXDict begin 72 81 bop 1700 299 a FF(6.5.)53 b(Optimal)26
b(Construction)g(of)h(A)n(tomic)e(Bits)155 b(72)523 506
y FK(Let)33 b FC(\032)g FK(b)r(e)g(the)g(subread)e(of)i
FC(r)r FK('s)g(return)g(v)-5 b(alue)32 b(as)g(ab)r(o)n(v)n(e.)f(Then)i
FC(f)9 b FK(\()p FC(\032)p FK(\))32 b FC(<)f(f)9 b FK(\()p
FC(r)r FK(\))34 b(and)e(b)n(y)523 606 y(de\014nition)26
b(of)f FC(\031)s FK(,)g FC(s)p FK(\()p FC(\031)s FK(\()p
FC(r)r FK(\)\))h FC(<)c(s)p FK(\()p FC(\031)1570 576
y Fw(0)1594 606 y FK(\()p FC(\032)p FK(\)\).)k(T)-7 b(ogether)24
b(these)h(imply)h(that)f FC(\032)e FD(!)g FC(\031)2968
576 y Fw(0)2992 606 y FK(\()p FC(\032)p FK(\),)j(whic)n(h)523
706 y(con)n(tradicts)g(the)i(safet)n(y)f(of)h FC(\031)1476
676 y Fw(0)1500 706 y FK(.)606 805 y FH(Pro)s(of)38 b(of)g(A1)c
FK(The)f(pro)r(of)g(is)g(again)f(b)n(y)h(con)n(tradiction.)f(Let)h
FC(r)2706 817 y FB(1)2744 805 y FC(;)14 b(r)2818 817
y FB(2)2888 805 y FD(2)33 b(R)g FK(b)r(e)h(suc)n(h)523
905 y(that)c FC(r)742 917 y FB(1)806 905 y FD(!)c FC(r)952
917 y FB(2)1019 905 y FK(and)k FC(\031)s FK(\()p FC(r)1302
917 y FB(2)1340 905 y FK(\))d FD(!)f FC(\031)s FK(\()p
FC(r)1627 917 y FB(1)1665 905 y FK(\).)k(By)f(the)h(\014nite)g(history)
f(assumption,)g(w)n(e)g(can)g(let)523 1005 y FC(r)560
1017 y FB(1)624 1005 y FK(b)r(e)d(the)g(\014rst)f(suc)n(h)h(read)f
(action.)g(Let)h FC(\032)1874 1017 y FA(i)1901 1005 y
FC(;)14 b(i)23 b FD(2)g(f)p FK(1)p FC(;)14 b FK(2)p FD(g)24
b FK(b)r(e)i(the)g(subread)f(from)g FC(V)45 b FK(of)26
b FC(r)3274 1017 y FA(i)3302 1005 y FK('s)523 1104 y(return)31
b(v)-5 b(alue.)32 b(F)-7 b(or)31 b(notational)g(con)n(v)n(enience,)f(w)
n(e)i(use)f(the)h(elemen)n(t-of-set)f(sym)n(b)r(ol)g
FD(2)523 1204 y FK(to)d(denote)g(that)g(a)f(safe)h(bit)g(access)f(is)h
(part)f(of)h(a)g(read)f(or)g(write)h(action.)f(Then)h
FC(\032)3143 1216 y FB(1)3204 1204 y FD(2)c FC(r)3320
1216 y FB(1)523 1303 y FK(if)34 b FC(r)642 1315 y FB(1)712
1303 y FK(returned)f(in)g(line)g(5)g(or)f(line)h(7.)g(Otherwise,)f(if)h
FC(r)2300 1315 y FB(1)2371 1303 y FK(returned)g(in)g(line)g(1,)g(then)g
(b)n(y)523 1403 y(the)c(minimalit)n(y)g(of)g FC(r)1221
1415 y FB(1)1259 1403 y FK(,)f FC(\032)1353 1415 y FB(1)1419
1403 y FK(is)h(in)g(the)g(immediately)g(preceding)f(read)g(action.)g
(De\014ning)523 1503 y FC(!)575 1515 y FA(i)636 1503
y FK(as)33 b FC(\031)794 1473 y Fw(0)817 1503 y FK(\()p
FC(\032)892 1515 y FA(i)920 1503 y FK(\),)h(w)n(e)f(also)g(ha)n(v)n(e)f
FC(!)1559 1515 y FA(i)1619 1503 y FD(2)h FC(\031)s FK(\()p
FC(r)1826 1515 y FA(i)1855 1503 y FK(\).)h(F)-7 b(rom)33
b FC(\031)s FK(\()p FC(r)2287 1515 y FB(2)2325 1503 y
FK(\))g FD(!)g FC(\031)s FK(\()p FC(r)2625 1515 y FB(1)2664
1503 y FK(\))h(follo)n(ws)e FC(!)3060 1515 y FB(2)3130
1503 y FD(!)h FC(!)3298 1515 y FB(1)3335 1503 y FK(.)523
1602 y(According)d(to)i(the)f(reader)f(proto)r(col,)g
FC(r)1820 1614 y FB(1)1888 1602 y FD(!)f FC(r)2037 1614
y FB(2)2106 1602 y FK(implies)j FC(\032)2435 1614 y FB(1)2501
1602 y FK(=)d FC(\032)2638 1614 y FB(2)2707 1602 y FK(or)h
FC(\032)2855 1614 y FB(1)2922 1602 y FD(!)f FC(\032)3077
1614 y FB(2)3114 1602 y FK(.)j(Since)523 1702 y FC(\031)s
FK(\()p FC(r)642 1714 y FB(1)680 1702 y FK(\))41 b FD(6)p
FK(=)f FC(\031)s FK(\()p FC(r)977 1714 y FB(2)1015 1702
y FK(\),)f FC(\032)1152 1714 y FB(1)1230 1702 y FD(6)p
FK(=)h FC(\032)1378 1714 y FB(2)1453 1702 y FK(hence)e
FC(\032)1737 1714 y FB(1)1815 1702 y FD(!)i FC(\032)1981
1714 y FB(2)2018 1702 y FK(.)f(Since)f(the)g(run)g(on)g
FC(V)57 b FK(is)38 b(regular,)e(w)n(e)523 1802 y(ha)n(v)n(e)e
FC(s)p FK(\()p FC(!)845 1814 y FB(1)882 1802 y FK(\))i
FC(<)g(f)9 b FK(\()p FC(\032)1176 1814 y FB(1)1213 1802
y FK(\))36 b(\(from)f(A0\))g(and)g FC(s)p FK(\()p FC(\032)1971
1814 y FB(2)2009 1802 y FK(\))h FC(<)f(f)9 b FK(\()p
FC(!)2311 1814 y FB(1)2348 1802 y FK(\))36 b(\(from)f(A2\).)h(Since)f
FC(!)3123 1814 y FB(1)3195 1802 y FK(lasts)523 1901 y(throughout)27
b(the)h(time)g(in)n(terv)-5 b(al)27 b([)p FC(f)9 b FK(\()p
FC(\032)1737 1913 y FB(1)1774 1901 y FK(\))p FC(;)14
b(s)p FK(\()p FC(\032)1957 1913 y FB(2)1995 1901 y FK(\)],)664
2084 y(all)28 b(reads)e(from)i FC(W)39 b FK(b)r(et)n(w)n(een)28
b FC(f)9 b FK(\()p FC(\032)1757 2096 y FB(1)1794 2084
y FK(\))28 b(and)f FC(s)p FK(\()p FC(\032)2129 2096 y
FB(2)2166 2084 y FK(\))h(obtain)g(the)g(same)f(v)-5 b(alue)p
FC(:)141 b FK(\(6.1\))606 2267 y(W)-7 b(e)28 b(no)n(w)f(consider)g(all)
g(three)g(p)r(ossible)h(cases)e(of)i(the)g(p)r(osition)f(of)g
FC(\032)2785 2279 y FB(1)2823 2267 y FK(.)523 2433 y
Fk(read)42 b(x)h(:=)g(V)31 b FH(in)h(line)e(2)42 b FK(Then)34
b FC(r)1677 2445 y FB(1)1749 2433 y FK(returned)g(in)g(line)h(7)e(of)i
(its)f(proto)r(col)f(execution,)731 2532 y(after)e(seeing)g
FC(W)42 b FD(6)p FK(=)29 b FC(R)j FK(in)g(line)g(5.)f(Ho)n(w)n(ev)n
(er,)f(at)i(that)f(p)r(oin)n(t,)h(the)g(v)-5 b(alue)32
b(of)f FC(R)i FK(is)731 2632 y(the)i(same)g(as)f(the)i(v)-5
b(alue)34 b(read)h(from)f FC(W)48 b FK(in)35 b(line)g(3.)g(Because)f
FC(\032)2836 2644 y FB(2)2908 2632 y FK(is)h(either)g(the)731
2731 y(read)25 b(in)i(line)g(6)f(of)g(this)h(proto)r(col)e(execution,)h
(or)g(a)g(later)f(read,)h(w)n(e)g(ha)n(v)n(e)f(found)i(a)731
2831 y(con)n(tradiction)f(with)i(\(6.1\))f(ab)r(o)n(v)n(e.)523
2997 y Fk(read)42 b(v)h(:=)g(V)31 b FH(in)h(line)e(4)42
b FK(Then)34 b FC(r)1677 3009 y FB(1)1749 2997 y FK(returned)g(in)g
(line)h(5)e(of)i(its)f(proto)r(col)f(execution,)731 3097
y(after)39 b(seeing)g FC(W)51 b FK(equal)39 b(to)h FC(R)q
FK(.)f(Since)h FC(\032)2068 3109 y FB(1)2148 3097 y FD(!)j
FC(\032)2317 3109 y FB(2)2354 3097 y FK(,)c FC(\032)2459
3109 y FB(2)2536 3097 y FK(m)n(ust)h(b)r(e)g(part)f(of)g(some)731
3196 y(later)29 b(read)g(action)g(whic)n(h)h(sees)f FC(W)42
b FK(di\013eren)n(t)30 b(from)f FC(R)i FK(in)f(line)g(1)f(of)h(its)g
(proto)r(col)731 3296 y(execution.)d(This)h(con)n(tradicts)e(\(6.1\))h
(again.)523 3462 y Fk(read)42 b(v)h(:=)g(V)31 b FH(in)h(line)e(6)42
b FK(Then)37 b FC(r)1680 3474 y FB(1)1754 3462 y FK(returned)f(in)h
(line)g(1)g(of)g(its)f(proto)r(col)g(execution)731 3562
y(\(whic)n(h)26 b(immediately)g(succeeds)g(that)g(of)g
FC(\032)2116 3574 y FB(1)2153 3562 y FK(\))g(after)g(seeing)f
FC(W)38 b FK(equal)26 b(to)g FC(R)q FK(.)g(This)731 3661
y(case)g(therefore)h(reduces)g(to)g(the)h(previous)f(one.)523
3827 y(W)-7 b(e)28 b(ha)n(v)n(e)e(sho)n(wn)h(that)h(the)g(assumed)f
(violation)g(of)g(A1)h(leads)f(to)g(a)g(con)n(tradiction.)606
3927 y FH(Pro)s(of)g(of)f(A2)d FK(The)g(pro)r(of)g(is)g(once)f(again)g
(b)n(y)h(con)n(tradiction.)f(Let)h FC(r)j FD(2)d(R)p
FC(;)14 b(w)26 b FD(2)d(W)30 b FK(b)r(e)523 4027 y(suc)n(h)24
b(that)h FC(\031)s FK(\()p FC(r)r FK(\))f FD(!)f FC(w)j
FD(!)d FC(r)r FK(.)i(Let)g FC(\032)f FK(b)r(e)h(the)g(read)e(from)h
FC(V)43 b FK(of)24 b FC(r)r FK('s)h(return)f(v)-5 b(alue)24
b(as)g(usual,)523 4126 y(and)31 b FC(!)j FK(the)d(write)g(to)g
FC(V)50 b FK(in)31 b FC(w)r FK(.)h(F)-7 b(rom)31 b FC(\031)1827
4096 y Fw(0)1850 4126 y FK(\()p FC(\032)p FK(\))f FD(2)f
FC(\031)s FK(\()p FC(r)r FK(\))k(and)e FC(!)g FD(2)e
FC(w)34 b FK(follo)n(ws)c FC(\031)3008 4096 y Fw(0)3032
4126 y FK(\()p FC(\032)p FK(\))f FD(!)g FC(!)s FK(.)523
4226 y(By)h(regularit)n(y)e(of)i FC(V)18 b FK(,)30 b
FD(:)p FK(\()p FC(!)h FD(!)c FC(\032)p FK(\),)j(in)g(other)f(w)n(ords,)
g FC(s)p FK(\()p FC(\032)p FK(\))e FC(<)g(f)9 b FK(\()p
FC(!)s FK(\).)30 b(Hence)g FC(\032)d FD(62)g FC(r)r FK(,)k(and)523
4325 y FC(r)k FK(m)n(ust)d(ha)n(v)n(e)f(returned)h(in)g(line)h(1)f(of)g
(its)g(proto)r(col)f(execution)h(after)g(seeing)f FC(W)43
b FK(=)30 b FC(R)q FK(.)523 4425 y(This)g(means)f(the)h(in)n(terv)-5
b(al)29 b([)p FC(f)9 b FK(\()p FC(!)s FK(\))p FC(;)14
b(s)p FK(\()p FC(r)r FK(\)])31 b(is)f(not)f(free)h(from)f(c)n(hanges)f
(to)i FC(R)q FK(,)f(since)h(write)523 4525 y FC(w)f FK(executes)d(line)
g(2)g(of)g(its)g(proto)r(col)f(in)i(this)f(in)n(terv)-5
b(al.)26 b(So)g FC(R)h FK(m)n(ust)f(ha)n(v)n(e)f(b)r(een)i(c)n(hanged)
523 4624 y(b)r(et)n(w)n(een)d FC(\032)g FK(and)g FC(r)r
FK(.)h(According)e(to)h(the)g(reader)f(proto)r(col,)g(this)h(is)g(done)
g(in)g(line)g(3,)g(and)g(is)523 4724 y(follo)n(w)n(ed)k(b)n(y)i(a)f
Fk(read)41 b(v)j(:=)e(V)29 b FK(statemen)n(t.)h(This)f(read)g(b)r(et)n
(w)n(een)g FC(\032)h FK(and)f FC(r)j FK(con)n(tradicts)523
4824 y(the)c(de\014nition)g(of)g FC(\032)p FK(.)f(This)h(completes)f
(the)h(pro)r(of.)f Fu(2)606 4923 y FK(Lemma)g(6.5)g(and)h(the)g(giv)n
(en)e(construction)h(pro)n(v)n(e)f(the)i(follo)n(wing)523
5089 y Fv(Theorem)k FH(6.6)41 b FK(3)c FG(safe)h(bits)g(ar)l(e)g(ne)l
(c)l(essary)g(and)f(su\016cient)h(to)f(c)l(onstruct)f(a)i(single-)523
5189 y(r)l(e)l(ader,)31 b(single-writer,)g(atomic)g(bit.)p
eop end
%%Page: 73 83
TeXDict begin 73 82 bop 1700 299 a FF(6.5.)53 b(Optimal)26
b(Construction)g(of)h(A)n(tomic)e(Bits)155 b(73)523 506
y FL(6.6)116 b(The)38 b(4-trac)m(k)f(Proto)s(col)523
706 y FK(W)-7 b(e)29 b(return)e(to)h(the)h(general)e(problem)g(of)h
(constructing)g(a)f FC(b)p FK(-bit)h(atomic)g(v)-5 b(ariable)27
b(with)523 805 y(a)f(linear)g(n)n(um)n(b)r(er)g(of)g(safe)g(bits.)g
(The)h(safe)f(bits)g(are)g(divided)g(in)n(to)g(a)g(n)n(um)n(b)r(er)g
(of)h FG(c)l(ontr)l(ol)523 905 y(bits)p FK(,)35 b(collectiv)n(ely)e
(called)h(the)g FG(switch)p FK(,)h(and)f(sev)n(eral)f
FC(b)p FK(-bit)h(trac)n(ks,)f(whose)g(purp)r(ose)h(is)523
1005 y(to)c(hold)g(the)g(v)-5 b(alues)30 b(of)g(the)g(atomic)f(v)-5
b(ariable,)29 b(W)-7 b(e)31 b(start)e(out)h(with)h(a)e(pro)r(of)h(of)g
(wh)n(y)f(4)523 1104 y(trac)n(ks)h(are)h(necessary)-7
b(,)31 b(a)g(result)h(due)g(to)g(Burns)f(and)h(P)n(eterson)e([3)o(].)i
(It)h(is)e(a)h(v)-5 b(ariation)523 1204 y(on)27 b(pro)r(ofs)g(sho)n
(wing)g(the)g(imp)r(ossibilit)n(y)h(of)g(w)n(ait-free)e(consensus)h
(\([6,)g(3,)h(1)o(]\).)606 1303 y(Consider)j(the)g(initial)h(state)f
(of)h(a)f(construction,)f(where)h(b)r(oth)h(the)g(reader)e(and)h(the)
523 1403 y(writer)25 b(are)f(ab)r(out)h(to)g(start)g(their)g(proto)r
(col.)f(In)i(this)f(state,)g(the)h(c)n(hoice)f(of)g(whic)n(h)g(trac)n
(k)523 1503 y(the)36 b(reader)e(will)i(access)e(in)i(its)g(next)f
(action)g(is)h(not)f(\014xed)h(y)n(et,)f(since)h(a)f(write)g(using)523
1602 y(a)k(new)h(trac)n(k)e(could)h(or)g(could)g(not)g(o)r(ccur)g(b)r
(efore)g(the)h(read)f(starts.)f(Suc)n(h)i(a)f(state,)523
1702 y(from)32 b(whic)n(h)h(di\013eren)n(t)f(runs)h(lead)f(to)g
(di\013eren)n(t)h(c)n(hoices)e(of)i(trac)n(k)e(to)i(read)f(are)f
(called)523 1802 y(`biv)-5 b(alen)n(t',)33 b(while)h(states)e(in)i
(whic)n(h)f(the)g(c)n(hoice)g(is)g(\014xed)g(are)f(called)h(`univ)-5
b(alen)n(t'.)33 b(F)-7 b(or)523 1901 y(a)38 b(w)n(ait-free)f
(construction,)g(there)h(can)g(b)r(e)h(no)f(in\014nite)h(run)f(of)g
(biv)-5 b(alen)n(t)38 b(states)g(\(w)n(e)523 2001 y(restrict)27
b(atten)n(tion)g(to)g(runs)f(where)h(the)h(read)e(and)h(write)g(proto)r
(col)f(are)g(eac)n(h)h(executed)523 2100 y(only)g(once\).)g(Th)n(us,)g
(from)g(the)h(starting)f(state)g(w)n(e)g(can)g(reac)n(h)f(a)h(biv)-5
b(alen)n(t)27 b(state,)h(sa)n(y)e FC(S)5 b FK(,)523 2200
y(b)r(oth)31 b(of)f(whose)g(successors)e FC(S)1510 2212
y FA(w)1594 2200 y FK(and)i FC(S)1809 2212 y FA(r)1876
2200 y FK(are)f(univ)-5 b(alen)n(t,)31 b(where)e(the)i(writer)f(p)r
(erforms)523 2300 y(subaction)e FC(x)h FK(to)f(get)h(from)f(state)g
FC(S)34 b FK(to)28 b FC(S)1855 2312 y FA(w)1909 2300
y FK(,)g(and)h(the)g(reader)e(p)r(erform)h(subaction)g
FC(y)j FK(to)523 2399 y(get)25 b(from)g(state)g FC(S)30
b FK(to)25 b FC(S)1285 2411 y FA(r)1322 2399 y FK(.)g(The)h(c)n(hoice)e
(of)h(trac)n(k)f(is)i(\014xed)f(in)g FC(S)2510 2411 y
FA(w)2590 2399 y FK(and)g FC(S)2800 2411 y FA(r)2836
2399 y FK(,)h(but)g(di\013eren)n(t.)523 2499 y(If)c FC(S)651
2511 y FA(w)r(r)759 2499 y FK(is)f(the)h(state)f(reac)n(hed)f(from)h
FC(S)1707 2511 y FA(w)1783 2499 y FK(b)n(y)g(a)g(subaction)g(of)g(the)h
(reader,)e(then)i(the)g(c)n(hoice)523 2599 y(of)h(trac)n(k)f(in)i
FC(S)965 2611 y FA(w)r(r)1075 2599 y FK(is)f(also)f(di\013eren)n(t)h
(from)g(that)h(in)f FC(S)2150 2611 y FA(r)2187 2599 y
FK(.)h(So)f(the)g(reader's)f(lo)r(cal)h(state)g(m)n(ust)523
2698 y(b)r(e)i(di\013eren)n(t)g(b)r(et)n(w)n(een)g FC(S)1327
2710 y FA(w)r(r)1438 2698 y FK(and)f FC(S)1647 2710 y
FA(r)1684 2698 y FK(.)h(It)g(follo)n(ws)f(that)h FC(x)g
FK(is)g(a)f(write)h(subaction,)f(and)h FC(y)i FK(a)523
2798 y(read)f(subaction,)g(accessing)f(the)i(same)f(v)-5
b(ariable.)25 b(T)-7 b(o)27 b(the)g(writer,)f(states)g
FC(S)2957 2810 y FA(w)3037 2798 y FK(and)g FC(S)3248
2810 y FA(r)r(w)3335 2798 y FK(,)523 2897 y(reac)n(hed)35
b(from)h FC(S)1090 2909 y FA(r)1163 2897 y FK(b)n(y)f(executing)h
FC(x)p FK(,)h(are)e(indistinguishable,)h(while)g(the)h(reader)e(will)
523 2997 y(access)26 b(di\013eren)n(t)i(trac)n(ks)e(from)h(these)h
(states.)f(If)h(the)g(writer)f(no)n(w)g(\014nishes)g(its)h(proto)r(col)
523 3097 y(and)k(pro)r(ceeds)e(to)i(write)g(a)f(second)g(and)h(third)f
(v)-5 b(alue,)32 b(then)g(these)g(2)f(new)h(v)-5 b(alues)32
b(will)523 3196 y(ha)n(v)n(e)c(to)i(b)r(e)f(written)h(to)f(new)h(trac)n
(ks)e(to)h(ensure)g(collision-freedom.)f(This)h(sho)n(ws)f(that)523
3296 y(4)f(trac)n(ks)f(are)h(necessary)-7 b(.)26 b(It)i(remains)f(to)g
(sho)n(w)g(that)h(4)f(trac)n(ks)f(su\016ce.)606 3396
y(W)-7 b(e)31 b(con)n(v)n(enien)n(tly)f(split)h(the)h(4)e(trac)n(ks)g
(in)n(to)g(2)h(groups)f FC(T)2454 3408 y FB(0)2490 3396
y FC(;)14 b(T)2576 3408 y FB(1)2644 3396 y FK(of)31 b(2)f(trac)n(ks)g
FC(T)3112 3408 y FA(i;)p FB(0)3192 3396 y FC(;)14 b(T)3278
3408 y FA(i;)p FB(1)523 3495 y FK(eac)n(h.)34 b(In)i(order)e(to)h(a)n
(v)n(oid)e(collisions,)h(the)i(writer)e(alw)n(a)n(ys)f(tries)i(to)g(go)
f(to)h(the)h(group)523 3595 y(other)d(than)h(where)g(it)g(sees)f(the)i
(reader.)d(The)i(reader)f(in)h(turn)g(w)n(an)n(ts)f(recen)n(t)g(v)-5
b(alues,)523 3694 y(hence)39 b(it)g(tries)f(to)h(go)f(to)g(the)h(group)
f(where)g(it)h(sees)f(the)h(writer.)g(Both)f(the)h(reader)523
3794 y(and)28 b(the)g(writer)g(use)g(part)f(of)h(the)g(switc)n(h)g(to)g
(signal)f(the)i(other)e(user)h(ab)r(out)f(the)i(group)523
3894 y(they)f(are)e(in.)i(F)-7 b(or)27 b(the)g(momen)n(t)h(this)f(in)n
(v)n(olv)n(es)f(an)h(atomic)g(bit)h FC(W)39 b FK(for)27
b(the)h(writer,)f(and)523 3993 y(an)32 b(atomic)g(bit)h
FC(R)g FK(for)f(the)g(reader.)f(In)i(addition,)f(the)h(switc)n(h)f(has)
g(t)n(w)n(o)f FG(tr)l(ackdisplays)523 4093 y FC(D)592
4105 y FB(0)629 4093 y FC(;)14 b(D)735 4105 y FB(1)772
4093 y FK(,)28 b(one)g(for)f(eac)n(h)g(group,)g(displa)n(ying)g(the)h
(most)g(recen)n(tly)f(completed)i(trac)n(k.)d(F)-7 b(or)523
4193 y(the)26 b(momen)n(t,)g(these)f(to)r(o)h(are)e(atomic)i(bits.)g
(Later)e(w)n(e)i(will)f(sho)n(w)g(ho)n(w)g(to)h(use)f(safe)g(bits)523
4292 y(instead.)k(No)n(w)g(when)h(the)g(writer)f(completes)g(a)g(write)
g(action,)g(the)h(new)f(v)-5 b(alue)30 b(will)f(b)r(e)523
4392 y(on)e(trac)n(k)g FC(T)900 4404 y FA(W)n(;D)1034
4412 y Fx(W)1103 4392 y FK(.)606 4491 y(In)h(summary)-7
b(,)27 b(the)i(arc)n(hitecture)d(consists)h(of)h(4)g(trac)n(ks)e(of)i
FC(b)g FK(safe)f(bits)i(eac)n(h)e(and)g(the)523 4591
y(follo)n(wing)g(4)g(atomic)g(bits,)h(whic)n(h)f(comprise)g(the)h
(switc)n(h:)523 4767 y FH(W)-8 b(riter)32 b FD(!)p 915
4685 191 4 v 915 4804 4 119 v 42 w FC(D)1026 4779 y FB(0)p
1102 4804 V 915 4807 191 4 v 1106 4767 a FD(!)f FH(Reader)1096
b FK(group)27 b(0)g(trac)n(kdispla)n(y)523 4967 y FH(W)-8
b(riter)32 b FD(!)p 915 4886 V 915 5005 4 119 v 42 w
FC(D)1026 4979 y FB(1)p 1102 5005 V 915 5008 191 4 v
1106 4967 a FD(!)f FH(Reader)1096 b FK(group)27 b(1)g(trac)n(kdispla)n
(y)523 5168 y FH(W)-8 b(riter)32 b FD(!)p 915 5086 V
915 5193 4 107 v 46 w FH(W)p 1102 5193 V 915 5196 191
4 v 46 w FD(!)f FH(Reader)1331 b FK(writer's)27 b(group)523
5356 y FH(W)-8 b(riter)32 b FD( )p 915 5274 V 915 5381
4 107 v 59 w FH(R)p 1102 5381 V 915 5384 191 4 v 60 w
FD( )f FH(Reader)1322 b FK(reader's)26 b(group)p eop
end
%%Page: 74 84
TeXDict begin 74 83 bop 2253 299 a FF(6.6.)53 b(The)26
b(4-trac)n(k)f(Proto)r(col)156 b(74)606 506 y FK(W)-7
b(e)21 b(can)f(no)n(w)g(informally)g(state)h(the)g(writer)f(proto)r
(col.)f(The)i(writer)f(starts)g(b)n(y)g(reading)523 606
y FC(R)q FK(,)30 b(the)g(group)f(that)h(the)g(reader)f(is)h(in,)g(and)g
(compares)e(it)j(to)e FC(W)12 b FK(,)30 b(the)h(writer's)e(group.)523
706 y(If)d(they)g(are)f(equal,)h(then)g(the)h(reader)d(m)n(ust)i(ha)n
(v)n(e)f(left)h(the)h(other)e(group,)g(so)g(the)h(writer)523
805 y(simply)36 b(writes)g(to)g(a)f(trac)n(k)g(in)i(that)f(other)f
(group,)g(and)h(c)n(hanges)f FC(W)48 b FK(afterw)n(ards.)34
b(It)523 905 y(c)n(ho)r(oses)26 b(the)i(displa)n(y)n(ed)f(trac)n(k)g
(so)g(that)h(it)g(do)r(esn't)g(ha)n(v)n(e)e(to)i(c)n(hange)f(the)h
(trac)n(kdispla)n(y)-7 b(.)523 1005 y(If)28 b FC(R)g
FK(is)f(di\013eren)n(t)h(from)f FC(W)12 b FK(,)27 b(then)h(the)f
(writer)g(writes)g(to)g(the)h(other)f(trac)n(k)f(in)h(its)h(group)523
1104 y(and)f(c)n(hanges)g(the)h(trac)n(kdispla)n(y)d
FC(D)1670 1116 y FA(W)1773 1104 y FK(afterw)n(ards.)606
1204 y(The)e(reader)f(proto)r(col)g(is)h(then)h(as)e(follo)n(ws:)h(The)
g(reader)f(starts)g(b)n(y)h(reading)f FC(W)35 b FK(to)23
b(see)523 1303 y(if)k(the)g(writer)e(has)h(v)-5 b(acated)26
b(the)g(reader's)f(group.)g(In)i(that)f(case)g(the)g(reader)f(c)n
(hanges)g FC(R)523 1403 y FK(and)i(follo)n(ws)g(the)h(writer)f(to)g
(the)h(other)f(group.)606 1503 y(Next,)33 b(the)h(reader)d(reads)h(the)
i(trac)n(kdispla)n(y)c FC(D)2157 1515 y FA(R)2245 1503
y FK(of)j(its)g(group.)f(It)h(then)h(reads)d(the)523
1602 y(trac)n(k)26 b FC(T)784 1614 y FA(R;D)908 1622
y Fx(R)987 1602 y FK(and)h(returns)g(the)h(obtained)g(v)-5
b(alue.)606 1702 y(In)30 b(a)g(programming)e(language,)g(not)i(unlik)n
(e)g(the)h(one)e(in)n(tro)r(duced)h(in)g(section)g(6.5.3,)523
1802 y(the)e(ab)r(o)n(v)n(e)e(proto)r(cols)g(lo)r(ok)h(lik)n(e:)523
1968 y Fk(WRITER)41 b(PROTOCOL)781 b(READER)41 b(PROTOCOL)523
2167 y(1.)i(if)g(R==w)e(then)783 b(if)43 b(W<>r)f(then)523
2267 y(2.)130 b(w)43 b(:=)g(1-w)958 b(r)43 b(:=)f(1-r)523
2366 y(3.)130 b(write)41 b(track)h(T[w,d[w]])389 b(change)41
b(R)523 2466 y(4.)130 b(change)41 b(W)871 b(endif)523
2565 y(5.)43 b(else)1131 b(read)42 b(d)h(:=)g(D[r])523
2665 y(6.)130 b(d[w])42 b(:=)h(1-d[w])607 b(read)42 b(track)g(T[r,d])
523 2765 y(7.)130 b(write)41 b(track)h(T[w,d[w]])523
2864 y(8.)130 b(change)41 b(D[w])523 2964 y(9.)i(endif)606
3130 y FK(The)e(lo)n(w)n(er-case)e(lo)r(cal)i(v)-5 b(ariables)40
b(hold)h(copies)g(of)g(the)h(similarly)e(named)i(shared)523
3230 y(bits.)27 b(An)f(arra)n(y)e(notation)i(is)g(used)g(for)g(the)g
(trac)n(ks)f(and)h(displa)n(ys)f(instead)h(of)g(the)h(index)523
3329 y(notation)20 b(that)h(w)n(e)f(reserv)n(e)f(for)h(the)h(text.)g
(The)g(access)e(to)h(a)h(trac)n(k)e(has)h(b)r(een)h(compressed)523
3429 y(to)27 b(a)f(single)g(statemen)n(t)g(since)h(w)n(e)f(can)g
(ignore)g(ho)n(w)g(man)n(y)g(bits)h(m)n(ust)f(b)r(e)h(c)n(hanged)f(and)
523 3528 y(in)j(what)g(order.)f(F)-7 b(or)28 b(notational)g(con)n(v)n
(enience,)g(w)n(e)g(do)h(not)g(men)n(tion)g(the)g(v)-5
b(alue)29 b(to)f(b)r(e)523 3628 y(written)k(in)g(the)h(writer)e(proto)r
(col)g(or)g(the)h(v)-5 b(alue)32 b(to)g(b)r(e)g(read)f(in)h(the)g
(reader)f(proto)r(col.)523 3728 y(Since)i(eac)n(h)g(proto)r(col)f
(execution)h(in)n(v)n(olv)n(es)e(exactly)i(one)g(trac)n(k)f(access,)g
(the)h(meaning)523 3827 y(should)27 b(b)r(e)h(ob)n(vious.)606
3927 y(Consider)k(a)g(run)h(of)g(the)g(ab)r(o)n(v)n(e)e(construction.)h
(Eac)n(h)g(action)g(con)n(tains)g(lo)n(w)n(er-lev)n(el)523
4027 y(accesses)h(to)i(the)g(atomic)f(bits)h(of)g(the)g(switc)n(h)f
(and)h(to)f(the)h(safe)g(bits)g(of)f(a)h(trac)n(k.)e(By)523
4126 y(de\014nition)42 b(6.1,)f(the)g(partial)g(order)f(on)h(the)h
(accesses)e(to)i(eac)n(h)e(atomic)h(bit)h(can)f(b)r(e)523
4226 y(extended)34 b(to)f(a)h(total)f(one.)g(In)n(tuitiv)n(ely)-7
b(,)34 b(the)g(accesses)e(to)h(di\013eren)n(t)h(atomic)f(bits)h(can)523
4325 y(then)d(also)e(b)r(e)h(totally)g(ordered.)f(In)h([1],)g(it)h(w)n
(as)e(sho)n(wn)h(that)g(this)g(is)h(indeed)f(the)h(case,)523
4425 y(if)26 b(the)g(precedence)f(relation)f(is)h(de\014ned)h(in)g
(terms)f(of)g(a)h(global)e(time)2710 4395 y FB(2)2747
4425 y FK(.)i(Using)f(this)h(total)523 4525 y(ordering)g(on)h(all)h
(atomic)f(bit)h(accesses,)e(w)n(e)i(can)f(mo)r(del)h(a)f(run)h(b)n(y)f
(a)g(sequence)g(of)h FG(state)523 4624 y(tr)l(ansitions)p
FK(,)19 b(eac)n(h)g(transition)f(corresp)r(onding)f(to)i(an)g(atomic)g
(bit)h(access.)e(In)h(this)h(mo)r(del,)523 4724 y(the)28
b(states)f(of)h(the)g(writer)f(are:)523 4890 y FH(0)41
b FK(idle,)28 b(i.e.,)g(b)r(efore)f(the)h(atomic)f(read)g(of)h
FC(R)g FK(in)g(line)g(1,)p 523 5010 499 5 v 627 5092
a Fm(2)661 5124 y FF(This)d Fg(glob)l(al)h(time)g(assumption)f
FF(is)f(equiv)l(alen)n(t)f(to)i(the)e Fg(interval)k(axiom)p
FF(:)d(if)h Fo(a)c Fp(!)g Fo(b)13 b Fp(^)h Fo(c)21 b
Fp(!)h Fo(d)p FF(,)523 5207 y(then)j Fo(a)c Fp(!)g Fo(d)26
b FF(or)g Fo(c)21 b Fp(!)g Fo(b)p FF(.)p eop end
%%Page: 75 85
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b(4-trac)n(k)f(Proto)r(col)156 b(75)523 2984 y @beginspecial
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 @endspecial 1078 3159 a(FIGURE)25 b(6.1.)i(state)f(diagram)h(of)g
(4-trac)n(k)e(construction)523 3420 y FH(1)41 b FK(b)r(et)n(w)n(een)31
b(the)f(atomic)g(read)g(of)g FC(R)h FK(and)f(the)h(atomic)f(c)n(hange)f
(of)h FC(W)42 b FK(in)31 b(line)g(4,)f(when)731 3520
y(it)e(is)f(writing)g(trac)n(k)g FC(T)1445 3532 y FB(1)p
Fw(\000)p FA(W)n(;D)1664 3540 y Fy(1)p Ft(\000)p Fx(W)1806
3520 y FK(,)523 3685 y FH(2)41 b FK(b)r(et)n(w)n(een)27
b(the)g(atomic)f(read)g(of)h FC(R)g FK(and)g(the)g(atomic)f(c)n(hange)g
(of)g FC(D)2723 3697 y FA(W)2825 3685 y FK(in)h(line)g(8,)g(when)731
3784 y(it)h(is)f(writing)g(trac)n(k)g FC(T)1445 3796
y FA(W)n(;)p FB(1)p Fw(\000)p FA(D)1664 3804 y Fx(W)1732
3784 y FK(.)523 3948 y(Th)n(us,)37 b(the)h(writer)f(is)g(alw)n(a)n(ys)f
(mo)n(ving)h(from)g(state)g(0)g(to)g(either)h(state)f(1)g(or)g(state)g
(2)523 4048 y(\(dep)r(ending)31 b(on)f(the)h(outcome)e(of)i(the)f
(test\),)h(and)f(then)h(bac)n(k)f(to)g(state)g(0.)g(The)g(states)523
4147 y(of)e(the)g(reader)e(are:)523 4311 y FH(0)41 b
FK(idle,)28 b(i.e.,)g(b)r(efore)f(the)h(atomic)f(read)g(of)h
FC(W)39 b FK(in)28 b(line)g(1,)523 4476 y FH(1)41 b FK(b)r(et)n(w)n
(een)28 b(the)g(atomic)f(read)g(of)g FC(W)40 b FK(and)27
b(the)h(atomic)f(c)n(hange)g(of)g FC(R)i FK(in)f(line)f(3,)523
4641 y FH(2)41 b FK(just)28 b(b)r(efore)g(the)g(atomic)f(read)g(of)g
FC(D)1789 4653 y FA(R)1871 4641 y FK(in)h(line)g(5,)523
4806 y FH(3)41 b FK(after)28 b(the)g(atomic)f(read)f(of)i
FC(D)1572 4818 y FA(R)1626 4806 y FK(,)g(when)g(it)g(is)f(reading)g
(trac)n(k)f FC(T)2617 4818 y FA(R;d)2725 4806 y FK(.)523
4969 y(Th)n(us)33 b(the)h(reader)d(is)j(alw)n(a)n(ys)d(mo)n(ving)h
(from)h(state)g(0)g(to)g(either)g(state)g(1)g(and)g(then)h(to)523
5069 y(state)27 b(2)h(or)e(directly)i(to)f(state)h(2,)f(then)h(on)f(to)
h(state)f(3,)g(and)h(\014nally)f(bac)n(k)g(to)g(state)h(0.)606
5169 y(No)n(w)23 b(\014gure)g(6.1)g(sho)n(ws)g(all)g(p)r(ossible)h
(transitions)e(in)i(a)g(run)f(of)h(the)g(4-trac)n(k)e(construc-)523
5268 y(tion.)29 b(It)g(can)f(b)r(e)h(easily)f(c)n(hec)n(k)n(ed)f(that)i
(the)g(in)n(v)-5 b(arian)n(ts)28 b(in)h(the)g(no)r(des)f(hold.)h(Note)f
(that)523 5368 y(it)g(is)g(imp)r(ossible)f(for)g(the)h(writer)f(and)g
(the)h(reader)f(to)g(b)r(e)h(in)g(state)f(1)g(sim)n(ultaneously)-7
b(.)p eop end
%%Page: 76 86
TeXDict begin 76 85 bop 2253 299 a FF(6.6.)53 b(The)26
b(4-trac)n(k)f(Proto)r(col)156 b(76)523 506 y Fv(Lemma)32
b FH(6.7)40 b FG(The)31 b FK(4)p FG(-tr)l(ack)e(c)l(onstruction)g(is)h
(c)l(ol)t(lision-fr)l(e)l(e.)606 672 y FH(Pro)s(of.)g
FK(W)-7 b(e)31 b(denote)g(the)g(com)n(bined)f(writer)g(and)g(reader)f
(state)i(in)g(a)f(pair)g(\()p FC(w)r(s;)14 b(r)r(s)p
FK(\).)523 772 y(Collisions)29 b(can)g(only)g(o)r(ccur)g(in)h(states)f
(\(1)p FC(;)14 b FK(3\))29 b(and)g(\(2)p FC(;)14 b FK(3\),)29
b(when)h(b)r(oth)g(the)g(writer)f(and)523 872 y(the)f(reader)e(are)h
(accessing)f(a)h(trac)n(k.)606 971 y(In)i(the)f(former)g(case,)f(the)i
(writer)f(is)g(in)h(group)e FC(w)g FK(=)d(1)18 b FD(\000)h
FC(W)12 b FK(,)28 b(while)h(the)f(reader)f(is)h(in)523
1071 y(group)c FC(r)h FK(=)e FC(R)q FK(.)i(F)-7 b(rom)24
b(the)h(diagram)e(w)n(e)i(see)f(that)h FC(W)35 b FK(=)23
b FC(R)i FK(in)g(state)g(\(1)p FC(;)14 b FK(3\),)24 b(so)h(the)g(users)
523 1171 y(are)i(accessing)f(trac)n(ks)g(in)i(di\013eren)n(t)f(groups.)
606 1270 y(In)34 b(state)f(\(2)p FC(;)14 b FK(3\),)33
b(the)g(writer)g(writes)g(on)g(trac)n(k)f FC(d)2198 1282
y FA(w)2284 1270 y FK(=)g(1)22 b FD(\000)g FC(D)2601
1282 y FA(W)2710 1270 y FK(in)33 b(group)f FC(w)k FK(=)c
FC(W)12 b FK(,)523 1370 y(while)25 b(the)g(reader)e(reads)g(from)h
(trac)n(k)g FC(d)h FK(in)f(group)g FC(R)q FK(.)g(The)h(diagram)e(sho)n
(ws)g(that)i(either)523 1469 y FC(W)35 b FD(6)p FK(=)23
b FC(R)28 b FK(or)f FC(d)c FK(=)g FC(D)1140 1481 y FA(R)1194
1469 y FK(,)28 b(so)f(the)g(users)g(are)g(again)f(accessing)g
(di\013eren)n(t)i(trac)n(ks.)e Fu(2)p FK(.)523 1719 y
Fl(6.6.1)99 b(Corr)-5 b(e)g(ctness)523 1868 y FK(Giv)n(en)33
b(lemma)h(6.7,)e(it)i(remains)f(to)g(sho)n(w)g(that)g(for)g(ev)n(ery)f
(run)i(\()p FD(A)p FC(;)14 b FD(!)p FK(\),)34 b(there)f(exists)523
1968 y(a)40 b(reading)f(function)i FC(\031)j FK(suc)n(h)c(that)h
FC(\033)47 b FK(=)d(\()p FD(A)p FC(;)14 b FD(!)p FC(;)g(\031)s
FK(\))42 b(satis\014es)d(the)i(three)f(atomicit)n(y)523
2067 y(conditions.)29 b(As)g(b)r(efore)f(w)n(e)h(ma)n(y)f(assume)g
(that)i(the)f(set)g(of)g(all)f(atomic)h(bit)g(accesses)f(is)523
2167 y(totally)f(ordered)g(b)n(y)g FD(!)p FK(,)h(hence)f(w)n(e)g(can)h
(use)f(the)h(state)f(mo)r(del.)606 2267 y(Lemma)i(6.7)f(allo)n(ws)f(us)
i(to)f(de\014ne)h(the)g(reading)f(function)h FC(\031)k
FK(as)28 b(the)h(\\union")f(of)g(the)523 2366 y(four)33
b(reading)f(functions)h(that)h(mak)n(e)e(eac)n(h)h(trac)n(k)f(atomic.)g
(This)i(means)e(that)i(a)f(read)523 2466 y(is)c(mapp)r(ed)h(to)g(the)g
(write)f(whic)n(h)h(w)n(as)e(the)i(last)f(to)h(access)e(the)i(trac)n(k)
f(from)g(whic)n(h)g(the)523 2565 y(read)e(obtained)g(its)h(v)-5
b(alue.)27 b(W)-7 b(e)28 b(no)n(w)f(pro)n(v)n(e)f(eac)n(h)h(of)h(the)g
(three)f(conditions)g(in)h(turn.)606 2665 y FH(Pro)s(of)k(of)g(A0)c
FK(The)f(reading)g(function)h(is)g(ob)n(viously)e(normal)h(b)n(y)g(the)
i(safet)n(y)e(of)g(the)523 2765 y(trac)n(k-bits.)606
2864 y FH(Pro)s(of)k(of)f(A2)d FK(The)f(pro)r(of)g(is)h(b)n(y)f(con)n
(tradiction.)f(Let)i FC(r)f FD(2)d(R)p FC(;)14 b(w)26
b FD(2)e(W)33 b FK(b)r(e)27 b(suc)n(h)f(that)523 2964
y FC(\031)s FK(\()p FC(r)r FK(\))f FD(!)e FC(w)j FD(!)d
FC(r)r FK(.)1060 2934 y FB(3)1125 2964 y FK(Assume)28
b(without)g(loss)f(of)g(generalit)n(y)f(that)i FC(w)i
FK(writes)d(on)h(trac)n(k)e FC(T)3268 2976 y FB(0)p FA(;)p
FB(0)523 3064 y FK(and)h(that)h FC(D)933 3076 y FB(1)993
3064 y FK(=)23 b(0)k(at)h(time)g FC(f)9 b FK(\()p FC(w)r
FK(\).)28 b(Then)g(at)g(the)g(same)f(time,)h FC(W)35
b FK(=)22 b(0)28 b(and)f FC(D)3049 3076 y FB(0)3109 3064
y FK(=)c(0.)606 3163 y(Consider)k(no)n(w)g(the)h(4)f(p)r(ossible)g
(trac)n(ks)f(that)i FC(r)j FK(can)c(read)f(from:)523
3329 y FC(T)572 3341 y FB(0)p FA(;)p FB(0)703 3329 y
FK(This)i(con)n(tradicts)e(the)i(assumption)f(that)h
FC(\031)s FK(\()p FC(r)r FK(\))h(precedes)e FC(w)r FK(.)523
3495 y FC(T)572 3507 y FB(0)p FA(;)p FB(1)703 3495 y
FK(In)c(this)f(case,)g FC(r)j FK(reads)c FC(d)j FK(=)e(1)g(from)g
FC(D)1904 3507 y FB(0)1941 3495 y FK(,)h(whic)n(h)f(requires)f(that)i
(the)f(writer)g(c)n(hanges)731 3595 y FC(D)800 3607 y
FB(0)856 3595 y FK(to)e(1)f(b)r(et)n(w)n(een)h FC(f)9
b FK(\()p FC(w)r FK(\))21 b(and)f(the)g(read)f(of)g FC(D)2141
3607 y FB(0)2198 3595 y FK(b)n(y)h FC(r)r FK(.)g(But)h(according)d(to)h
(the)h(writer)731 3694 y(proto)r(col,)31 b(this)i(c)n(hange)f(is)g
(preceded)g(b)n(y)h(the)g(writing)f(of)h(trac)n(k)e FC(T)2896
3706 y FB(0)p FA(;)p FB(1)2986 3694 y FK(,)i(implying)731
3794 y FC(w)25 b FD(!)e FC(\031)s FK(\()p FC(r)r FK(\))30
b(and)d(hence)h(leading)f(to)g(a)g(con)n(tradiction.)731
3927 y(The)e(last)g(t)n(w)n(o)f(cases)g(are)g(similar)g(and)h(w)n(e)g
(need)g(only)g(sho)n(w)f(that)h(the)h(trac)n(k)e(read)731
4027 y(b)n(y)j FC(r)j FK(m)n(ust)e(ha)n(v)n(e)e(b)r(een)i(written)g
(after)g FC(w)r FK(.)523 4193 y FC(T)572 4205 y FB(1)p
FA(;)p FB(0)703 4193 y FK(In)f(this)g(case,)e FC(r)30
b FK(reads)25 b(1)h(from)g FC(W)12 b FK(,)27 b(whic)n(h)f(requires)f
(that)i(the)g(writer)f(c)n(hanges)f FC(W)731 4292 y FK(to)i(1.)g(This)h
(is)f(preceded)h(b)n(y)f(the)h(writing)f(of)h(trac)n(k)e
FC(T)2444 4304 y FB(1)p FA(;)p FB(0)2534 4292 y FK(.)523
4458 y FC(T)572 4470 y FB(1)p FA(;)p FB(1)703 4458 y
FK(In)d(this)f(case,)g FC(r)j FK(reads)c FC(d)j FK(=)e(1)g(from)g
FC(D)1904 4470 y FB(1)1941 4458 y FK(,)h(whic)n(h)f(requires)f(that)i
(the)f(writer)g(c)n(hanges)731 4558 y FC(D)800 4570 y
FB(1)864 4558 y FK(to)28 b(1.)f(This)h(is)f(preceded)g(b)n(y)g(the)h
(writing)g(of)f(trac)n(k)g FC(T)2578 4570 y FB(1)p FA(;)p
FB(1)2667 4558 y FK(.)523 4724 y(In)38 b(all)g(three)g(cases,)f(w)n(e)g
(see)h(that)g FC(r)j FK(cannot)d(read)f(a)g(v)-5 b(alue)38
b(older)g(than)g(that)g(of)g FC(w)r FK(,)523 4824 y(b)r(ecause)31
b(the)i(displa)n(y)e(\()p FC(W)n(;)14 b(D)1482 4836 y
FB(0)1519 4824 y FC(;)g(D)1625 4836 y FB(1)1662 4824
y FK(\))32 b(do)r(esn't)g(c)n(hange)f(un)n(til)h(the)g(new)g(trac)n(k)f
(has)g(b)r(een)523 4923 y(written.)k(In)f(other)g(w)n(ords,)f(once)g
(the)i(displa)n(y)f(is)g(set,)g(ev)n(ery)f(new)h(read)g(action)g(m)n
(ust)523 5023 y(read)27 b(either)g(the)h(trac)n(k)f(on)g(displa)n(y)g
(or)f(a)i(more)e(recen)n(tly)h(written)h(one.)p 523 5143
499 5 v 627 5225 a Fm(3)661 5256 y FF(It)23 b(will)h(b)r(e)e(clear)i
(from)g(con)n(text)e(whether)g(w)n(e)i(mean)f(the)f(write)h(action)h
Fo(w)h FF(or)e(the)f(similarly)523 5339 y(denoted)j(writer's)i(lo)r
(cal)h(cop)n(y)d(of)h Fo(W)11 b FF(.)p eop end
%%Page: 77 87
TeXDict begin 77 86 bop 2253 299 a FF(6.6.)53 b(The)26
b(4-trac)n(k)f(Proto)r(col)156 b(77)606 506 y FH(Pro)s(of)33
b(of)g(A1)c FK(W)-7 b(e)30 b(claim)e(that)h(A1)g(follo)n(ws)f(from)g
(A2)h(and)g(sho)n(w)f(this)h(b)n(y)g(deriving)523 606
y(a)g(violation)g(of)h(A2)f(from)h(a)f(violation)g(of)h(A1.)f(Let)h
FC(r)2227 618 y FB(1)2265 606 y FC(;)14 b(r)2339 618
y FB(2)2403 606 y FD(2)27 b(R)j FK(b)r(e)h(suc)n(h)e(that)h
FC(r)3109 618 y FB(1)3173 606 y FD(!)d FC(r)3320 618
y FB(2)523 706 y FK(and)i FC(\031)s FK(\()p FC(r)805
718 y FB(2)843 706 y FK(\))d FD(!)g FC(\031)s FK(\()p
FC(r)1129 718 y FB(1)1167 706 y FK(\).)k(By)e(de\014nition)i(of)f
FC(\031)k FK(and)c(lemma)g(6.7,)f FC(r)2557 718 y FB(1)2624
706 y FK(accesses)f(some)i(trac)n(k,)523 805 y(sa)n(y)i
FC(T)720 817 y FB(0)p FA(;)p FB(0)810 805 y FK(,)i(after)f
FC(\031)s FK(\()p FC(r)1186 817 y FB(1)1224 805 y FK(\))h(do)r(es)f
(so.)g(But)g(since)h FC(r)2026 817 y FB(1)2096 805 y
FK(cannot)f(access)f(the)i(trac)n(k)e(un)n(til)i(after)523
905 y FC(\031)s FK(\()p FC(r)642 917 y FB(1)680 905 y
FK(\))23 b(c)n(hanges)e(an)i(atomic)f(bit)h(\()p FC(W)35
b FK(or)21 b FC(D)1849 917 y FB(0)1886 905 y FK(\),)i(w)n(e)f(ha)n(v)n
(e)g(that)g FC(f)9 b FK(\()p FC(\031)s FK(\()p FC(r)2643
917 y FB(1)2682 905 y FK(\)\))23 b FC(<)g(f)9 b FK(\()p
FC(r)2976 917 y FB(1)3013 905 y FK(\))24 b FC(<)e(s)p
FK(\()p FC(r)3264 917 y FB(2)3302 905 y FK(\),)523 1005
y(hence)28 b(with)g FC(w)d FK(=)e FC(\031)s FK(\()p FC(r)1234
1017 y FB(1)1272 1005 y FK(\),)28 b FC(\031)s FK(\()p
FC(r)1474 1017 y FB(2)1513 1005 y FK(\))23 b FD(!)g FC(w)j
FD(!)d FC(r)1902 1017 y FB(2)1940 1005 y FK(,)k(violating)g(A2.)h
Fu(2)523 1254 y Fl(6.6.2)99 b(Sp)-5 b(ac)g(e)34 b(Complexity)523
1403 y FK(No)n(w)k(that)g(the)h(4-trac)n(k)d(construction)i(has)f(b)r
(een)i(pro)n(v)n(en)e(correct,)g(w)n(e)g(consider)h(its)523
1503 y(\\space)24 b(complexit)n(y)-7 b(.")24 b(Using)h(the)h(3)e(safe)h
(bit)h(construction)e(to)h(implemen)n(t)h(eac)n(h)e(of)h(the)523
1602 y(four)c(atomic)g(bits,)g(w)n(e)g(see)g(that)h(12)e(bits)i
(su\016ce)f(for)g(the)h(switc)n(h.)f(But)g(w)n(e)g(can)g(do)g(b)r
(etter,)523 1702 y(b)r(ecause)26 b(those)g(atomic)f(bits)i(are)e(used)h
(in)h(a)f(sp)r(ecial)g(w)n(a)n(y)-7 b(.)25 b(In)h(particular,)f(since)h
(the)h FC(W)523 1802 y FK(and)33 b FC(R)i FK(bits)f(are)e(used)i(for)f
(handshaking,)f(there)i(is)f(exactly)g(one)g(atomic)g(read)g(of)h
FC(W)523 1901 y FK(b)r(et)n(w)n(een)c(an)f(atomic)g(c)n(hange)g(of)g
FC(W)42 b FK(and)29 b(an)h(atomic)f(c)n(hange)f(of)i
FC(R)g FK(\(and)g(vice)f(v)n(ersa\).)523 2001 y(Hence)21
b(there)g(is)g(at)g(most)g(one)g(atomic)g(c)n(hange)e(of)j
FC(W)33 b FK(b)r(et)n(w)n(een)21 b(t)n(w)n(o)f(consecutiv)n(e)g(atomic)
523 2100 y(reads)k(of)h FC(W)37 b FK(\(and)25 b(vice)g(v)n(ersa\).)f
(With)i(the)f(trac)n(kdispla)n(y)e(bits)j FC(D)2613 2112
y FB(0)2650 2100 y FC(;)14 b(D)2756 2112 y FB(1)2818
2100 y FK(the)25 b(situation)g(is)523 2200 y(more)f(complicated.)h
(When)h(the)f(reader)f(c)n(hanges)g(groups)f(\(sa)n(y)-7
b(,)25 b(to)g(0\),)g(and)g(atomically)523 2300 y(reads)20
b FC(D)803 2312 y FB(0)840 2300 y FK(,)h(there)g(can)g(b)r(e)h(at)f
(most)g(one)f(atomic)h(c)n(hange)f(of)h FC(D)2471 2312
y FB(0)2529 2300 y FK(b)r(efore)g(the)h(writer)e(lea)n(v)n(es)523
2399 y(group)26 b(0.)606 2499 y(W)-7 b(e)25 b(will)h(sho)n(w)e(that,)h
(b)r(ecause)g(of)f(these)h(prop)r(erties,)g(w)n(e)f(can)h(implemen)n(t)
g(an)n(y)f(of)h(the)523 2599 y(four)d(atomic)g(bits,)g(call)g(it)h
FC(B)t FK(,)f(with)h(2)f(safe)g(bits)g FC(B)2100 2611
y FB(0)2137 2599 y FC(;)14 b(B)2237 2611 y FB(1)2274
2599 y FK(.)23 b(The)f(problem)g(with)h(safe)e(bits)i(is)523
2698 y(their)g(\015ic)n(k)n(ering.)f(If,)h(for)g(example,)g
FC(R)h FK(w)n(as)e(only)g(a)h(safe)g(bit,)h(then)f(while)h(b)r(eing)f
(c)n(hanged)523 2798 y(b)n(y)i(the)h(reader,)e(the)i(writer)f(could)h
(\014rst)f(see)g(the)h(new)g(v)-5 b(alue,)25 b(c)n(hange)g(groups,)f
(then)i(see)523 2897 y(the)34 b(old)f(v)-5 b(alue)34
b(and)f(write)g(to)h(the)g(displa)n(y)n(ed)e(trac)n(k)g(in)i(the)g(old)
f(group.)g(With)h(2)f(safe)523 2997 y(bits,)h(the)h(follo)n(wing)e(sc)n
(heme)g(can)h(b)r(e)g(applied)g(to)g(alleviate)f(the)i(\015ic)n(k)n
(ering)d(problem.)523 3097 y(W)-7 b(e)27 b(represen)n(t)f(the)h(v)-5
b(alue)27 b(of)g(atomic)g(bit)g FC(B)k FK(as)c(the)g(exclusiv)n(e-or)e
(of)i(2)f(safe)h(bit)g(v)-5 b(alues:)523 3196 y FC(B)27
b FK(=)c FC(B)764 3208 y FB(0)817 3196 y FD(\010)16 b
FC(B)961 3208 y FB(1)998 3196 y FK(.)26 b(The)h(c)n(hange)e(of)h
(atomic)g(bit)h FC(B)k FK(is)26 b(then)h(replaced)f(b)n(y)g(a)g(c)n
(hange)f(of)h(safe)523 3296 y(bit)20 b FC(B)707 3308
y FA(b)741 3296 y FK(,)f(where)h FC(b)f FK(is)h(the)g(old)f(v)-5
b(alue)20 b(of)f FC(B)t FK(.)h(Th)n(us,)g FC(B)2104 3308
y FB(0)2161 3296 y FK(and)f FC(B)2377 3308 y FB(1)2434
3296 y FK(are)g(c)n(hanged)g(alternatingly)-7 b(.)523
3396 y(F)g(or)33 b(the)g(purp)r(ose)g(of)g(reading)f
FC(B)t FK(,)i(t)n(w)n(o)e(lo)r(cal)h(copies)f FC(b)2319
3408 y FB(0)2356 3396 y FC(;)14 b(b)2429 3408 y FB(1)2499
3396 y FK(of)33 b FC(B)2662 3408 y FB(0)2733 3396 y FK(and)g
FC(B)2963 3408 y FB(1)3034 3396 y FK(are)f(k)n(ept.)523
3495 y(Normally)f(then,)i(an)f(atomic)f(read)h(of)g FC(B)k
FK(is)c(replaced)f(b)n(y)h(a)g(safe)g(read)f(of)h FC(B)3027
3507 y FA(b)3093 3495 y FK(in)n(to)g FC(b)3302 3507 y
FA(b)3335 3495 y FK(,)523 3595 y(where)i FC(b)f FK(=)h
FC(b)974 3607 y FB(0)1033 3595 y FD(\010)23 b FC(b)1157
3607 y FB(1)1228 3595 y FK(is)34 b(the)g(old)g(v)-5 b(alue)34
b(of)g FC(B)t FK(.)h(In)f(this)h(case,)e(new-old)h(in)n(v)n(ersions)e
(are)523 3694 y(eliminated,)27 b(since)f(the)h(\015ic)n(k)n(ering)e
(bit)i(is)f(no)g(longer)f(examined)h(once)g(the)h(new)g(v)-5
b(alue)26 b(is)523 3794 y(obtained.)h(This)g(pro)r(cedure)f(su\016ces)h
(for)g(reading)f FC(W)39 b FK(and)27 b FC(R)q FK(,)g(since)g(the)h
(handshaking)523 3894 y(ensures)d(that)h(eac)n(h)f(safe)g(bit)i(c)n
(hange)d(is)i(noticed)g(b)n(y)f(the)i(other)e(user.)g(It)h(also)f
(su\016ces)g(if)523 3993 y(the)d(reader)e(sees)i(the)g(writer)f(in)h
(the)g(same)f(group)g(and)g(w)n(an)n(ts)g(to)h(read)f(the)h(trac)n
(kdispla)n(y)-7 b(,)523 4093 y(b)r(ecause)27 b(the)h(writer)f(will)h(c)
n(hange)f(the)h(displa)n(y)e(at)i(most)f(once)h(\(b)r(efore)f(mo)n
(ving)g(to)g(the)523 4193 y(other)22 b(group\).)f(If)i(on)f(the)h
(other)f(hand)g(the)h(reader)e(sees)g(the)i(writer)f(in)g(the)h(other)f
(group,)523 4292 y(then)k(an)n(y)e(lo)r(cal)g(copies)h(it)g(w)n(ould)g
(ha)n(v)n(e)e(of)i(the)h(trac)n(kdispla)n(y)d(bits)i(in)g(that)g(other)
g(group)523 4392 y(migh)n(t)36 b(b)r(e)h(out)f(of)g(date.)h(In)f(this)h
(case)e(it)i(can)e(simply)i(read)e(b)r(oth)i(safe)f(bits)g(of)g(that)
523 4491 y(displa)n(y)25 b(one)g(after)h(the)g(other,)f(b)r(ecause)g
(again)f(the)j(writer)e(will)g(c)n(hange)g(the)h(displa)n(y)f(at)523
4591 y(most)i(once)h(b)r(efore)f(mo)n(ving)f(to)i(the)g(other)f(group.)
606 4691 y(The)h(new)f(arc)n(hitecture)g(of)g(the)h(switc)n(h)g(is)f
(as)g(follo)n(ws:)523 4857 y FH(W)-8 b(riter)32 b FD(!)p
915 4775 191 4 v 915 4905 4 131 v 17 w FC(D)1001 4869
y FB(0)p FA(;)p FB(0)p 1102 4905 V 915 4908 191 4 v 1106
4857 a FD(!)f FH(Reader)523 5068 y(W)-8 b(riter)32 b
FD(!)p 915 4987 V 915 5117 4 131 v 17 w FC(D)1001 5080
y FB(0)p FA(;)p FB(1)p 1102 5117 V 915 5120 191 4 v 1106
5068 a FD(!)f FH(Reader)1096 b FK(group)27 b(0)g(trac)n(kdispla)n(y)523
5280 y FH(W)-8 b(riter)32 b FD(!)p 915 5198 V 915 5329
4 131 v 17 w FC(D)1001 5292 y FB(1)p FA(;)p FB(0)p 1102
5329 V 915 5332 191 4 v 1106 5280 a FD(!)f FH(Reader)p
eop end
%%Page: 78 88
TeXDict begin 78 87 bop 2059 299 a FF(6.7.)52 b(The)26
b(A)n(tomicit)n(y)f(Automaton)154 b(78)523 508 y FH(W)-8
b(riter)32 b FD(!)p 915 427 191 4 v 915 557 4 131 v 17
w FC(D)1001 520 y FB(1)p FA(;)p FB(1)p 1102 557 V 915
560 191 4 v 1106 508 a FD(!)f FH(Reader)1096 b FK(group)27
b(1)g(trac)n(kdispla)n(y)523 720 y FH(W)-8 b(riter)32
b FD(!)p 915 638 V 915 757 4 119 v 37 w FC(W)1030 732
y FB(0)p 1102 757 V 915 760 191 4 v 1106 720 a FD(!)f
FH(Reader)523 920 y(W)-8 b(riter)32 b FD(!)p 915 839
V 915 958 4 119 v 37 w FC(W)1030 932 y FB(1)p 1102 958
V 915 961 191 4 v 1106 920 a FD(!)f FH(Reader)1331 b
FK(writer's)27 b(group)523 1121 y FH(W)-8 b(riter)32
b FD( )p 915 1039 V 915 1158 4 119 v 45 w FC(R)1023 1133
y FB(0)p 1102 1158 V 915 1161 191 4 v 1106 1121 a FD( )f
FH(Reader)523 1321 y(W)-8 b(riter)32 b FD( )p 915 1239
V 915 1358 4 119 v 45 w FC(R)1023 1333 y FB(1)p 1102
1358 V 915 1361 191 4 v 1106 1321 a FD( )f FH(Reader)1322
b FK(reader's)26 b(group)606 1487 y(The)i(corresp)r(onding)d(proto)r
(cols)i(are:)523 1637 y Fk(WRITER)41 b(PROTOCOL)781 b(READER)41
b(PROTOCOL)523 1836 y(1.)i(if)g(R[1-w]==W[1-w])37 b(then)347
b(if)43 b(W[r]<>R[r])d(then)523 1935 y(2.)130 b(w)43
b(:=)g(1-w)958 b(change)41 b(R[r])523 2035 y(3.)130 b(write)41
b(track)h(T[w,x[w]])389 b(r)43 b(:=)f(1-r)523 2135 y(4.)130
b(change)41 b(W[1-w])739 b(read)41 b(d[0])h(:=)h(D[r,0])523
2234 y(5.)g(else)1219 b(read)41 b(d[1])h(:=)h(D[r,1])523
2334 y(6.)130 b(x[w])42 b(:=)h(1-x[w])607 b(else)42 b(read)g(d[d[0])p
FD(\010)p Fk(d[1]])523 2434 y(7.)130 b(write)41 b(track)h(T[w,x[w]])519
b(:=)43 b(D[r,d[0])p FD(\010)p Fk(d[1]])523 2533 y(8.)130
b(change)41 b(D[w,1-x[w]])431 b(endif)523 2633 y(9.)43
b(endif)1087 b(read)42 b(track)g(T[r,d[0])p FD(\010)p
Fk(d[1]])606 2782 y FK(The)26 b(writer's)f(lo)r(cal)h(v)-5
b(ariables)25 b FC(d)1655 2794 y FB(0)1692 2782 y FC(;)14
b(d)1772 2794 y FB(1)1836 2782 y FK(ha)n(v)n(e)25 b(b)r(een)h(renamed)g
(to)g FC(x)2703 2794 y FB(0)2740 2782 y FC(;)14 b(x)2824
2794 y FB(1)2888 2782 y FK(to)26 b(emphasize)523 2882
y(that)36 b FC(x)758 2894 y FA(i)821 2882 y FK(no)n(w)f(represen)n(ts)f
(the)i(eXclusiv)n(e-or)e(of)h FC(D)2200 2894 y FA(i;)p
FB(0)2316 2882 y FK(and)g FC(D)2554 2894 y FA(i;)p FB(1)2634
2882 y FK(.)h(The)g(reader's)d(lo)r(cal)523 2982 y(v)-5
b(ariable)31 b FC(d)h FK(has)g(b)r(een)g(replaced)f(b)n(y)h
FC(d)1762 2994 y FB(0)1831 2982 y FK(and)g FC(d)2040
2994 y FB(1)2077 2982 y FK(,)h(where)e FC(d)2420 2994
y FA(i)2480 2982 y FK(is)h(mean)n(t)f(to)h(hold)g(a)g(cop)n(y)523
3081 y(of)26 b FC(D)685 3093 y FA(r)n(;i)761 3081 y FK(.)g(All)g
(shared)f(and)h(lo)r(cal)f(v)-5 b(ariables)25 b(are)g(initialized)h(to)
f(0)h(as)f(usual.)h(Because)e(the)523 3181 y(switc)n(h)j(no)n(w)g
(consists)g(of)h(eigh)n(t)f(safe)g(bits,)h(w)n(e)f(call)h(it)g(the)g
(\\Safe)f(Byte)g(Switc)n(h.")606 3280 y(W)-7 b(e)28 b(can)f(no)n(w)g
(state)h(the)g(main)f(theorem:)523 3446 y Fv(Theorem)32
b FH(6.8)41 b FG(A)26 b(single-r)l(e)l(ader,)i(single-writer)e
FC(b)p FG(-bits)g(atomic)h(variable)h(c)l(an)f(b)l(e)f(c)l(on-)523
3546 y(structe)l(d)j(fr)l(om)h FK(4)p FC(b)18 b FK(+)g(8)29
b FG(safe)i(bits)f(\()p FK(4)f FG(tr)l(acks)h(and)g(a)g(safe)h(byte\).)
606 3712 y FK(W)-7 b(e)20 b(p)r(ostp)r(one)f(the)h(pro)r(of)f(of)h
(correctness)d(of)j(the)g(new)f(construction)g(to)g(Section)h(6.7.3.)
523 4019 y FL(6.7)116 b(The)38 b(A)m(tomicit)m(y)e(Automaton)523
4219 y FK(In)31 b(this)g(and)f(the)h(next)g(few)g(sections)f(w)n(e)g
(discuss)h(the)g(use)f(of)h(mac)n(hines)f(\(computers\))523
4318 y(as)d(an)g(aid)h(in)g(designing)e(and)i(v)n(erifying)e(atomic)h
(v)-5 b(ariable)27 b(constructions.)606 4418 y(The)j(v)n(eri\014cation)
e(is)i(based)f(on)h(a)f(generic)g(automaton)f(whic)n(h)i(em)n(b)r(o)r
(dies)g(the)g(three)523 4517 y(atomicit)n(y)36 b(prop)r(erties)g(of)g
(system)h(executions)f(\(in)h(the)g(single-reader,)d(single-writer)523
4617 y(case\).)k(Figure)f(6.2)h(sho)n(ws)f(a)g(picture)i(of)f(the)g
(automaton.)g(The)g(transitions)f(of)h(this)523 4717
y(automaton)30 b(represen)n(t)g(the)h(starts)f(and)h(ends)g(of)g(read)f
(and)h(write)g(actions,)f(while)h(the)523 4816 y(no)r(des)37
b(represen)n(t)g(the)h(\\atomicit)n(y)e(state")h(of)h(a)f(run)h(on)f
(the)h(atomic)f(v)-5 b(ariable.)37 b(The)523 4916 y(latter)e(corresp)r
(onds)f(to)i(the)g(set)f(of)h(v)-5 b(alues)35 b(that)h(the)g
(next-ending)f(read)g(can)g(return)523 5016 y(without)28
b(violating)f(atomicit)n(y|its)g(size)g(is)h(sho)n(wn)e(inside)i(eac)n
(h)f(no)r(de.)606 5115 y(The)g(no)r(des)f(can)h(b)r(e)g(divided)g(in)n
(to)f(four)g(groups,)g(dep)r(ending)h(on)f(whether)h(eac)n(h)e(user)523
5215 y(is)k(idle)f(or)g(busy)g(accessing)g(the)g(v)-5
b(ariable.)28 b(When)h(b)r(oth)g(users)f(are)g(idle,)g(the)h(atomicit)n
(y)523 5314 y(state)34 b(of)g(the)g(run)g(is)g(\014xed)g(b)n(y)g(the)g
(curren)n(t)g(v)-5 b(alue)34 b(of)g(the)g(v)-5 b(ariable|this)33
b(b)r(eing)h(the)p eop end
%%Page: 79 89
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 @endspecial 1152 2796 a(FIGURE)25 b(6.2.)i(the)f(general)g(atomicit)n
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b(that)h(a)f(newly)g(started)g(read)g(is)g(allo)n(w)n(ed)f(to)h
(return.)g(This)h(explains)e(the)523 3159 y(single)27
b(no)r(de)h(in)g(this)g(group.)606 3259 y(When)d(the)g(writer)f(is)g
(busy)g(and)g(the)h(reader)e(idle,)i(w)n(e)f(can)g(distinguish)g(b)r
(et)n(w)n(een)h(t)n(w)n(o)523 3359 y(states:)36 b(either)g(the)h
(reader)e(has)h(read)g(the)h(new)f(v)-5 b(alue)36 b(that)h(is)g(b)r
(eing)f(written,)h(or)e(it)523 3458 y(hasn't.)25 b(Hence)h(there)f(are)
f(t)n(w)n(o)h(no)r(des)g(in)h(this)g(group.)e(In)h(the)h(former)f(case)
f(subsequen)n(t)523 3558 y(read)35 b(actions)h(m)n(ust)g(return)f(the)i
(same)e(v)-5 b(alue)36 b(as)g(the)g(last)g(read)f(action)h(in)g(order)f
(to)523 3657 y(prev)n(en)n(t)27 b(new-old)g(in)n(v)n(ersions)e
(\(condition)j(A1\).)g(Hence)g(the)g(set)f(size)h(of)f(one.)606
3757 y(When)k(the)h(reader)d(is)i(busy)-7 b(,)30 b(there)h(are)f(man)n
(y)g(p)r(ossible)g(states,)h(dep)r(ending)g(on)f(the)523
3857 y(n)n(um)n(b)r(er)36 b(of)g(writes)f(that)i(o)n(v)n(erlap)d(the)i
(read.)f(As)h(the)h(no)r(des)e(progress)f(to)i(the)g(righ)n(t,)523
3956 y(the)f(set)f(of)g(v)-5 b(alues)34 b(that)h(the)g(curren)n(t)e
(read)g(action)h(is)g(allo)n(w)n(ed)f(to)h(return)g(gro)n(ws.)f(Of)523
4056 y(course,)g(while)i(the)f(picture)h(suggests)d(an)i(in\014nite)h
(progression)d(of)i(no)r(des,)g(its)h(size)f(is)523 4156
y(in)c(fact)f(limited)i(b)n(y)e(the)h(n)n(um)n(b)r(er)f(of)g(v)-5
b(alues)30 b(that)f(the)h(atomic)f(v)-5 b(ariable)29
b(can)g(hold)g(\(its)523 4255 y(domain-size\).)606 4355
y(In)c(the)g(group)e(where)h(only)g(the)h(reader)e(is)i(busy)-7
b(,)24 b(there)g(are)g(t)n(w)n(o)g(start-of-write)f(tran-)523
4454 y(sitions)41 b(emanating)f(do)n(wn)n(w)n(ard)f(from)h(eac)n(h)g
(no)r(de.)h(As)g(can)f(b)r(e)i(deduced)f(from)f(the)523
4554 y(resulting)e(set)g(sizes,)g(the)g(upp)r(er)g(transition)g
(corresp)r(onds)e(to)i(the)h(write)f(of)g(a)g(v)-5 b(alue)523
4654 y(already)32 b(in)i(the)g(set)g(of)f(p)r(ermitted)i(return)e(v)-5
b(alues.)33 b(In)h(this)g(case,)f(while)g(the)h(set)g(size)523
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(required)h(to)g(map)g(the)h(read)e(action)h(to)g(the)523
4853 y(curren)n(t)23 b(write)h(action)g(\(if)h(the)g(read)e(action)h
(decides)g(to)g(return)f(its)i(v)-5 b(alue\).)24 b(This)g(means)523
4953 y(that)30 b(the)f(next)h(read)e(action)h(will)g(not)h(b)r(e)f
(able)g(to)g(com)n(bine)g(with)h(the)g(curren)n(t)e(one)h(to)523
5052 y(create)e(a)g(new-old)g(in)n(v)n(ersion.)606 5152
y(Alternativ)n(ely)-7 b(,)20 b(if)h(the)g(v)-5 b(alue)20
b(of)g(the)h(new)f(write)g(is)h(outside)f(the)h(set,)f(then)h(this)g(v)
-5 b(alue)20 b(is)523 5251 y(added)i(to)g(it,)h(but)g(if)g(no)n(w)f
(the)h(read)e(decides)h(to)h(return)e(the)i(new)g(v)-5
b(alue,)22 b(then)h(the)g(atom-)523 5351 y(icit)n(y)31
b(state)g(represen)n(ted)f(b)n(y)h(the)h(middle-left)g(no)r(de)f(is)g
(reac)n(hed.)f(The)h(other)g(left)n(w)n(ard)p eop end
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(ottom)g(no)r(de)f(corresp)r(ond)f(to)523 606 y(return)j(v)-5
b(alues)27 b(written)h(b)n(y)g(earlier)e(write)h(actions.)523
857 y Fl(6.7.1)99 b(Using)35 b(the)f(A)n(utomaton)h(for)g(V)-7
b(eri\014c)i(ation)34 b(of)g(a)h(given)f(R)n(un)523 1006
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(state)g(information)f(the)h(actual)f(set)h(of)f(v)-5
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b(for)f(the)g(next)h(ending)g(read.)523 1205 y(The)22
b(v)-5 b(alue)22 b(returned)f(b)n(y)h(a)f(read)g(can)h(then)g(b)r(e)h
(v)n(eri\014ed)e(as)g(follo)n(ws:)g(If)h(it)h(is)f(in)g(the)g(ab)r(o)n
(v)n(e)523 1305 y(set,)i(then)h(w)n(e)f(tak)n(e)f(the)i(left)n(w)n(ard)
e(transition)g(to)h(the)h(the)f(b)r(ottom-left)h(or)e(top-left)i(no)r
(de)523 1405 y(\(in)j(case)e(writer)g(is)h(idle\),)h(and)f(reduce)f
(the)h(set)h(to)e(con)n(tain)h(only)f(the)i(v)-5 b(alue)27
b(of)g(the)g(last)523 1504 y(completed)35 b(write.)f(Otherwise,)g(if)i
(it)f(is)g(the)g(v)-5 b(alue)34 b(curren)n(tly)g(b)r(eing)h(written)g
(b)n(y)f(the)523 1604 y(writer,)26 b(then)g(w)n(e)g(tak)n(e)g(the)g
(left)n(w)n(ard)f(transition)h(to)g(the)g(middle-left)h(no)r(de)g(and)f
(empt)n(y)523 1703 y(the)38 b(set)g(\(since)g(no)n(w)f(ev)n(en)g(the)h
(v)-5 b(alue)37 b(of)h(the)g(last)g(completed)f(write)h(is)f(in)n(v)-5
b(alid\).)38 b(If)523 1803 y(the)29 b(returned)g(v)-5
b(alue)29 b(satis\014es)f(neither)h(of)g(these)g(cases,)f(then)i(the)f
(run)g(is)g(non-atomic.)523 1903 y(The)34 b(set)f(is)h(further)f(main)n
(tained)g(at)h(the)f(completion)h(of)f(a)g(write,)h(b)n(y)f(either)g
(adding)523 2002 y(the)i(written)f(v)-5 b(alue)34 b(to)g(the)g(set)g
(if)h(the)f(reader)f(is)h(busy)-7 b(,)34 b(or)f(c)n(hanging)g(the)h
(set)g(to)g(the)523 2102 y(singleton)27 b(with)h(that)g(v)-5
b(alue)27 b(if)i(the)f(reader)e(is)h(idle.)606 2202 y(F)-7
b(or)29 b(atomic)f(bit)i(constructions,)e(w)n(e)h(kno)n(w)f(that)i(the)
f(v)-5 b(alues)29 b(written)h(are)e(alternat-)523 2301
y(ingly)j(1)h(and)f(0.)h(This)f(means)h(that)g(the)g(size)f(of)h(the)g
(set)g(of)f(p)r(ermitted)i(return)e(v)-5 b(alues)523
2401 y(is)27 b(b)r(ounded)h(b)n(y)f(2.)g(The)g(size)g(of)g(the)h
(atomicit)n(y)e(automaton)h(shrinks)f(acccordingly)-7
b(.)26 b(In)523 2500 y(the)35 b(group)f(with)h(only)g(the)g(reader)f
(busy)-7 b(,)35 b(there)f(are)g(only)h(t)n(w)n(o)f(essen)n(tially)g
(di\013eren)n(t)523 2600 y(no)r(des|either)c(the)i(reader)d(can)h
(return)h(the)g(curren)n(t)f(v)-5 b(alue)31 b(of)f(the)i(atomic)e(bit,)
h(or)f(it)523 2700 y(can)k(return)h(b)r(oth)g(0)f(and)h(1.)f(In)h(the)h
(group)d(where)h(b)r(oth)i(users)e(are)f(busy)-7 b(,)35
b(there)g(are)523 2799 y(only)24 b(three)g(no)r(des.)h(In)f(one,)h(it)g
(can)f(return)g(either)g(the)h(old)f(or)g(the)h(new)f(v)-5
b(alue)25 b(with)g(t)n(w)n(o)523 2899 y(di\013eren)n(t)34
b(transitions.)f(In)h(the)h(second,)e(it)i(can)e(return)h(b)r(oth)g(0)g
(and)f(1)h(as)f(old)h(v)-5 b(alues)523 2999 y(so)27 b(there)h(is)g
(only)g(one)f(suc)n(h)h(transition.)f(In)i(the)f(third)g(no)r(de,)g(it)
h(m)n(ust)f(return)f(the)i(new)523 3098 y(v)-5 b(alue.)22
b(Figure)g(6.3)f(sho)n(ws)h(this)g(reduced)g(automaton,)g(with)h
(explicit)f(men)n(tion)h(of)f(whic)n(h)523 3198 y(v)-5
b(alue)26 b(is)f(returned)h(b)n(y)f(a)g(read)g(\(if)i(the)f(writer)f
(is)h(idle,)g(then)g(\\new")f(means)g(\\curren)n(t."\))523
3297 y(There)30 b(are)g(t)n(w)n(o)h(no)r(des)f(from)h(whic)n(h)g(a)f
(single)h(new-lab)r(elled)f(transition)h(emanates)f(to)523
3397 y(the)36 b(left.)g(F)-7 b(rom)35 b(these)h(no)r(des)f(atomicit)n
(y)g(can)g(b)r(e)h(violated)f(if)h(the)g(read)f(returns)f(the)523
3497 y(other)27 b(v)-5 b(alue.)523 3747 y Fl(6.7.2)99
b(V)-7 b(erifying)34 b(the)h(A)n(tomic)g(Bit)f(Construction)523
3897 y FK(A)e(program)e(has)h(b)r(een)h(written)g(to)g(systematically)e
(searc)n(h)h(all)g(states)g(of)h(the)g(atomic)523 3996
y(bit)c(construction.)f(The)h(state)f(information)g(in)n(v)n(olv)n(es)f
(the)i(follo)n(wing:)648 4157 y FD(\017)41 b FK(p)r(osition)27
b(of)h(writer)e(in)i(its)g(proto)r(col,)f(i.e.,)g(writer)g(state)648
4320 y FD(\017)41 b FK(p)r(osition)27 b(of)h(reader)e(in)i(its)f(proto)
r(col,)g(i.e.,)h(reader)e(state)648 4483 y FD(\017)41
b FK(v)-5 b(alues)27 b(of)g(the)h(reader's)e(lo)r(cal)h(v)-5
b(ariables)648 4646 y FD(\017)41 b FK(v)-5 b(alues)27
b(of)g(the)h(three)g(safe)f(bits)648 4809 y FD(\017)41
b FK(p)r(osition)27 b(in)h(automaton,)f(i.e.,)g(atomicit)n(y)g(state)
606 4969 y(W)-7 b(e)30 b(no)n(w)g(explain)f(ho)n(w)g(the)i(safet)n(y)e
(of)h(the)g(shared)f(bits)h(is)g(mo)r(delled.)g(A)g(safe)g(write)523
5069 y(is)c(mo)r(delled)g(b)n(y)g(t)n(w)n(o)g(separate)e(transitions)h
(represen)n(ting)g(the)h(start)g(and)g(the)g(\014nish)h(of)523
5169 y(that)c(write.)f(A)h(read,)f(on)g(the)h(other)e(hand,)i(is)f
(represen)n(ted)f(b)n(y)i(a)f(single)g(transition,)f(as)h(if)523
5268 y(it)28 b(o)r(ccurred)e(in)h(a)g(single)g(time)h(instan)n(t.)f
(This)g(can)g(b)r(e)h(done)f(for)f(the)i(follo)n(wing)e(reason.)523
5368 y(If)e(a)g(read)f(from)g(a)h(safe)f(bit)h(o)n(v)n(erlaps)e(a)h
(write)h(on)f(the)i(same)e(bit,)h(then)h(either)e(0)h(or)f(1)g(can)p
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 @endspecial 1270 2545 a(FIGURE)25 b(6.3.)i(the)e(atomic)i(bit)e
(automaton)523 2794 y FK(b)r(e)33 b(returned,)g(so)f(the)i(read)e(migh)
n(t)h(as)f(w)n(ell)h(ha)n(v)n(e)f(o)r(ccurred)g(completely)h(within)g
(that)523 2893 y(write.)f(If)h(no)f(write)g(o)n(v)n(erlaps)e(the)j
(read,)f(then)h(the)g(v)-5 b(alue)32 b(returned)g(m)n(ust)g(b)r(e)h
(that)g(of)523 2993 y(the)27 b(last)f(preceding)g(write,)h(and)f(it)i
(clearly)d(do)r(esn't)i(matter)f(ho)n(w)g(long)g(the)h(read)f(lasts.)
606 3093 y(In)31 b(summary)-7 b(,)31 b(if)g(a)g(read)f(o)r(ccurs)g(b)r
(et)n(w)n(een)h(t)n(w)n(o)f(consecutiv)n(e)g(writes,)h(then)g(there)g
(is)523 3192 y(a)h(single)h(transition)f(corresp)r(onding)e(to)j(the)g
(return)f(of)h(the)g(curren)n(t)f(v)-5 b(alue,)33 b(and)f(if)h(it)523
3292 y(o)r(ccurs)k(b)r(et)n(w)n(een)h(the)h(start)e(and)h(\014nish)g
(of)g(a)g(write,)g(then)g(there)g(are)f(t)n(w)n(o)h(di\013eren)n(t)523
3392 y(transitions,)27 b(one)g(for)g(eac)n(h)g(v)-5 b(alue)27
b(that)h(can)f(b)r(e)h(returned.)606 3491 y(This)h(mo)r(del)h(captures)
f(the)g(essence)g(of)h(safe)f(bits.)g(It)h(leads)f(to)g(3)g(writer)g
(states)g(and)523 3591 y(7)c(reader)e(states.)i(The)g(program)e(starts)
h(b)n(y)h(putting)g(the)h(initial)f(state)g(in)g(an)g(otherwise)523
3690 y(empt)n(y)f(set.)g(Then)g(it)h(rep)r(eatedly)e(tak)n(es)g(an)h
(elemen)n(t)g(from)g(the)g(set,)g(and)g(replaces)f(it)h(b)n(y)523
3790 y(all)d(states)f(that)h(result)f(from)h(the)g(remo)n(v)n(ed)e(one)
i(b)n(y)f(a)g(single)h(transition)f(and)g(are)g(not)h(y)n(et)523
3890 y(in)26 b(the)h(set.)f(Additionally)-7 b(,)26 b(the)g(program)e(k)
n(eeps)h(trac)n(k)g(of)h(the)g(shortest)g(path)g(from)f(the)523
3989 y(initial)e(state)f(to)g(eac)n(h)g(visited)h(one.)f(If)h(some)f
(transition)g(is)g(the)h(return)f(of)g(a)g(v)-5 b(alue)23
b(whic)n(h)523 4089 y(is)d(in)n(v)-5 b(alid)19 b(according)f(to)i(the)g
(automaton,)f(then)h(the)g(program)e(prin)n(ts)h(out)h(a)f(description)
523 4189 y(of)41 b(the)g(shortest)f(path)h(to)g(the)g(failing)g(state,)
f(rev)n(ealing)f(the)i(shortcomings)f(of)g(the)523 4288
y(construction)c(b)r(eing)g(v)n(eri\014ed.)g(Otherwise,)f(if)i(the)g
(set)f(b)r(ecomes)g(empt)n(y)-7 b(,)37 b(then)f(some)523
4388 y(statistics)25 b(are)f(prin)n(ted)h(suc)n(h)g(as)f(the)i(n)n(um)n
(b)r(er)f(of)g(visited)g(states)g(for)f(eac)n(h)h(com)n(bination)523
4488 y(of)j(writer)f(state)g(and)g(reader)f(state.)606
4587 y(The)36 b(program)e(help)r(ed)i(the)h(design)e(of)h(the)g(atomic)
f(bit)i(construction)e(b)n(y)g(making)523 4687 y(it)30
b(easy)e(to)h(try)g(out)h(v)-5 b(arious)28 b(alternativ)n(es,)f(and)j
(immediately)f(getting)g(a)g(diagnosis)f(of)523 4786
y(p)r(ossible)f(problems.)523 5035 y Fl(6.7.3)99 b(V)-7
b(erifying)34 b(the)h(Safe)f(Byte)h(Switch)g(Construction)523
5185 y FK(Lik)n(e)30 b(in)h(the)g(pro)r(of)f(of)h(the)g(4-trac)n(k)e
(construction)h(with)h(the)g(4)f(atomic)g(bits,)h(w)n(e)g(m)n(ust)523
5285 y(\014rst)24 b(establish)g(that)g(the)h(new)f(construction)g(is)g
(collision)f(free,)h(that)h(is,)f(w)n(e)g(m)n(ust)g(pro)n(v)n(e)p
eop end
%%Page: 82 92
TeXDict begin 82 91 bop 2059 299 a FF(6.7.)52 b(The)26
b(A)n(tomicit)n(y)f(Automaton)154 b(82)523 506 y FK(lemma)24
b(6.7)g(again.)f(F)-7 b(or)24 b(this)h(w)n(e)f(again)f(need)i(an)f(in)n
(v)-5 b(arian)n(t)23 b(to)h(hold)h(under)f(all)g(p)r(ossible)523
606 y(runs)32 b(based)h(on)f(a)h(state)f(diagram.)g(In)h(this)g(case)f
(ho)n(w)n(ev)n(er,)f(w)n(e)h(cannot)g(assume)g(that)523
706 y(the)42 b(switc)n(h)f(bit)h(accesses)e(can)h(b)r(e)g(linearized,)g
(since)g(they)h(are)e(only)h(safe.)g(Instead)523 805
y(w)n(e)c(adopt)g(the)h(safe)g(bit)g(mo)r(del)f(of)h(the)g(previous)e
(section.)i(This)f(en)n(tails)g(rede\014ning)523 905
y(the)i(reader)e(and)i(writer)f(states.)g(Since)h(the)g(simplicit)n(y)f
(of)h(the)g(in)n(v)-5 b(arian)n(ts)37 b(dep)r(ends)523
1005 y(rather)e(hea)n(vily)f(on)h(the)h(exact)f(form)h(of)f(the)h
(proto)r(cols.)e(w)n(e)h(base)g(the)h(pro)r(of)f(on)g(the)523
1104 y(follo)n(wing)c(reform)n(ulated)g(proto)r(cols,)f(that)j(are)e
(seman)n(tically)g(equiv)-5 b(alen)n(t)31 b(to)h(those)g(of)523
1204 y(Section)26 b(6.6.2)e(\(lo)r(cal)h(assignmen)n(ts)g(ha)n(v)n(e)f
(mo)n(v)n(ed)h(and)g(indices)h(c)n(hanged)e(accordingly\):)523
1370 y Fk(WRITER)41 b(PROTOCOL)781 b(READER)41 b(PROTOCOL)523
1569 y(1.)i(if)g(R[1-w]==W[1-w])37 b(then)347 b(if)43
b(W[r]<>R[r])d(then)523 1669 y(2.)130 b(write)41 b(track)h
(T[1-w,x[1-w]])213 b(r)43 b(:=)f(1-r)523 1768 y(3.)130
b(w)43 b(:=)g(1-w)958 b(change)41 b(R[1-r])523 1868 y(4.)130
b(change)41 b(W[1-w])739 b(read)41 b(d[0])h(:=)h(D[r,0])523
1968 y(5.)g(else)1219 b(read)41 b(d[1])h(:=)h(D[r,1])523
2067 y(6.)130 b(write)41 b(track)h(T[w,1-x[w]])213 b(else)42
b(read)g(d[d[0])p FD(\010)p Fk(d[1]])523 2167 y(7.)130
b(x[w])42 b(:=)h(1-x[w])825 b(:=)43 b(D[r,d[0])p FD(\010)p
Fk(d[1]])523 2267 y(8.)130 b(change)41 b(D[w,1-x[w]])431
b(endif)523 2366 y(9.)43 b(endif)1087 b(read)42 b(track)g(T[r,d[0])p
FD(\010)p Fk(d[1]])606 2532 y FK(Note)30 b(that)h(when)f(a)g(safe)f
(bit)i(is)f(c)n(hanged,)f(the)i(lo)r(cal)e(cop)n(y)h(already)e(holds)i
(the)h(new)523 2632 y(v)-5 b(alue|this)22 b(is)f(the)h(prop)r(ert)n(y)f
(that)g(ensures)g(the)h(most)f(simple)h(in)n(v)-5 b(arian)n(ts.)20
b(W)-7 b(e)22 b(pro)r(ceed)523 2731 y(to)28 b(en)n(umerate)e(the)i
(essen)n(tial)f(p)r(ositions)g(of)h(the)g(users)e(in)i(their)g(proto)r
(cols.)606 2831 y(The)g(states)f(of)g(the)h(writer)f(are:)523
2997 y FH(0)41 b FK(idle,)28 b(i.e.,)g(b)r(efore)f(the)h(safe)f(read)g
(of)h FC(R)1848 3009 y FB(1)p Fw(\000)p FA(w)2014 2997
y FK(in)g(line)g(1)523 3163 y FH(1)41 b FK(b)r(et)n(w)n(een)27
b(the)h(safe)e(read)g(of)h FC(R)1579 3175 y FB(1)p Fw(\000)p
FA(w)1745 3163 y FK(and)g(the)h(safe)e(c)n(hange)g(of)h
FC(W)2658 3175 y FB(1)p Fw(\000)p FA(w)2824 3163 y FK(in)h(line)f(4,)g
(when)731 3263 y(it)h(is)f(writing)g(trac)n(k)g FC(T)1445
3275 y FB(1)p Fw(\000)p FA(w)r(;x)1638 3283 y Fy(1)p
Ft(\000)p Fx(w)523 3429 y FH(2)41 b FK(c)n(hanging)27
b(safe)g(bit)h FC(W)1332 3441 y FB(1)p Fw(\000)p FA(w)1499
3429 y FK(in)g(line)f(4)523 3595 y FH(3)41 b FK(b)r(et)n(w)n(een)36
b(the)g(safe)f(read)g(of)h FC(R)1623 3607 y FB(1)p Fw(\000)p
FA(w)1797 3595 y FK(and)g(the)g(safe)f(c)n(hange)g(of)g
FC(D)2744 3607 y FA(w)r(;)p FB(1)p Fw(\000)p FA(x)2937
3615 y Fx(w)3023 3595 y FK(in)h(line)g(8,)731 3694 y(when)27
b(it)h(is)g(writing)f(trac)n(k)f FC(T)1661 3706 y FA(w)r(;)p
FB(1)p Fw(\000)p FA(x)1854 3714 y Fx(w)523 3861 y FH(4)41
b FK(c)n(hanging)27 b(safe)g(bit)h FC(D)1323 3873 y FA(w)r(;)p
FB(1)p Fw(\000)p FA(x)1516 3881 y Fx(w)1594 3861 y FK(in)f(line)h(8)523
4027 y(Th)n(us,)34 b(the)g(writer)f(is)h(alw)n(a)n(ys)e(mo)n(ving)g
(from)i(state)f(0)h(to)g(either)f(state)h(1)f(follo)n(w)n(ed)g(b)n(y)
523 4126 y(state)23 b(2)g(or)f(to)h(state)g(3)g(follo)n(w)n(ed)f(b)n(y)
h(state)g(4)g(\(dep)r(ending)g(on)g(the)h(outcome)e(of)i(the)f(test\),)
523 4226 y(and)k(then)i(bac)n(k)d(to)i(state)f(0.)606
4325 y(The)h(states)f(of)g(the)h(reader)f(are:)523 4491
y FH(0)41 b FK(idle,)28 b(i.e.,)g(b)r(efore)f(the)h(safe)f(read)g(of)h
FC(W)1863 4503 y FA(r)1928 4491 y FK(in)f(line)h(1)523
4658 y FH(1)41 b FK(b)r(et)n(w)n(een)32 b(the)f(safe)g(read)g(of)g
FC(W)1616 4670 y FA(r)1685 4658 y FK(and)g(the)h(safe)e(read)h(of)g
FC(D)2520 4670 y FA(r)n(;)p FB(0)2637 4658 y FK(in)h(line)g(4,)f(when)g
(it)h(is)731 4757 y(c)n(hanging)26 b(safe)h(bit)h FC(R)1435
4769 y FB(1)p Fw(\000)p FA(r)523 4923 y FH(2)41 b FK(b)r(et)n(w)n(een)
28 b(the)g(safe)f(read)g(of)g FC(D)1588 4935 y FA(r)n(;)p
FB(0)1702 4923 y FK(and)g(the)h(safe)f(read)g(of)h FC(D)2519
4935 y FA(r)n(;)p FB(1)2632 4923 y FK(in)g(line)g(5)523
5089 y FH(3)41 b FK(b)r(et)n(w)n(een)28 b(the)g(safe)f(read)g(of)g
FC(W)1597 5101 y FA(r)1662 5089 y FK(and)h(the)g(safe)f(read)f(of)i
FC(D)2479 5101 y FA(r)n(;d)2563 5109 y Fy(0)2594 5101
y Fw(\010)p FA(d)2681 5109 y Fy(1)2745 5089 y FK(in)g(line)g(7)523
5255 y FH(4)41 b FK(reading)27 b(trac)n(k)f FC(T)1169
5267 y FA(r)n(;d)1253 5275 y Fy(0)1284 5267 y Fw(\010)p
FA(d)1371 5275 y Fy(1)1435 5255 y FK(in)i(line)g(9)p
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%%EndDocument
 @endspecial 711 2576 a(FIGURE)25 b(6.4.)i(state)f(diagram)h(of)f
(4-trac)n(k)g(construction)g(with)g(safe)h(b)n(yte)d(switc)n(h)523
2838 y FK(Th)n(us)33 b(the)g(reader)f(is)h(alw)n(a)n(ys)e(mo)n(ving)i
(from)f(state)h(0)g(to)g(either)g(state)g(1,)g(follo)n(w)n(ed)f(b)n(y)
523 2937 y(state)27 b(2)h(or)e(to)i(state)f(3,)g(then)i(on)e(to)g
(state)h(4,)f(and)g(\014nally)h(bac)n(k)f(to)g(state)g(0.)606
3037 y(Altogether,)e(there)g(are)f(no)n(w)h(5)13 b FD(\002)h
FK(5)22 b(=)h(25)h(states,)h(whic)n(h)h(are)e(pictured)h(in)h(\014gure)
f(6.4,)523 3137 y(along)j(with)i(all)f(p)r(ossible)h(transitions.)e
(Eac)n(h)g(no)r(de)i(con)n(tains)f(a)g(\(p)r(ossibly)g(empt)n(y\))h
(set)523 3236 y(of)25 b(form)n(ulas,)g(whic)n(h)g(are)f(to)i(b)r(e)g
(conjuncted,)g(together)e(with)i(the)g(in)n(v)-5 b(arian)n(t)24
b FC(W)3076 3248 y FA(w)3153 3236 y FK(=)f FC(R)3304
3248 y FA(w)523 3336 y FK(whic)n(h)d(holds)g(for)f(all)h(no)r(des.)g(T)
-7 b(o)19 b(solv)n(e)g(the)h(p)r(oten)n(tial)g(am)n(biguit)n(y)g(of)f
(this)i(form)n(ula)e(whic)n(h)523 3436 y(arises)33 b(when)h(the)h
(reader)d(is)i(c)n(hanging)f FC(R)1898 3448 y FA(w)1986
3436 y FK(\(the)i(writer)e(c)n(hanges)g FC(W)2811 3448
y FB(1)p Fw(\000)p FA(w)2950 3436 y FK(\),)i(w)n(e)f(mak)n(e)523
3535 y(the)28 b(follo)n(wing)f(de\014nition:)523 3701
y Fv(Definition)32 b FH(6.9)41 b FK(If)27 b(a)g(safe)g(bit)g
FC(B)32 b FK(is)27 b(b)r(eing)g(c)n(hanged)f(then)h(in)h(form)n(ulas,)e
FC(B)31 b FK(refers)26 b(to)523 3800 y(its)i FG(new)g
FK(v)-5 b(alue.)606 3966 y(The)38 b(notation)f FC(D)r
FK(\()p FC(i)p FK(\))h(is)f(used)h(as)e(an)i(abbreviation)e(of)h
FC(d)2500 3978 y FA(i)2568 3966 y FK(=)i FC(D)2741 3978
y FA(r)n(;i)2855 3966 y FK(and)e(expresses)523 4065 y(that)32
b(a)f(lo)r(cal)f(displa)n(y)h(bit)h(of)f(the)h(reader)e(matc)n(hes)h
(the)g(shared)g(one)g(\(in)h(the)f(reader's)523 4165
y(group\).)h(A)i(greater-than)d(sign)h(\()p FC(>)p FK(\))i(is)f(used)g
(to)f(denote)h(implication.)h(Starting)e(from)523 4264
y(the)24 b(initial)g(state)f(\(0)p FC(;)14 b FK(0\))23
b(\(b)r(oth)h(users)e(idle\),)j(eac)n(h)d(in)n(v)-5 b(arian)n(t)22
b(can)h(b)r(e)h(man)n(ually)f(c)n(hec)n(k)n(ed)523 4364
y(b)n(y)k(considering)g(the)h(p)r(ossible)f(predecessors)e(of)j(a)f(no)
r(de.)606 4464 y(Another)e(state)g(space)g(searc)n(h)f(program)f(w)n
(as)h(used)h(to)g(construct)g(this)h(diagram.)e(It's)523
4563 y(state)e(also)f(includes)h(information)f(on)h(the)g(v)-5
b(alidit)n(y)22 b(of)g(the)g(trac)n(ks)e(\(relativ)n(e)h(to)h(the)h
(next)523 4663 y(end-of-read\),)30 b(as)f(deriv)n(ed)h(from)h(the)g
(atomicit)n(y)f(automaton.)f(With)j(the)f(help)g(of)f(this)523
4763 y(information,)18 b(the)h(program)e(actually)h(v)n(eri\014es)g
(the)h(correctness)e(of)h(the)i(construction.)e(In)523
4862 y(this)24 b(c)n(hapter)e(ho)n(w)n(ev)n(er,)g(w)n(e)h(neglect)g
(this)g(extra)g(state,)g(since)g(including)h(it)g(w)n(ould)f(mak)n(e)
523 4962 y(the)j(diagram)f(o)n(v)n(erly)f(complex,)h(making)h(man)n
(ual)f(insp)r(ection)h(practically)f(imp)r(ossible.)606
5061 y(It)41 b(is)g(no)n(w)g(easy)f(to)h(see)f(from)h(the)g(diagram)f
(that)h(lemma)g(6.7)f(holds.)h(P)n(oten)n(tial)523 5161
y(collisions)28 b(can)h(o)r(ccur)g(in)h(states)f(\(1)p
FC(;)14 b FK(4\))29 b(and)g(\(3)p FC(;)14 b FK(4\).)29
b(In)h(the)g(\014rst)f(case,)f FC(w)h FK(=)d FC(r)32
b FK(and)d(the)523 5261 y(writer)18 b(writes)g(on)g(a)g(trac)n(k)g(in)h
(group)e(1)p FD(\000)p FC(w)r FK(,)i(so)f(there)g(is)h(no)f(collision.)
g(In)h(the)g(second)f(case,)523 5360 y(the)27 b(writer)e(writes)h(on)g
(trac)n(k)f FC(T)1526 5372 y FA(w)r(;)p FB(1)p Fw(\000)p
FA(x)1719 5380 y Fx(w)1769 5360 y FK(.)h(If)h FC(w)f
FD(6)p FK(=)d FC(r)r FK(,)k(then)g(w)n(e)e(are)h(done.)g(Otherwise,)f
(the)p eop end
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b(and)g(writer)h(are)e(in)j(the)f(same)f(group,)g(so)g
FC(w)g FK(=)c FC(r)r FK(,)29 b(implying)f FC(d)2785 518
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b FC(D)3126 518 y FA(r)n(;)p FB(1)p Fw(\000)p FA(x)3298
526 y Fx(r)3335 506 y FK(.)523 606 y(Along)29 b(with)h
FC(d)1001 618 y FB(1)p Fw(\000)p FA(x)1124 626 y Fx(r)1188
606 y FK(=)c FC(D)1348 618 y FA(r)n(;)p FB(1)p Fw(\000)p
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y FB(0)2921 606 y FD(\010)19 b FC(d)3048 618 y FB(1)3086
606 y FK(,)29 b(so)g(the)523 706 y(reader)d(is)i(on)f(the)h(other)f
(trac)n(k|no)f(collision.)523 1013 y FL(6.8)116 b(Correctness)37
b(of)h(Safe)g(Byte)h(Switc)m(h)e(Construction)523 1212
y FK(Giv)n(en)c(lemma)h(6.7,)e(it)i(remains)f(to)g(sho)n(w)g(that)g
(for)g(ev)n(ery)f(run)i(\()p FD(A)p FC(;)14 b FD(!)p
FK(\),)34 b(there)f(exists)523 1312 y(a)40 b(reading)f(function)i
FC(\031)j FK(suc)n(h)c(that)h FC(\033)47 b FK(=)d(\()p
FD(A)p FC(;)14 b FD(!)p FC(;)g(\031)s FK(\))42 b(satis\014es)d(the)i
(three)f(atomicit)n(y)523 1411 y(conditions.)24 b(W)-7
b(e)24 b(c)n(ho)r(ose)f(the)h(reading)f(function)h FC(\031)j
FK(to)d(map)g(a)f(read)g(action)h(to)g(the)g(write)523
1511 y(action)d(that)h(last)g(writes)f(to)h(the)g(trac)n(k)f(b)r(efore)
g(the)h(read)f(action)h(reads)e(from)i(that)g(trac)n(k.)523
1611 y(As)31 b(b)r(efore,)g(this)g(can)g(b)r(e)h(view)n(ed)e(as)h(the)g
(union)g(of)g(the)h(four)e(reading)g(functions)h(that)523
1710 y(mak)n(e)c(eac)n(h)g(trac)n(k)f(atomic,)h(according)f(to)i(lemma)
f(6.7.)606 1810 y(W)-7 b(e)28 b(no)n(w)f(pro)n(v)n(e)f(eac)n(h)h(of)g
(the)h(three)g(conditions)f(in)h(turn.)606 1910 y FH(Pro)s(of)k(of)g
(A0)c FK(The)f(reading)g(function)h(is)g(ob)n(viously)e(normal)h(b)n(y)
g(the)i(safet)n(y)e(of)g(the)523 2009 y(trac)n(k-bits.)606
2109 y FH(Pro)s(of)41 b(of)h(A1)35 b FK(The)h(pro)r(of)g(is)g(b)n(y)f
(con)n(tradiction.)g(Let)h FC(r)2504 2121 y FB(1)2542
2109 y FC(;)14 b(r)2616 2121 y FB(2)2690 2109 y FD(2)37
b(R)g FK(b)r(e)f(suc)n(h)f(that)523 2208 y FC(r)560 2220
y FB(1)636 2208 y FD(!)k FC(r)795 2220 y FB(2)870 2208
y FK(are)d(t)n(w)n(o)g(consecutiv)n(e)g(read)h(actions)f(and)h
FC(\031)s FK(\()p FC(r)2408 2220 y FB(2)2446 2208 y FK(\))i
FD(!)g FC(\031)s FK(\()p FC(r)2758 2220 y FB(1)2796 2208
y FK(\).)e(Assume)g(that)523 2308 y FC(\031)s FK(\()p
FC(r)642 2320 y FB(1)680 2308 y FK(\))j(writes)g(on)f(trac)n(k)f
FC(T)1410 2320 y FB(0)p FA(;)p FB(0)1500 2308 y FK(,)i(that)g
FC(W)1833 2320 y FB(0)1914 2308 y FK(=)i FC(W)2099 2320
y FB(1)2180 2308 y FK(=)h FC(R)2351 2320 y FB(0)2431
2308 y FK(=)g(0)c(\(using)h(the)g(in)n(v)-5 b(arian)n(t)523
2408 y FC(W)601 2420 y FA(w)678 2408 y FK(=)23 b FC(R)829
2420 y FA(w)883 2408 y FK(\))28 b(and)f(that)h FC(D)1353
2420 y FB(1)p FA(;)p FB(0)1466 2408 y FK(=)22 b FC(D)1622
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772 y([10])41 b(G.L.)36 b(P)n(eterson)e(and)i(J.E.)f(Burns,)g
FG(Concurr)l(ent)i(r)l(e)l(ading)h(while)h(writing)f(II:)f(the)694
872 y(multiwriter)f(c)l(ase)p FK(,)g(Pro)r(c.)e(28th)h(IEEE)e(Symp)r
(osium)j(on)f(F)-7 b(oundations)34 b(of)h(Com-)694 971
y(puter)27 b(Science,)h(pp.)g(383{392,)c(1987.)523 1137
y([11])41 b(G.L.)21 b(P)n(eterson)d(and)j(J.E.)f(Burns,)g
FG(The)k(A)n(mbiguity)g(of)g(Cho)l(osing)p FK(,)f(Pro)r(c.)c(8th)i(A)n
(CM)694 1237 y(Symp)r(osium)28 b(on)f(Principles)g(of)g(Distributed)h
(Computing,)g(pp.)g(145{157,)c(1989.)523 1403 y([12])41
b(G.L.)36 b(P)n(eterson,)e FG(Concurr)l(ent)j(r)l(e)l(ading)h(while)h
(writing)p FK(,)e(A)n(CM)f(T)-7 b(ransactions)34 b(on)694
1503 y(Programming)25 b(Languages)g(and)i(Systems,)h(v)n(ol.5,)e(No.1,)
h(1983,)f(pp.)i(46{55)523 1669 y([13])41 b(J.E.)31 b(Burns,)g(G.L.)h(P)
n(eterson,)e FG(Sharp)k(Bounds)g(for)h(Concurr)l(ent)e(R)l(e)l(ading)g
(While)694 1768 y(Writing)p FK(,)47 b(T)-7 b(ec)n(hnical)46
b(Rep)r(ort,)g(Georgia)f(Institute)j(of)e(T)-7 b(ec)n(hnology)45
b(GIT-ICS-)694 1868 y(87/31)523 2034 y([14])c(R.)26 b(Sc)n(ha\013er,)g
FG(On)i(the)h(c)l(orr)l(e)l(ctness)f(of)i(atomic)f(multi-writer)g(r)l
(e)l(gisters)p FK(,)d(T)-7 b(ec)n(hnical)694 2134 y(Rep)r(ort)27
b(MIT/LCS/TM-364,)e(MIT)j(lab.)f(for)g(Computer)h(Science,)f(June)h
(1988.)523 2300 y([15])41 b(P)-7 b(.M.B.)36 b(Vit\023)-42
b(an)n(yi,)37 b(B.)g(Aw)n(erbuc)n(h,)f FG(A)n(tomic)j(Shar)l(e)l(d)g(R)
l(e)l(gister)f(A)l(c)l(c)l(ess)g(by)g(Asyn-)694 2399
y(chr)l(onous)25 b(Har)l(dwar)l(e)p FK(,)e(Pro)r(c.)f(27th)g(IEEE)e
(Symp)r(osium)j(on)f(F)-7 b(oundations)22 b(of)g(Com-)694
2499 y(puter)27 b(Science,)h(pp.)g(233{243,)c(1986.)i(\(Errata,)g
(Ibid.,1987\))p eop end
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972 y(V)g(ariables)523 2549 y FL(7.1)116 b(In)m(tro)s(duction)523
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(atomic)g(v)-5 b(ariable|one)33 b(whose)h(accesses)e(ap-)523
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h(one)g(writer)f(and)h(one)f(reader,)f(acting)i(asyn-)523
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(constructed)e(from)g(lo)n(w)n(er)g(lev)n(el)g(hardw)n(are)523
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(yp)r(e)f(\(see)h([20)o(],)g(on)f(whic)n(h)g(the)523
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(and)g(deriv)n(e)f(the)i(implemen)n(tations)523 3445
y(b)n(y)e(transformations)f(from)h(the)h(sp)r(eci\014cation.)523
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(Main)f(R)-5 b(esult)523 3844 y FK(Usually)e(,)24 b(with)g(async)n
(hronous)e(readers)g(and)h(writers,)h(atomicit)n(y)f(of)h(op)r
(erations)e(is)i(sim-)523 3944 y(ply)30 b(assumed)g(or)f(enforced)g(b)n
(y)h(sync)n(hronization)e(primitiv)n(es)i(lik)n(e)f(semaphores.)g(Ho)n
(w-)523 4043 y(ev)n(er,)41 b(activ)n(e)f(serialization)g(of)i(async)n
(hronous)d(concurren)n(t)h(actions)h(alw)n(a)n(ys)f(implies)523
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(con)n(trast,)e(our)g(aim)h(is)g(to)g(realize)f(the)h(maxi-)523
4242 y(m)n(um)23 b(amoun)n(t)g(of)g(parallelism)e(inheren)n(t)i(in)h
(concurren)n(t)d(systems)i(b)n(y)f(a)n(v)n(oiding)f(w)n(aiting)523
4342 y(altogether)35 b(in)h(our)f(algorithms.)g(In)h(suc)n(h)g(a)f
(setting,)h(serializabilit)n(y)f(is)h FG(not)g FK(activ)n(ely)523
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(y)e(the)j(executions)e(of)g(the)h(algorithm)f(b)n(y)g(the)523
4541 y(v)-5 b(arious)25 b(pro)r(cesses)f(in)n(teract.)i(An)n(y)g(one)f
(of)h(the)h(references,)e(sa)n(y)g([10)o(])h(or)f([20)o(],)h(describ)r
(es)523 4641 y(the)i(problem)f(area)f(in)i(some)f(detail.)606
4741 y(The)c(p)r(oin)n(t)g(of)f(departure)g(is)h(the)g(solution)f(of)h
(the)g(follo)n(wing)f(problem.)g(\(W)-7 b(e)23 b(k)n(eep)f(the)523
4840 y(discussion)e(informal.\))h(Consider)f(t)n(w)n(o)g(pro)r(cessors)
e(that)j(are)f(async)n(hronous)f(and)h(do)h(not)523 4940
y(w)n(ait)28 b(for)f(one)h(another.)f(A)i(\015ip-\015op)f(is)g(a)g(b)r
(o)r(olean)f(v)-5 b(ariable)27 b(that)i(can)e(b)r(e)i(read)e(b)n(y)h
(one)523 5039 y(pro)r(cessor)g(and)j(written)g(b)n(y)f(the)h(other.)f
(Supp)r(ose,)h(one)f(is)g(giv)n(en)g(atomic)g(\015ip-\015ops)g(as)523
5139 y(building)25 b(blo)r(c)n(ks,)f(and)g(is)g(ask)n(ed)g(to)g
(implemen)n(t)h(a)f FC(k)s FK(-bit)h(atomic)f(v)-5 b(ariable,)23
b(that)i(can)f(b)r(e)523 5239 y(written)29 b(b)n(y)f(one)g(pro)r
(cessor)e(and)i(read)g(b)n(y)g(the)h(other.)e(Of)i(course,)e(a)h
(bu\013er)h(consisting)523 5338 y(of)37 b FC(k)j FK(\015ip-\015ops)c
(su\016ces)h(to)f(hold)h(suc)n(h)g(a)f(v)-5 b(alue.)37
b(If,)h(ho)n(w)n(ev)n(er,)c(the)k(implemen)n(tation)p
eop end
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(for)g(writing,)h(then)g(either)g(the)g(writer)f(or)g(the)523
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g(other)f(is)g(sleeping)h(halfw)n(a)n(y)-7 b(.)22 b(Th)n(us,)h(the)523
805 y(reader)18 b(w)n(ould)g(obtain)h(a)g(v)-5 b(alue)19
b(consisting)f(of)h(half)g(the)h(new)f(v)-5 b(alue)19
b(and)g(half)g(the)g(old)g(one.)523 905 y(Ob)n(viously)-7
b(,)19 b(this)h(violates)f(atomicit)n(y)-7 b(.)20 b(The)g(problem)g
(then)g(is)g(to)g(design)g(a)g(proto)r(col)e(that)523
1005 y(pro)n(vides)28 b(exclusiv)n(e)h(access)g(to)g(bu\013ers)h
(without)g(w)n(aiting.)f(Correct)g(implemen)n(tations)523
1104 y(of)f(atomic)f(m)n(ulti-bit)h(v)-5 b(ariables)26
b(from)i(single)f(bits)g(can)h(b)r(e)g(found)g(in)f([15,)g(10)o(,)h(26)
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1303 y FC(n)p FK(-user)d(v)-5 b(ariable;)23 b(a)f(v)-5
b(ariable)23 b(shared)f(b)r(et)n(w)n(een)h FC(n)g FK(users)g(eac)n(h)f
(of)i(whic)n(h)f(can)g(atomically)523 1403 y(execute)28
b(b)r(oth)g(read)e(and)i(write)f(op)r(erations.)606 1503
y(A)n(t)h(the)g(outset)f(w)n(e)h(state)f(our)g(main)h(result:)523
1669 y Fv(Theorem)k FH(7.1)41 b FG(A)n(n)24 b(atomic)i
FC(n)p FG(-user)e(variable)j(is)e(implemente)l(d)h(wait-fr)l(e)l(e)g
(fr)l(om)f FC(O)r FK(\()p FC(n)3287 1639 y FB(2)3325
1669 y FK(\))523 1768 y FG(atomic)31 b FK(1)p FG(-r)l(e)l(ader)f
FK(1)p FG(-writer)g(variables)i(e)l(ach)f(with)g FC(O)r
FK(\()p FC(n)p FK(\))g FG(c)l(ontr)l(ol)f(bits;)h(using)f
FC(O)r FK(\()p FC(n)p FK(\))h FG(ac-)523 1868 y(c)l(esses)f(p)l(er)g(R)
l(e)l(ad/Write)g(running)f(in)h FC(O)r FK(\(1\))h FG(p)l(ar)l(al)t(lel)
g(time.)606 2034 y FK(Our)f(notion)g(of)h(parallel)f(time)h(allo)n(ws)e
(a)i(set)f(of)h(accesses)e(to)i(di\013eren)n(t)f(v)-5
b(ariables)30 b(to)523 2134 y(pro)r(ceed)d(in)h(arbitrary)d(order)i(in)
g(one)h(time-unit.)523 2383 y Fl(7.1.2)99 b(Comp)-5 b(arison)33
b(with)i(R)-5 b(elate)g(d)34 b(Work.)523 2532 y FK(Related)21
b(constructions)f(are)g(giv)n(en)g(b)n(y)g([17,)h(9)o(,)g(3,)g(13)o(,)g
(7)o(])h(for)e(the)h(single-reader)e(to)i(m)n(ulti-)523
2632 y(reader)h(case,)h(and)g(b)n(y)g([20)o(,)h(14)o(,)f(16)o(,)h(8])f
(for)g(the)h(m)n(ulti-reader)e(to)h(m)n(ulti-writer)g(case.)g(\(see)523
2731 y(also)h(Section)h(7.6.\))g(The)g(latter)g(problem)f(is)h(the)h
(more)e(di\016cult)i(one.)f(The)g(solutions)f(in)523
2831 y(references)k([20)o(,)h(14)o(])g(are)f(kno)n(wn)h(to)f(b)r(e)i
(incorrect)e([16)o(].)h(There)f(has)h(b)r(een)g(no)g(previous)523
2931 y(attempt)f(to)g(implemen)n(t)g(an)f FC(n)p FK(-user)g(v)-5
b(ariable)26 b(directly)i(from)f(single)g(reader)f(v)-5
b(ariables.)606 3030 y(The)34 b(algorithm)g(uses)f FC(O)r
FK(\()p FC(n)p FK(\))j(accesses)d(to)h(single-reader)e(v)-5
b(ariables)33 b(p)r(er)h(op)r(eration,)523 3130 y(and)23
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(copies)g(of)h(the)g(v)-5 b(alue)22 b(of)h(the)g(constructed)523
3230 y(v)-5 b(ariable)20 b(together)f(with)i FC(O)r FK(\()p
FC(n)p FK(\))h(bits)f(of)f(con)n(trol)f(information.)h(In)h(order)e(to)
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(single-reader)e(to)i(m)n(ulti-reader)f(algorithm.)h(This)g(do)r(esn't)
h(increase)523 3528 y(the)i(n)n(um)n(b)r(er)f(of)h(v)-5
b(ariables)22 b(used,)i(since)f(one)h(single-reader)d(register)h(from)h
(eac)n(h)g(pro)r(cess)523 3628 y(to)j(eac)n(h)e(other)i(pro)r(cess)e
(is)i(b)r(oth)g(necessary)e(and)h(su\016cien)n(t.)h(It)g(do)r(es)f(ho)n
(w)n(ev)n(er)f(increase)523 3728 y(their)34 b(size,)g(as)g(w)n(ell)g
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(b)n(y)g(a)f(factor)h(of)g FC(n)p FK(.)523 3827 y(This)28
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(elo)n(w.)606 3927 y(The)32 b(m)n(ulti-writer)f(algorithm)f(in)i([16)o
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FC(n)2970 3897 y FB(2)3006 3927 y FK(\))g(accesses)523
4027 y(to)h(m)n(ulti-reader)f(v)-5 b(ariables)31 b(p)r(er)i(op)r
(eration.)f(F)-7 b(ollo)n(wing)32 b([7],)h(recen)n(t)f(w)n(ork)g([4)o
(,)h(21)o(,)h(22)o(,)523 4126 y(23)o(,)42 b(24)o(])f(pro)n(vides)f(a)h
(general)f(metho)r(d)i(for)f(replacing)g(un)n(b)r(ounded)g(timestamps)h
(b)n(y)523 4226 y(b)r(ounded)26 b(timestamps)f(in)g(concurren)n(t)f
(systems)g(of)i(m)n(ulti-reader)d(v)-5 b(ariables.)24
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FC(n)p FK(\),)g(and)g(\002\()p FC(n)p FK(\))g(accesses)523
4425 y(p)r(er)e(op)r(eration,)e(and)i(can)f(b)r(e)h(applied)f(directly)
h(to)f(giv)n(e)g(a)g(m)n(ulti-writer)g(v)-5 b(ariable.)606
4525 y(A)32 b(more)f(recen)n(t)g(construction)g(than)h(ours,)f([8],)h
(comes)f(up)h(with)g(a)g(direct)f(solution)523 4624 y(that)i(is)f
(optimal)g(in)g(space)g(\(logarithmic)f(con)n(trol)g(bit)i(complexit)n
(y\))f(as)f(w)n(ell)h(as)g(n)n(um-)523 4724 y(b)r(er)f(of)g(v)-5
b(ariable)30 b(accesses)f(p)r(er)i(read/write)e(\(linear\).)i(They)g
(do)f(not)h(ho)n(w)n(ev)n(er)e(ac)n(hiev)n(e)523 4824
y(constan)n(t)k(parallel)g(time)i(and)e(ha)n(v)n(e)g(a)h(rather)f(more)
g(complicated)g(proto)r(col.)g(W)-7 b(e)35 b(b)r(e-)523
4923 y(liev)n(e)23 b(the)h(construction)f(presen)n(ted)g(here)g(is)h
(relativ)n(ely)e(simple)i(and)g(transparen)n(t.)e(Both)523
5023 y(problems)27 b(ab)r(o)n(v)n(e)f(are)h(solv)n(ed)f(b)n(y)i
(simpli\014cations)f(of)g(our)g(main)h(solution.)606
5122 y(The)d(basis)e(of)i(our)f(pro)r(of-tec)n(hnique)f(w)n(as)h(dev)n
(elop)r(ed)g(in)h([1)o(].)g(Our)f(mo)r(del)h(and)f(termi-)523
5222 y(nology)e(is)h(based)g(on)g([6)o(],)h(whic)n(h)f(de\014nes)g(and)
g(motiv)-5 b(ates)23 b(the)h(notion)f(of)g(linearizabilit)n(y)-7
b(.)523 5322 y(Blo)r(om)35 b([2])h(presen)n(ted)f(an)g(elegan)n(t)g
(2-writer)g(construction.)g(In)h([5)o(],)g(Herlih)n(y)g(consid-)p
eop end
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(tations)g(from)523 606 y(v)-5 b(ariables.)523 857 y
Fl(7.1.3)99 b(Multi-user)36 b(V)-7 b(ariable)34 b(Construction)523
1006 y FK(In)25 b(this)f(section)g(w)n(e)g(consider)f(the)i(problem)f
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(construction)523 1205 y(has)f(to)h(satisfy)-7 b(.)606
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(pro)r(cedures,)523 1903 y(in)k(the)h(case)e(of)h(Read)f(ha)n(ving)g
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b(alue)33 b(read)523 2002 y(from)27 b Fv(abs)p FK(.)606
2102 y(A)37 b(pro)r(cedure)e(con)n(tains)g(a)h(declaration)e(of)i(lo)r
(cal)g(v)-5 b(ariables)35 b(and)h(a)f(b)r(o)r(dy)-7 b(.)37
b(A)f(lo)r(cal)523 2202 y(v)-5 b(ariable)30 b(app)r(earing)g(in)i(b)r
(oth)g(pro)r(cedures)e(can)h(b)r(e)g(declared)g FG(static)p
FK(,)g(whic)n(h)h(means)e(it)523 2301 y(retains)g(its)h(v)-5
b(alue)30 b(b)r(et)n(w)n(een)h(pro)r(cedure)f(in)n(v)n(o)r(cations.)f
(The)i(b)r(o)r(dy)g(is)f(a)g(program)f(frag-)523 2401
y(men)n(t)i(comprised)g(of)g(atomic)f(statemen)n(ts.)h(Access)g(to)g
(shared)f(v)-5 b(ariables)30 b(is)h(naturally)523 2500
y(restricted)26 b(to)g(assignmen)n(ts)g(from)g FC(R)1710
2512 y FA(j;i)1811 2500 y FK(to)h(lo)r(cal)f(v)-5 b(ariables)25
b(and)i(assignmen)n(ts)e(from)h(lo-)523 2600 y(cal)f(v)-5
b(ariables)24 b(to)h FC(R)1154 2612 y FA(i;j)1233 2600
y FK(,)g(for)g(an)n(y)g FC(j)30 b FK(\(recall)25 b(that)g
FC(i)g FK(is)h(the)f(index)h(of)f(the)h(executing)f(user\).)523
2700 y(No)35 b(other)f(means)h(of)g(in)n(ter-pro)r(cess)e(comm)n
(unication)h(is)h(allo)n(w)n(ed.)e(In)i(particular,)f(no)523
2799 y(sync)n(hronization)26 b(primitiv)n(es)i(can)g(b)r(e)g(used.)g
(Assignmen)n(ts)g(to)g(and)g(from)g(shared)f(v)-5 b(ari-)523
2899 y(ables)28 b(are)g(called)h(writes)g(and)f(reads)g(resp)r(ectiv)n
(ely)-7 b(,)29 b(alw)n(a)n(ys)e(in)i(lo)n(w)n(er)e(case.)h(The)i(space)
523 2999 y(complexit)n(y)24 b(of)h(a)f(construction)g(is)h(the)g(maxim)
n(um)f(size,)h(in)g(bits,)g(of)g(a)f(shared)g(v)-5 b(ariable.)606
3098 y(A)24 b(lo)r(op)f(of)h(the)g(t)n(yp)r(e)g FH(for)32
b FC(j)c FD(2)c FC(I)31 b FK(signi\014es)23 b(a)g(parallel)g(lo)r(op)g
(whose)g(iterations)3064 3068 y FB(1)3124 3098 y FK(can)g(b)r(e)523
3198 y(executed)k(in)f(arbitrary)f(order.)g(Shared)h(v)-5
b(ariables)25 b(accessed)h(in)h(di\013eren)n(t)f(iterations)g(of)523
3297 y(a)h(parallel)f(lo)r(op)h(m)n(ust)g(b)r(e)h(disjoin)n(t)f(since)g
(a)g(user)g(can)g(only)g(execute)g(one)g(op)r(eration)f(at)523
3397 y(a)e(time)g(on)g(a)f(giv)n(en)g(shared)g(v)-5 b(ariable.)23
b(As)h(the)g(time)g(complexit)n(y)g(of)g(the)g(Read)g(or)f(W)-7
b(rite)523 3497 y(pro)r(cedure)34 b(w)n(e)g(tak)n(e)g(the)h(maxim)n(um)
g(n)n(um)n(b)r(er)f(of)h(shared)e(v)-5 b(ariable)34 b(accesses)f
(outside)523 3596 y(parallel)26 b(lo)r(ops)g(plus)h(for)f(eac)n(h)g
(suc)n(h)g(lo)r(op)h(the)g(maxim)n(um)g(n)n(um)n(b)r(er)f(of)h(shared)f
(v)-5 b(ariable)523 3696 y(accesses)26 b(in)i(a)f(single)g(iteration)g
(of)h(that)g(lo)r(op.)606 3796 y(A)g(construction)f(m)n(ust)g(satisfy)h
(the)g(follo)n(wing)e(constrain)n(t.)565 3971 y FH(W)-8
b(ait-F)g(reeness)26 b FK(Eac)n(h)g(pro)r(cedure)h(m)n(ust)h(b)r(e)g
(free)f(from)g(un)n(b)r(ounded)h(lo)r(ops.)606 4146 y(Giv)n(en)c(a)h
(construction,)e(w)n(e)h(are)g(in)n(terested)g(in)h(prop)r(erties)f(of)
g(its)h(executions,)f(whic)n(h)523 4246 y(the)h(follo)n(wing)e(notions)
h(help)g(form)n(ulate.)g(A)g FG(state)g FK(is)h(a)e(con\014guration)g
(of)h(the)h(construc-)523 4345 y(tion,)36 b(comprising)e(v)-5
b(alues)36 b(of)g(all)f(shared)g(and)g(lo)r(cal)h(v)-5
b(ariables,)34 b(as)h(w)n(ell)h(as)f(program)523 4445
y(coun)n(ters.)i(Note)h(that)h(w)n(e)e(need)i(a)e(somewhat)h(lib)r
(eral)g(notion)f(of)i(program)d(coun)n(ter)523 4544 y(to)30
b(c)n(haracterize)d(the)k(execution)e(of)h(a)f(parallel)g(lo)r(op.)h
(In)g(b)r(et)n(w)n(een)f(in)n(v)n(o)r(cations)g(of)g(the)523
4644 y(Read)i(and)f(W)-7 b(rite)32 b(pro)r(cedure,)e(a)g(user)g(is)h
(said)g(to)f(b)r(e)i(idle,)f(and)g(its)g(program)e(coun)n(ter)523
4744 y(has)g(the)h(v)-5 b(alue)30 b(`idle',)g(One)f(state)g(is)h
(designated)f(as)g FG(initial)j(state)p FK(.)e(All)g(users)f(m)n(ust)g
(b)r(e)523 4843 y(idle)f(in)g(this)g(state.)606 4943
y(A)k(state)g FC(t)g FK(is)g(an)f FG(imme)l(diate)k(suc)l(c)l(essor)d
FK(of)g(a)g(state)f FC(s)h FK(if)h FC(t)f FK(can)f(b)r(e)i(reac)n(hed)d
(from)i FC(s)523 5043 y FK(through)i(the)h(execution)g(of)f(a)h(pro)r
(cedure)e(statemen)n(t)i(b)n(y)g(some)f(user)g(in)h(accordance)p
523 5162 499 5 v 627 5244 a Fm(1)661 5276 y FF(This)e(is)f(a)g(sligh)n
(t)g(abuse)g(of)h(the)e(term,)h(since)g(the)g(w)n(ord)g(iteration)g
(suggests)h(sequen)n(tial)523 5359 y(b)r(eha)n(viour.)p
eop end
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TeXDict begin 91 100 bop 2536 299 a FF(7.1.)53 b(In)n(tro)r(duction)153
b(91)523 506 y FK(with)37 b(its)f(program)e(coun)n(ter.)h(Recall)h
(that)g FC(n)g FK(denotes)g(the)g(n)n(um)n(b)r(er)g(of)g(users)f(of)h
(the)523 606 y(constructed)24 b(v)-5 b(ariable)23 b Fv(abs)p
FK(.)h(A)g(state)g(has)g(at)f(least)h FC(n)g FK(immediate)g
(successors:)e(If)j(a)e(user)523 706 y(is)h(idle,)g(it)g(can)g(in)n(v)n
(ok)n(e)e(either)h(the)i(Read)e(or)g(W)-7 b(rite)24 b(pro)r(cedure.)f
(And)h(if)h(it)f(is)g(within)g(one)523 805 y(of)h(these)g(pro)r
(cedures,)f(there)h(is)g(at)g(least)g(one)f(atomic)h(statemen)n(t)g(to)
g(b)r(e)g(executed)h(next)523 905 y(\(p)r(ossibly)h(more)g(during)g
(the)h(execution)g(of)f(a)g(parallel)g(lo)r(op\).)606
1005 y(A)19 b FG(history)g FK(of)g(the)f(construction)g(is)g(a)g
(\014nite)h(or)f(in\014nite)h(sequence)f(of)g(states)g
FC(t)3017 1017 y FB(0)3054 1005 y FC(;)c(t)3121 1017
y FB(1)3158 1005 y FC(;)g(t)3225 1017 y FB(2)3262 1005
y FC(;)g(:)g(:)g(:)523 1104 y FK(suc)n(h)28 b(that)g
FC(t)921 1116 y FB(0)986 1104 y FK(is)g(the)h(initial)f(state)g(and)g
FC(t)1850 1116 y FA(i)p FB(+1)1990 1104 y FK(is)g(an)f(immediate)i
(successor)d(of)i FC(t)3078 1116 y FA(i)3106 1104 y FK(.)g(T)-7
b(ran-)523 1204 y(sitions)30 b(b)r(et)n(w)n(een)g(successiv)n(e)g
(states)g(are)f(called)h(the)h FG(events)f FK(of)h(a)f(history)-7
b(.)29 b(With)j(eac)n(h)523 1303 y(ev)n(en)n(t)41 b(is)f(asso)r(ciated)
g(the)i(index)f(of)g(the)g(executing)g(user,)f(the)h(relev)-5
b(an)n(t)41 b(pro)r(cedure)523 1403 y(statemen)n(t,)28
b(and)f(the)h(v)-5 b(alues)27 b(manipulated)h(b)n(y)f(the)h(execution)g
(of)f(the)h(statemen)n(t.)606 1503 y(An)e(ev)n(en)n(t)f
FC(a)h FG(pr)l(e)l(c)l(e)l(des)g FK(an)f(ev)n(en)n(t)g
FC(b)h FK(in)g(history)e FC(h)p FK(,)i FC(a)d FD(\036)2335
1515 y FA(h)2401 1503 y FC(b)p FK(,)i(if)h FC(a)g FK(o)r(ccurs)e(b)r
(efore)i FC(b)f FK(in)h FC(h)p FK(.)523 1602 y(The)i(subscript)g
FC(h)g FK(is)f(omitted)i(when)f(clear)f(from)g(con)n(text.)h(Call)g(a)f
(\014nite)i(set)f(of)g(ev)n(en)n(ts)523 1702 y(of)k(a)g(history)g(an)g
(ev)n(en)n(t-set.)f(Then)i(w)n(e)f(similarly)f(sa)n(y)g(that)i(an)f(ev)
n(en)n(t-set)f FC(a)i FK(precedes)523 1802 y(an)25 b(ev)n(en)n(t-set)f
FC(b)g FK(in)h(a)g(history)-7 b(,)24 b FC(a)f FD(\036)1628
1814 y FA(h)1694 1802 y FC(b)p FK(,)h(when)h(eac)n(h)g(ev)n(en)n(t)f
(in)h FC(a)g FK(o)r(ccurs)f(b)r(efore)h(all)f(those)523
1901 y(in)34 b FC(b)p FK(.)f(The)h(relation)e FD(\036)1270
1913 y FA(h)1347 1901 y FK(on)h(ev)n(en)n(t-sets)f(constitutes)i(what)f
(is)h(kno)n(wn)f(as)g(an)g FG(interval)523 2001 y(or)l(der)p
FK(.)27 b(That)g(is,)f(a)g(partial)g(order)f(satisfying)h(the)g(in)n
(terv)-5 b(al)26 b(axiom)g FC(a)d FD(\036)f FC(b)16 b
FD(^)h FC(c)23 b FD(\036)f FC(d)17 b FD(^)f FC(c)23 b
FD(6\036)523 2100 y FC(b)i FD(\))g FC(a)h FD(\036)e FC(d)p
FK(.)30 b(This)f(implication)g(can)f(b)r(e)i(seen)e(to)h(hold)g(b)n(y)g
(considering)e(the)j(last)e(ev)n(en)n(t)523 2200 y(of)h
FC(c)g FK(and)h(the)f(earliest)g(ev)n(en)n(t)f(of)h FC(b)p
FK(.)g(See)h([10)o(])f(for)g(an)g(extensiv)n(e)f(discussion)h(on)g(mo)r
(dels)523 2300 y(of)f(time.)606 2399 y(Of)21 b(particular)e(in)n
(terest)h(are)g(the)h(sets)g(consisting)f(of)g(all)h(ev)n(en)n(ts)f(of)
g(a)h(single)f(pro)r(cedure)523 2499 y(in)n(v)n(o)r(cation,)i(whic)n(h)
h(w)n(e)f(call)h(an)g FG(op)l(er)l(ation)p FK(.)h(An)g(op)r(eration)e
(is)h(either)f(a)h(Read)g(op)r(eration)523 2599 y(or)37
b(a)g(W)-7 b(rite)38 b(op)r(eration.)e(It)i(is)f FG(c)l(omplete)i
FK(if)f(it)g(includes)f(the)h(execution)f(of)h(the)g(\014nal)523
2698 y FH(return)20 b FK(statemen)n(t)g(of)g(the)g(pro)r(cedure.)e
(Otherwise)h(it)h(is)f(said)h(to)f(b)r(e)h FG(p)l(ending)p
FK(.)h(A)f(history)523 2798 y(is)30 b(complete)f(if)i(all)e(its)h(op)r
(erations)f(are)f(complete.)i(Note)g(that)g(in)g(the)g(\014nal)g(state)
f(of)h(a)523 2897 y(complete)24 b(\014nite)h(history)-7
b(,)24 b(all)g(users)g(are)f(idle.)i(The)g FG(value)g
FK(of)f(an)g(op)r(eration)g(is)g(the)h(v)-5 b(alue)523
2997 y(written)31 b(to)f Fv(abs)h FK(in)g(the)g(case)f(of)g(a)h(W)-7
b(rite,)31 b(or)e(the)i(v)-5 b(alue)31 b(read)f(from)g
Fv(abs)h FK(in)g(the)g(case)523 3097 y(of)d(a)f(Read.)606
3196 y(The)32 b(follo)n(wing)g(crucial)f(de\014nition)h(expresses)f
(the)i(idea)f(that)g(the)h(op)r(erations)e(in)h(a)523
3296 y(history)23 b(app)r(ear)h(to)g(tak)n(e)f(place)h(instan)n
(taneously)f(somewhere)g(during)h(their)g(execution)523
3396 y(in)n(terv)-5 b(al.)31 b(A)h(more)f(general)f(v)n(ersion)g(of)i
(this)g(is)f(presen)n(ted)h(and)f(motiv)-5 b(ated)32
b(in)g([6)o(].)g(T)-7 b(o)523 3495 y(a)n(v)n(oid)35 b(sp)r(ecial)h
(cases,)g(w)n(e)g(in)n(tro)r(duce)g(the)h(notion)f(of)h(a)f
FG(pr)l(op)l(er)i FK(history)d(as)h(one)g(that)523 3595
y(starts)27 b(with)h(an)f(initializing)h(W)-7 b(rite)28
b(op)r(eration)e(that)i(precedes)f(all)g(other)g(op)r(erations.)565
3778 y FH(Linearizabilit)m(y)20 b FK(A)g(complete)f(prop)r(er)g
(history)g FC(h)g FK(is)h FG(line)l(arizable)i FK(if)e(the)g(partial)f
(order)731 3877 y FD(\036)796 3889 y FA(h)863 3877 y
FK(on)24 b(the)i(set)e(of)h(op)r(erations)e(can)i(b)r(e)g(extended)g
(to)f(a)h(total)f(order)g(whic)n(h)g(ob)r(eys)731 3977
y(the)36 b(seman)n(tics)e(of)i(a)f(v)-5 b(ariable.)35
b(That)g(is,)h(eac)n(h)f(Read)g(op)r(erations)f(returns)h(the)731
4076 y(v)-5 b(alue)27 b(written)h(b)n(y)g(that)g(W)-7
b(rite)28 b(op)r(eration)f(whic)n(h)h(last)f(precedes)g(it)i(in)f(the)g
(total)731 4176 y(order.)606 4359 y(A)39 b(construction)f(is)g
FG(c)l(orr)l(e)l(ct)g FK(if)h(it)g(satis\014es)f(W)-7
b(ait-F)g(reeness)38 b(and)g(all)g(its)h(complete)523
4458 y(prop)r(er)27 b(histories)f(are)h(linearizable.)523
4707 y Fl(7.1.4)99 b(The)34 b(T)-7 b(ag)34 b(F)-7 b(unction)523
4857 y FK(While)21 b(the)g(de\014nition)g(of)g(linearizabilit)n(y)f(is)
g(quite)h(clear,)f(it)h(is)g(con)n(v)n(enien)n(t)e(to)i(transform)523
4956 y(it)i(in)n(to)e(an)h(equiv)-5 b(alen)n(t)22 b(sp)r(eci\014cation)
g(from)g(whic)n(h)g(the)g(\014rst)g(algorithm)f(can)h(b)r(e)g(directly)
523 5056 y(deriv)n(ed.)27 b(The)h(idea)g(b)r(ehind)g(the)h(follo)n
(wing)e(lemma)g(w)n(as)g(\014rst)h(expressed)f(b)n(y)g(Lamp)r(ort)523
5156 y(in)k(Prop)r(osition)e(3)h(of)h([10)o(],)g(for)f(the)h(case)e(of)
i(a)f(single)g(writer.)g(In)h([17)o(],)g(the)g(equiv)-5
b(alen)n(t)523 5255 y(conditions)32 b(giv)n(en)g(b)n(y)h(Lamp)r(ort's)f
(prop)r(osition)f(are)h(in)h(fact)g(tak)n(en)f(as)g(the)h(de\014nition)
523 5355 y(of)24 b(linearizabilit)n(y)f(\(often)i(called)e(atomicit)n
(y)h(in)g(the)g(register)f(construction)g(literature\).)p
eop end
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TeXDict begin 92 101 bop 1747 299 a FF(7.2.)53 b(The)26
b(Basic)h(Un)n(b)r(ounded)d(Construction)154 b(92)523
506 y FK(The)27 b(A)n(tomicit)n(y)g(Criterion)f(of)h([1])g(is)g(the)g
(\014rst)g(generalization)e(of)i(Lamp)r(ort's)f(prop)r(osi-)523
606 y(tion)e(to)g(the)g(case)g(of)g(m)n(ultiple)g(readers.)f(A)h
(further)g(generalization)e(app)r(ears)h(in)h([18)o(])g(for)523
706 y(the)h(case)f(of)h(a)g(v)-5 b(ariable)24 b(ha)n(ving)g(sev)n(eral)
f(\014elds)i(whic)n(h)g(can)f(b)r(e)i(written)f(indep)r(enden)n(tly)-7
b(.)523 863 y Fv(Lemma)32 b FH(7.2)40 b FG(A)22 b(c)l(omplete)i(pr)l
(op)l(er)f(history)g FC(h)g FG(is)f(line)l(arizable)j(i\013)d(ther)l(e)
h(exist)f(a)h(function)523 962 y(mapping)31 b(e)l(ach)f(op)l(er)l
(ation)g(in)f FC(h)g FG(to)g(a)h(r)l(ational)g(numb)l(er,)f(c)l(al)t
(le)l(d)h(its)f(tag,)h(such)f(that)g(the)523 1062 y(fol)t(lowing)j(3)e
(c)l(onditions)h(ar)l(e)f(satis\014e)l(d:)523 1209 y
FH(Uniqueness)41 b FG(di\013er)l(ent)30 b(Write)g(op)l(er)l(ations)g
(have)h(di\013er)l(ent)f(tags.)523 1366 y FH(In)m(tegrit)m(y)42
b FG(for)25 b(e)l(ach)f(R)l(e)l(ad)f(op)l(er)l(ation)i(ther)l(e)e
(exists)g(a)h(Write)f(op)l(er)l(ation)i(with)f(the)f(same)731
1466 y(tag)30 b(and)g(value,)g(that)g(it)g(do)l(esn)-8
b('t)30 b(pr)l(e)l(c)l(e)l(de.)523 1623 y FH(Precedence)42
b FG(if)30 b(one)g(op)l(er)l(ation)g(pr)l(e)l(c)l(e)l(des)g(another,)h
(then)e(the)g(tag)h(of)g(the)g(latter)f(is)h(at)731 1722
y(le)l(ast)f(that)h(of)h(the)f(former.)523 1927 y Fv(Pr)n(oof.)23
b FD(\))h FK(Let)h(a)e(complete)h(prop)r(er)f(history)g
FC(h)h FK(b)r(e)h(linearizable.)d(Then)j(there)e(is)h(some)523
2027 y(total)d(order)f FC(<)h FK(extending)g FD(\036)1451
2039 y FA(h)1494 2027 y FK(.)g(Assign)g(to)g(eac)n(h)f(op)r(eration)h
(a)f(tag)h(whic)n(h)g(is)g(the)h(n)n(um)n(b)r(er)523
2126 y(of)h(W)-7 b(rite)22 b(op)r(erations)g(preceding)f(it)i(in)g
FC(<)p FK(.)f(This)h(clearly)e(satis\014es)h(Uniqueness.)h(F)-7
b(or)22 b(an)n(y)523 2226 y(Read)32 b(op)r(eration)f
FC(R)q FK(,)i(the)f(W)-7 b(rite)33 b(op)r(eration)e FC(W)45
b FK(that)32 b(precedes)g(it)h(last)f(in)h FC(<)f FK(has)f(the)523
2326 y(same)22 b(tag.)g(Also,)g(b)r(ecause)g FC(<)h FK(ob)r(eys)f(the)h
(seman)n(tics)e(of)i(a)f(v)-5 b(ariable,)22 b FC(W)34
b FK(and)23 b FC(R)g FK(ha)n(v)n(e)e(the)523 2425 y(same)30
b(v)-5 b(alue.)31 b(F)-7 b(rom)31 b(the)g(facts)g(that)g
FC(<)g FK(extends)g FD(\036)2193 2437 y FA(h)2236 2425
y FK(,)g FC(W)40 b(<)29 b(R)q FK(,)i(and)f FC(<)h FK(is)g(acyclic,)f(w)
n(e)523 2525 y(conclude)22 b(that)h FD(:)p FC(R)h FD(\036)1241
2537 y FA(h)1307 2525 y FC(W)12 b FK(.)23 b(So)f(In)n(tegrit)n(y)f(is)i
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FK([)p FC(s)p FK(][)p FC(j)5 b FK(])23 b FD(\025)g FK(3)710
3308 y FH(then)k(return)i FG(tmp)5 b FK([)p FC(j)g FK(])p
FC(:)p FG(pr)l(ev)710 3408 y FK(select)27 b FG(max)38
b FK(suc)n(h)27 b(that)h FD(8)p FC(j)g FK(:)23 b FG(fr)l(om)7
b FK([)p FG(max)j FK(])p FC(:)p FG(tag)31 b FD(\025)22
b FG(fr)l(om)7 b FK([)p FC(j)e FK(])p FC(:)p FG(tag)710
3508 y(me)h FC(:)p FG(pr)l(ev)k FC(;)k FG(me)6 b FC(:)p
FG(value)h FC(;)14 b FG(me)6 b FC(:)p FG(tag)i FC(;)14
b FG(me)6 b FC(:)p FG(ss)30 b FK(:=)936 3607 y FG(me)7
b FC(:)p FG(value)f FC(;)14 b FG(fr)l(om)7 b FK([)p FG(max)j
FK(])p FC(:)p FG(value)d FC(;)14 b FG(fr)l(om)7 b FK([)p
FG(max)j FK(])p FC(:)p FG(tag)d FC(;)14 b(s)551 3707
y FK(p:)90 b FH(for)28 b FC(j)g FD(2)23 b FC(I)35 b FH(do)27
b FC(R)1256 3719 y FA(i;j)1357 3707 y FK(:=)c FG(me)710
3806 y FK(\(Read)k(only\))h FH(return)g FG(me)6 b FC(:)p
FG(value)710 3906 y FH(for)28 b FC(j)g FD(2)23 b FC(I)7
b(;)14 b(s)23 b FD(2)g(f)p FK(0)p FC(::)p FK(1)p FD(g)j
FH(do)854 4006 y(if)36 b FG(me)6 b FC(:)p FG(sho)l(ot)j
FK([)p FC(s)p FK(][)p FC(j)c FK(])18 b FD(\000)g FG(fr)l(om)7
b FK([)p FC(j)e FK(])p FC(:)p FG(he)l(al)10 b FK([)p
FC(s)p FK(][)p FC(i)p FK(])23 b FC(<)g FK(6)854 4105
y FH(then)k FG(me)7 b FC(:)p FG(sho)l(ot)h FK([)p FC(s)p
FK(][)p FC(j)d FK(]+)23 b(:=)g(1)710 4205 y(select)k
FC(t)h FK(suc)n(h)f(that)h FC(t)18 b FD(\000)g FG(me)7
b FC(:)p FG(tag)30 b FD(2)24 b(f)p FK(1)p FC(;)14 b(:)g(:)g(:)e(;)i(n)p
FD(g)k(^)h FC(t)k FD(\021)g FC(i)82 b FK(\(mo)r(d)28
b FC(n)p FK(\))710 4305 y FG(me)6 b FC(:)p FG(value)h
FC(;)14 b FG(me)6 b FC(:)p FG(tag)31 b FK(:=)23 b FC(v)s(;)14
b(t)551 4404 y FK(w:)76 b FH(for)28 b FC(j)g FD(2)23
b FC(I)35 b FH(do)27 b FC(R)1256 4416 y FA(i;j)1357 4404
y FK(:=)c FG(me)551 4504 y FH(end)p 2902 4559 V 523 4562
2383 4 v 1453 4704 a FF(FIGURE)i(7.2.)i(Construction)f(1)p
eop end
%%Page: 96 106
TeXDict begin 96 105 bop 2393 299 a FF(7.3.)53 b(Solution)26
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(Construction)g(0.)h(The)f FG(pr)l(ev)39 b FK(\014eld)523
606 y(is)32 b(used)f(to)h(remem)n(b)r(er)f(v)-5 b(alues)31
b(of)h(former)f(op)r(erations,)f(whic)n(h)i(are)f(used)g(b)n(y)h(ab)r
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g(coun)n(ters,)e FG(he)l(al)10 b FK([0][0)p FC(::n)5
b FD(\000)g FK(1])19 b(and)h FG(he)l(al)10 b FK([1][0)p
FC(::n)5 b FD(\000)g FK(1],)523 805 y(are)28 b(used)g(to)h(hold)g
(targets.)e(The)i FG(ss)35 b FK(\(sho)r(ot-selector\))28
b(\014eld)h(selects)f(whic)n(h)h(of)f(the)i(t)n(w)n(o)523
905 y(sets)h(holds)f(the)i(target)e(asso)r(ciated)f(with)j(the)f
(curren)n(t)f(v)-5 b(alue-tag)30 b(pair.)g(A)h(second)g(set)523
1005 y(is)25 b(needed)g(since)f(new)h(op)r(erations)e(m)n(ust)i(set)g
(up)g(a)g(target)e(b)r(efore)i(they)g(can)f(compute)h(a)523
1104 y(new)20 b(tag.)g(T)-7 b(ogether)19 b(with)h(the)h(heal)e(coun)n
(ters,)g(the)i(shot)f(coun)n(ters,)f FG(sho)l(ot)8 b
FK([0)p FC(::)p FK(1][0)p FC(::n)s FD(\000)s FK(1],)523
1204 y(implemen)n(t)29 b(the)f(sho)r(oting)g(mec)n(hanism.)g(User)g
FC(j)33 b FK(sho)r(ots)27 b(at)h(a)g(target)g FG(he)l(al)9
b FK([)p FC(s)p FK(][0)p FC(::n)19 b FD(\000)f FK(1])523
1303 y(of)26 b(user)f FC(k)j FK(b)n(y)e(making)e(his)i(coun)n(ter)f
FG(sho)l(ot)8 b FK([)p FC(s)p FK(][)p FC(k)s FK(])26
b(larger)e(than)i(the)g(coun)n(ter)e FG(he)l(al)10 b
FK([)p FC(s)p FK(][)p FC(j)5 b FK(])26 b(of)523 1403
y(user)h FC(k)s FK(,)h(up)f(to)h(a)f(maxim)n(um)h(of)f(6.)606
1503 y(No)n(w)33 b(let's)i(consider)d(the)j(pro)r(cedures.)d(Note)i
(that)h(the)f(lines)g(are)f(no)g(longer)g(n)n(um-)523
1602 y(b)r(ered.)c(Instead,)g(the)h(lines)f(in)n(v)n(olving)f(shared)g
(v)-5 b(ariable)28 b(access)g(are)h(iden)n(ti\014ed)g(b)n(y)g(one)523
1702 y(of)35 b(the)g(c)n(haracters)d(s,h,r,t,p)j(and)f(w,)h(whic)n(h)f
(are)g(mnemonic)h(shorthands)e(for)h(setup,)523 1802
y(heal,)27 b(read,)g(test,)h(propagate)e(and)h(write,)h(resp)r(ectiv)n
(ely)-7 b(.)606 1901 y(A)n(t)36 b(the)h(start)e(of)h(an)g(op)r
(eration,)f(call)h(it)h FC(a)p FK(,)f(user)f FC(i)h FK(sets)g(up)g(a)g
(new)g(target)f(in)h(the)523 2001 y(a)n(v)-5 b(ailable)36
b(heal)h(coun)n(ter)g(set)g(\(1)25 b FD(\000)f FG(me)6
b FC(:)p FG(ss)h FK(\))38 b(b)n(y)f(catc)n(hing)f(up)i(with)g(eac)n(h)f
(user's)f(shot)523 2100 y(coun)n(ter.)24 b(It)h(then)g(writes)f(out)h
(the)g(target)f(in)h(line)f(h)h(so)f(that)h(the)g(other)f(users)g(can)g
(start)523 2200 y(sho)r(oting)29 b(it.)g(After)h(collecting)f(ev)n(ery)
e(one's)i(data)g(in)g(line)h(r,)f(it)g(pro)r(ceeds)g(to)g(test)g(ho)n
(w)523 2300 y(man)n(y)e(times)g(it's)h(target)e(has)h(b)r(een)g(shot.)g
(More)g(precisely)-7 b(,)26 b(if)i(an)n(y)f(user)f(has)h(increased)523
2399 y(its)k(shot)g(coun)n(ter)f(at)h(least)g(3)g(times)g(since)g(it)g
(w)n(as)g(previously)e(read)i(in)g(line)g(s,)g(then)h
FC(a)523 2499 y FK(will)d(ab)r(ort.)f(The)h FC(j)34 b
FK(in)29 b(the)g FH(return)g FK(statemen)n(t)g(is)f(mean)n(t)h(to)g(b)r
(e)g(an)n(y)f FC(j)33 b FK(satisfying)28 b(the)523 2599
y(condition)33 b(of)h(the)f(test,)h(but)g(could)f(b)r(e)h(c)n(hosen)f
(as)f(the)i(minimal)g(index)f(for)g(the)h(sak)n(e)523
2698 y(of)28 b(de\014niteness.)h(It)f(can)g(b)r(e)h(sho)n(wn)e(that)i
FC(a)f FK(completely)g(`con)n(tains')g(an)g(op)r(eration)f
FC(b)h FK(of)523 2798 y(user)j FC(j)37 b FK(with)32 b(the)h(v)-5
b(alue)31 b FG(tmp)5 b FK([)p FC(j)g FK(])p FC(:)p FG(pr)l(ev)10
b FK(.)32 b(Th)n(us,)g FC(a)f FK(can)h(b)r(e)g(imagined)g(to)f(ha)n(v)n
(e)g(o)r(ccurred)523 2897 y(righ)n(t)d(b)r(efore)g(or)g(after)g
FC(b)g FK(in)h(a)f(linearization,)f(dep)r(ending)i(on)g(whether)f(it's)
h(a)f(W)-7 b(rite)29 b(or)523 2997 y(Read)f(op)r(eration.)f(If)i(no)f
(user)g(shot)g(the)g(target)g(3)g(times,)g(then)h(user)f
FC(i)g FK(sets)g FG(max)38 b FK(to)28 b(an)523 3097 y(index)f(of)h(the)
f(largest)f(visible)h(tag.)g(It)h(then)g(sa)n(v)n(es)d(the)j(old)f(v)-5
b(alue)27 b(in)h FG(pr)l(ev)9 b FK(,)28 b(c)n(hanges)e(its)523
3196 y(v)-5 b(alue)27 b(and)g(tag)g(to)g(that)g(of)g
FG(max)10 b FK(,)27 b(and)g(asso)r(ciates)f(its)h(target)f(with)i(the)g
(new)f(v)-5 b(alue-tag)523 3296 y(pair.)28 b(In)h(line)g(p,)g(record)e
FG(me)36 b FK(is)28 b(written)h(out.)g(The)g(purp)r(ose)f(of)h(the)g(W)
-7 b(rite)29 b(op)r(erations)523 3396 y(propagating)17
b(the)j(v)-5 b(alue-tag)19 b(pair)f(of)i FG(max)29 b
FK(is)20 b(to)f(ensure)g(that)h(the)g(maxim)n(um)f(tag)g(visible)523
3495 y(to)k(one)g(user)g(is)g(at)h(most)f FC(n)g FK(larger)f(than)h
(the)h(maxim)n(um)f(tag)g(visible)h(to)f(an)n(y)f(other)h(user.)523
3595 y(The)j(Read)f(pro)r(cedure)g(ends)g(after)h(line)g(p)f(b)n(y)h
(returning)f(the)h(v)-5 b(alue)25 b(copied)g(from)h FG(max)10
b FK(.)606 3694 y(The)20 b(W)-7 b(rite)20 b(pro)r(cedure)f(con)n(tin)n
(ues)g(b)n(y)g(sho)r(oting)g(all)g(visible)h(targets,)f(that)h(is,)f
(increas-)523 3794 y(ing)34 b(all)g(its)h(shot)f(coun)n(ters)f(that)h
(are)g(not)g(already)f(6)h(ahead)f(of)h(their)h(corresp)r(onding)523
3894 y(heal)f(coun)n(ter.)g(User)g FC(i)g FK(next)h(c)n(ho)r(oses)d(a)j
(tag)e(unique)i(to)f(it)h(whic)n(h)g(is)f(larger)e(than)j(all)523
3993 y(visible)30 b(ones.)g(This)h(is)f(paired)g(with)h(the)g(argumen)n
(t)e FC(v)34 b FK(the)d(W)-7 b(rite)30 b(pro)r(cedure,)g(and)g(all)523
4093 y(is)e(written)f(out)h(in)g(line)g(w.)523 4342 y
Fl(7.3.2)99 b(Notational)35 b(Conventions)523 4491 y
FK(The)29 b(follo)n(wing)e(notions)h(are)g(used)h(in)g(the)g(pro)r(of.)
f(Where)g(necessary)-7 b(,)27 b(assume)h(a)h(\014xed,)523
4591 y(but)20 b(arbitrary)e(history)-7 b(.)19 b(The)h
FC(m)p FK('th)h(non-ab)r(orting)d(op)r(eration)h(of)g(user)h
FC(i)f FK(is)h(denoted)g FC(N)3272 4561 y FA(m)3263 4613
y(i)3335 4591 y FK(.)523 4691 y(If)30 b FC(a)d FK(=)f
FC(N)846 4661 y FA(m)837 4712 y(i)939 4691 y FK(then)k
FC(a)1174 4661 y FB(+)p FA(r)1292 4691 y FK(denotes)f
FC(N)1670 4655 y FA(m)p FB(+)p FA(r)1661 4714 y(i)1817
4691 y FK(,)h(i.e.)g(the)g FC(r)r FK(-th)h(next)f(non-ab)r(orting)e(op)
r(eration)523 4790 y(b)n(y)i(user)g FC(i)g FK(follo)n(wing)f
FC(a)p FK(,)i(assuming)e(it)i(exists.)f(If)h FC(a)d FK(=)f
FC(N)2360 4760 y FA(m)2351 4812 y(i)2453 4790 y FK(then)k
FC(a)2689 4760 y Fw(\000)p FA(r)2808 4790 y FK(denotes)f
FC(N)3187 4755 y FA(m)p Fw(\000)p FA(r)3178 4814 y(i)3335
4790 y FK(,)523 4890 y(i.e.)c(the)h FC(r)r FK(-th)g(previous)e(non-ab)r
(orting)g(op)r(eration)g(b)n(y)h(user)g FC(i)f FK(preceding)h
FC(a)p FK(.)g(Use)g(of)g(this)523 4990 y(notation)32
b(dep)r(ends)i(on)e(the)i(assumption)e(that)h FC(r)i(<)d(m)p
FK(.)h(Since)g(all)f(shot)h(coun)n(ters)f(are)523 5089
y(initialized)25 b(to)g(0,)g(and)g(increase)f(at)h(most)g(b)n(y)g(one)g
(p)r(er)g(non-ab)r(orting)e(W)-7 b(rite)26 b(op)r(eration,)523
5189 y(the)c(v)-5 b(alue)22 b(assigned)f(b)n(y)g(an)h(op)r(eration)e
FC(a)i FK(to)g(one)f(of)h(its)g(shot)g(coun)n(ters)f(pro)n(vides)f(a)h
(lo)n(w)n(er)523 5289 y(b)r(ound)26 b(on)f FC(m)p FK(.)g(W)-7
b(e'll)25 b(use)h(the)f(notation)g(only)g(where)f(it)i(is)f
(justi\014ed)h(on)f(these)g(grounds.)p eop end
%%Page: 97 107
TeXDict begin 97 106 bop 2393 299 a FF(7.3.)53 b(Solution)26
b(Metho)r(d)154 b(97)606 506 y FK(The)30 b(ev)n(en)n(ts)e(of)i(an)f(op)
r(eration)f FC(a)i FK(in)n(v)n(olving)e(shared)g(v)-5
b(ariable)28 b(access)h(constitute)g(up)523 606 y(to)f(6)f(ev)n(en)n
(ts-sets,)f(or)h FG(phases)p FK(:)1326 805 y FC(a:)p
FK(s)c FD(\036)g FC(a:)p FK(h)g FD(\036)g FC(a:)p FK(r)f
FD(\036)h FC(a:)p FK(t)g FD(\036)g FC(a:)p FK(p)g FD(\036)f
FC(a:)p FK(w)p FC(;)606 988 y FK(in)e(accordance)e(with)i(the)g(lab)r
(elled)g(lines)g(of)f(the)h(Read)g(and)f(W)-7 b(rite)20
b(pro)r(cedure.)f(Ab)r(ort-)523 1087 y(ing)36 b(op)r(erations)f
(consist)h(of)g(only)g(the)h(\014rst)f(3)f(phases,)h(while)g(a)g
(non-ab)r(orting)f(Read)523 1187 y(op)r(eration)28 b(has)h(the)h
(\014rst)g(5.)f(The)g FC(n)h FK(ev)n(en)n(ts)f(in)g(a)g(phase)g
FC(a:)p FK(c)h(\(c)g(one)f(of)g(s,h,r,t,p)h(or)e(w\))523
1286 y(are)f(denoted)g FC(a:)p FK(c)1079 1298 y FA(j)1142
1286 y FK(with)h FC(j)g FD(2)23 b FC(I)7 b FK(,)28 b(and)f(are)g
(called)g(c-ev)n(en)n(ts.)606 1386 y(F)-7 b(or)41 b(a)g(shared)f(v)-5
b(ariable)41 b(read)f(ev)n(en)n(t)h FC(e)p FK(,)g(de\014ne)h(p)p
FG(-L)l(ast)o FK(\()p FC(e)p FK(\))g(to)f(b)r(e)h(the)g(op)r(eration)
523 1486 y(con)n(taining)25 b(the)h(last)g(p-ev)n(en)n(t)f(preceding)g
FC(e)h FK(that)g(accesses)e(the)j(same)e(shared)g(v)-5
b(ariable.)523 1585 y(If)21 b(suc)n(h)e(an)h(ev)n(en)n(t)g(do)r(es)g
(not)g(exist,)g(then)h(p)p FG(-L)l(ast)o FK(\()p FC(e)p
FK(\))g(is)f(de\014ned)g(to)g(b)r(e)h(the)f(non-op)r(eration)523
1685 y FD(?)p FK(.)606 1785 y(F)-7 b(or)24 b FC(a)g FK(an)g(op)r
(eration)g(and)g FG(exp)30 b FK(an)24 b(expression)f(consisting)g(of)i
(\(sym)n(b)r(olic)f(or)f(explicit\))523 1884 y(constan)n(ts)g(and)g(lo)
r(cal)g(v)-5 b(ariables.)22 b(de\014ne)i FG(exp)5 b FK(@)p
FC(a)23 b FK(as)g(the)h(\014nal)g(v)-5 b(alue)23 b(of)g(that)h
(expression)523 1984 y(in)33 b(the)g(pro)r(cedure)f(in)n(v)n(o)r
(cation)f(corresp)r(onding)g(to)h FC(a)p FK(.)h(Arra)n(y)e(indices)i
FC(i;)14 b(j;)g(k)s(;)g(s;)g(t)31 b FK(refer)523 2083
y(to)c(sym)n(b)r(olic)f(constan)n(ts)g(de\014ned)i(in)f(the)g(con)n
(text,)g(not)g(to)g(the)g(lo)r(cal)f(v)-5 b(ariables.)26
b(De\014ne)656 2266 y FG(value)7 b FK(@)p FD(?)22 b FK(=)h
FG(pr)l(ev)9 b FK(@)p FD(?)22 b FK(=)h FG(tag)8 b FK(@)p
FD(?)22 b FK(=)g FG(ss)7 b FK(@)p FD(?)22 b FK(=)h FG(sho)l(ot)8
b FK([][]@)p FD(?)23 b FK(=)f FG(he)l(al)10 b FK([][]@)p
FD(?)23 b FK(=)f(0)p FC(;)523 2449 y FK(in)i(accordance)e(with)j(the)g
(initialization)e(of)h(the)h(construction.)e(De\014ne)i
FG(exp)5 b FK(@)p FC(a:)p FK(c)24 b(\(c)g(one)523 2548
y(of)31 b(s,h,r,t,p)g(or)f(w\))h(as)g(the)g(v)-5 b(alue)31
b(of)g(the)g(expression)f FG(exp)36 b FK(after)31 b(completion)g(of)g
(line)g(c)523 2648 y(of)d(the)g(pro)r(cedure)e(in)n(v)n(o)r(cation)h
(corresp)r(onding)e(to)j FC(a)p FK(.)g(By)f(con)n(v)n(en)n(tion,)f(the)
i(pre\014x)f FG(me)7 b FC(:)523 2748 y FK(is)28 b(omitted)g(when)f
FG(exp)33 b FK(is)28 b(a)f(\014eld)h(of)f FG(me)7 b FK(.)523
2997 y Fl(7.3.3)99 b(Corr)-5 b(e)g(ctness)34 b(of)g(Construction)h(1)
523 3146 y FK(Construction)44 b(1)g(clearly)f(satis\014es)g(W)-7
b(ait-F)g(reeness)44 b(as)g(all)g(lo)r(ops)f(range)g(o)n(v)n(er)g
FC(I)58 b FK(=)523 3246 y FD(f)p FK(0)p FC(;)14 b(:)g(:)g(:)f(;)h(n)6
b FD(\000)g FK(1)p FD(g)p FK(.)20 b(T)-7 b(o)22 b(pro)n(v)n(e)d
(correctness,)h(w)n(e)h(m)n(ust)h(therefore)f(sho)n(w)f(that)i(eac)n(h)
f(complete)523 3345 y(prop)r(er)k(history)h(is)g(linearizable.)f(The)h
(pro)r(of)g(is)g(based)g(on)g(Lemma)g(7.2.)g(First)g(w)n(e)g(need)523
3445 y(some)h(preparatory)e(claims.)523 3611 y Fv(Claim)32
b FH(7.3)40 b FK(All)e(shared)e(tag,)g(heal)h(and)g(shot)f(coun)n(ters)
g(are)g(nondecreasing)f(in)i(the)523 3711 y(course)26
b(of)i(a)f(history)-7 b(.)523 3877 y Fv(Pr)n(oof.)25
b FK(A)i(shared)e(v)-5 b(ariable)25 b FC(R)1559 3889
y FA(i;j)1663 3877 y FK(is)h(c)n(hanged)f(only)h(when)g
FG(me)33 b FK(is)26 b(written)g(to)g(it,)h(in)f(an)523
3976 y(h-,)19 b(p-,)h(or)e(w-ev)n(en)n(t)h(of)g(user)g
FC(i)p FK(,)g(so)f(a)h(nondecreasing)f(coun)n(ter)g(in)i(the)g(static)f
(lo)r(cal)g(v)-5 b(ariable)523 4076 y FG(me)29 b FK(of)23
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%%Page: 98 108
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%%Page: 99 109
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%%Page: 100 110
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%%Page: 101 111
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FK(,)h FG(alive)7 b FK(\()p FG(max)j FK(\)@)p FC(a:)p
FK(p.)606 5264 y(This)26 b(sho)n(ws)e(that)i(the)g(c)n(hoice)e(of)i
FG(max)35 b FK(can)25 b(b)r(e)h(restricted)f(to)h(those)f
FC(j)j FD(2)23 b FC(I)33 b FK(for)25 b(whic)n(h)523 5363
y FG(alive)8 b FK(\()p FC(j)d FK(\))28 b(holds.)p eop
end
%%Page: 102 112
TeXDict begin 102 111 bop 2157 299 a FF(7.4.)53 b(Bounding)26
b(the)f(coun)n(ters)153 b(102)523 506 y Fl(7.4.2)99 b(R)-5
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FC(m)h FK(in)g(the)h(next)f(lemma)g(serv)n(es)f(to)g(prepare)g(for)h(a)
f(later)h(simpli\014cation)523 756 y(of)k(the)g(construction,)e(for)h
(the)h(case)f(of)h(only)f(one)g(writer.)523 914 y Fv(Lemma)32
b FH(7.15)40 b FG(L)l(et)32 b FC(a)h FG(b)l(e)g(a)g(non-ab)l(orting)g
(op)l(er)l(ation)h(by)g(user)e FC(i)p FG(.)h(L)l(et)g
FC(j)g FD(2)c FC(I)40 b FG(and)33 b FC(s)c FK(=)523 1014
y FG(fr)l(om)7 b FK([)p FC(j)e FK(])p FC(:)p FG(ss)i
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FC(m)i FD(\024)e FC(n)32 b FG(b)l(e)g(the)h(numb)l(er)e(of)i(users)e
(with)i(Write)f(op)l(er)l(ations.)523 1114 y(If)e(alive)8
b FK(\()p FC(j)d FK(\)@)p FC(a:)p FK(r)29 b FG(then)1097
1278 y(fr)l(om)7 b FK([)p FG(max)j FK(])p FC(:)p FG(tag)e
FK(@)p FC(a:)p FK(p)18 b FD(\000)g FG(fr)l(om)7 b FK([)p
FC(j)e FK(])p FC(:)p FG(tag)i FK(@)p FC(a:)p FK(p)23
b FD(\024)g FK(10)p FC(mn)523 1442 y Fv(Pr)n(oof.)e FK(Assume,)h(to)g
(the)h(con)n(trary)-7 b(,)20 b(that)i FG(fr)l(om)7 b
FK([)p FG(max)j FK(])p FC(:)p FG(tag)d FK(@)p FC(a:)p
FK(p)23 b FC(>)g FG(fr)l(om)7 b FK([)p FC(j)e FK(])p
FC(:)p FG(tag)i FK(@)p FC(a:)p FK(p)g(+)523 1542 y(10)p
FC(mn)p FK(.)39 b(Let)g FC(W)52 b FK(b)r(e)40 b(the)g(set)g(of)f(all)h
(non-ab)r(orting)e(W)-7 b(rite)40 b(op)r(erations)e FC(w)k
FK(suc)n(h)d(that)523 1642 y FG(tag)8 b FK(@)p FC(w)25
b(>)e FG(fr)l(om)6 b FK([)p FC(j)f FK(])p FC(:)p FG(tag)j
FK(@)p FC(a:)p FK(p)k FD(^)g(:)p FK(\()p FC(a:)p FK(r)24
b FD(\036)f FC(w)r FK(\).)i(Since)g FC(a)f FK(reads)g(a)g(tag)g(whic)n
(h)g(is)h(more)f(than)523 1741 y(10)p FC(mn)g FK(greater)f(than)i
FC(j)5 b FK('s,)24 b(and)h(new)g(tags)f(are)g(c)n(hosen)g(in)h
(incremen)n(ts)f(of)h(at)g(most)f FC(n)p FK(,)h(w)n(e)523
1841 y(m)n(ust)h(ha)n(v)n(e)f FD(j)p FC(W)12 b FD(j)24
b FC(>)e FK(10)p FC(m)p FK(.)j(Therefore,)h(some)f(user,)h(sa)n(y)f
FC(k)s FK(,)h(has)g(at)g(least)g(11)f(op)r(erations,)523
1941 y(sa)n(y)h FC(w)725 1953 y FB(0)786 1941 y FD(\036)d
FC(w)933 1953 y FB(1)994 1941 y FD(\036)f(\001)14 b(\001)g(\001)23
b(\036)g FC(w)1348 1953 y FB(10)1419 1941 y FK(,)k(in)h
FC(W)12 b FK(.)606 2040 y(W)-7 b(e)28 b(claim)f(that)1277
2205 y FG(fr)l(om)7 b FK([)p FC(j)e FK(])p FC(:)p FG(he)l(al)k
FK([)p FC(s)p FK(][)p FC(k)s FK(]@)p FC(w)2007 2217 y
FB(1)2068 2205 y FD(\025)22 b FG(he)l(al)10 b FK([)p
FC(s)p FK(][)p FC(k)s FK(]@)p FC(b:)582 b FK(\(7.2\))523
2369 y(Otherwise)27 b(corollary)f(7.4)h(giv)n(es)g FC(b)c
FD(6)p FK(=)h FD(?)k FK(and)f FC(w)2060 2381 y FB(0)2122
2369 y FD(\036)c FC(w)2269 2381 y FB(1)2307 2369 y FC(:)p
FK(r)2363 2381 y FA(j)2421 2369 y FD(\036)h FC(b:)p FK(h)2615
2381 y FA(k)2679 2369 y FD(\036)g FC(b:)p FK(r)2859 2381
y FA(k)2900 2369 y FK(,)k(from)g(whic)n(h)523 2469 y(corollary)37
b(7.4)j(and)f(claim)h(7.7)f(imply)i FG(tag)7 b FK(@)p
FC(w)2064 2481 y FB(0)2145 2469 y FD(\024)43 b FG(fr)l(om)7
b FK([)p FC(k)s FK(])p FC(:)p FG(tag)h FK(@)p FC(b:)p
FK(r)42 b FD(\024)i FG(tag)7 b FK(@)p FC(b:)p FK(p)43
b FD(\024)523 2568 y FG(fr)l(om)7 b FK([)p FC(j)e FK(])p
FC(:)p FG(tag)j FK(@)p FC(a:)p FK(p)o(,)36 b(con)n(tradictory)d(to)i
(the)g(de\014nition)h(of)f FC(w)2473 2580 y FB(0)2546
2568 y FD(2)h FC(W)12 b FK(.)35 b(As)h(the)f(pro)r(of)g(of)523
2668 y(claim)30 b(7.10)f(sho)n(ws,)h(7.2)f(implies)i
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FC(w)2098 2680 y FB(6)2163 2668 y FD(\025)27 b FG(he)l(al)10
b FK([)p FC(s)p FK(][)p FC(k)s FK(]@)p FC(b)19 b FK(+)h(6.)30
b(On)h(the)f(other)523 2768 y(hand,)23 b(the)h(assumption)f
FG(alive)8 b FK(\()p FC(j)d FK(\)@)p FC(a:)p FK(r)22
b(and)i(claim)f(7.7)f(giv)n(e)h FG(fr)l(om)6 b FK([)p
FC(k)s FK(])p FC(:)p FG(sho)l(ot)j FK([)p FC(s)p FK(][)p
FC(j)c FK(]@)p FC(a:)p FK(r)22 b FC(<)523 2867 y FG(fr)l(om)7
b FK([)p FC(j)e FK(])p FC(:)p FG(he)l(al)10 b FK([)p
FC(s)p FK(][)p FC(k)s FK(]@)p FC(a:)p FK(r)t(+)c(6)22
b(=)h FG(he)l(al)9 b FK([)p FC(s)p FK(][)p FC(k)s FK(]@)p
FC(b)d FK(+)g(6.)19 b(T)-7 b(ogether,)20 b(using)h(corollary)d(7.4,)j
(these)523 2967 y(inequalities)27 b(sho)n(w)g(that)1422
3131 y FC(a:)p FK(h)1535 3143 y FA(k)1599 3131 y FD(\036)c
FC(a:)p FK(r)1786 3143 y FA(k)1850 3131 y FD(\036)g FC(w)1997
3143 y FB(6)2034 3131 y FC(:)p FK(w)2117 3143 y FA(i)2168
3131 y FD(\036)g FC(w)2315 3143 y FB(7)2352 3131 y FC(:)p
FK(r)2408 3143 y FA(i)2436 3131 y FC(:)523 3296 y FK(Since)28
b FC(a)f FK(do)r(esn't)h(ab)r(ort,)f(claim)g(7.10)g(implies)1474
3460 y FC(a:)p FK(r)c FD(\036)g FC(a:)p FK(t)1784 3472
y FA(k)1848 3460 y FD(\036)f FC(w)1994 3472 y FB(9)2032
3460 y FC(:)p FK(w)2115 3472 y FA(i)2166 3460 y FD(\036)g
FC(w)2312 3472 y FB(10)2383 3460 y FC(;)523 3625 y FK(in)28
b(con)n(tradiction)e(to)i(the)g(de\014nition)g(of)f FC(w)1892
3637 y FB(10)1986 3625 y FD(2)c FC(W)12 b FK(.)1119 b
Fu(2)606 3767 y FK(The)26 b(lemma)g(sho)n(ws)f(that)h(all)g(aliv)n(e)f
(tags)g(are)g(from)h(10)p FC(mn)f FK(to)h(0)f(less)h(than)g(the)g
(maxi-)523 3866 y(m)n(um.)i(The)g(follo)n(wing)e(is)i(an)f(easy)g
(consequence.)523 4025 y Fv(Cor)n(ollar)-6 b(y)31 b FH(7.16)41
b FK(Let)24 b FC(a)g FK(b)r(e)h(a)e(non-ab)r(orting)g(op)r(eration)g(b)
n(y)h(user)g FC(i)p FK(.)g(Let)g(1)f FD(\024)f FC(m)h
FD(\024)g FC(n)523 4125 y FK(b)r(e)c(the)g(n)n(um)n(b)r(er)f(of)g
(users)g(with)g(W)-7 b(rite)19 b(op)r(erations.)e(Let)i
FC(j;)14 b(k)26 b FD(2)d FC(I)j FK(suc)n(h)18 b(that)g
FG(alive)8 b FK(\()p FC(j)d FK(\)@)p FC(a:)p FK(r)p FD(^)523
4224 y FG(alive)j FK(\()p FC(k)s FK(\)@)p FC(a:)p FK(r)o(.)28
b(Then)1129 4389 y FD(\000)p FK(10)p FC(mn)22 b FD(\024)g
FG(fr)l(om)7 b FK([)p FC(k)s FK(])p FC(:)p FG(tag)26
b FD(\000)18 b FG(fr)l(om)7 b FK([)p FC(j)e FK(])p FC(:)p
FG(tag)31 b FD(\024)23 b FK(10)p FC(mn:)523 4806 y Fl(7.4.3)99
b(Bounds)34 b(on)g(p)-5 b(er)g(c)g(eive)g(d)34 b(shots)523
4956 y Fv(Lemma)e FH(7.17)40 b FG(L)l(et)d FC(a)g FG(b)l(e)g(a)g
(non-ab)l(orting)g(op)l(er)l(ation)i(by)e(user)g FC(i)p
FG(.)g(L)l(et)g FC(j;)14 b(k)39 b FD(2)e FC(I)7 b FG(,)37
b(and)523 5055 y FC(s)23 b FK(=)g FG(fr)l(om)7 b FK([)p
FC(j)e FK(])p FC(:)p FG(ss)h FK(@)p FC(a:)p FK(r)o FG(.)31
b(Then)913 5220 y FD(\000)p FK(4)22 b FD(\024)g FG(fr)l(om)7
b FK([)p FC(k)s FK(])p FC(:)p FG(sho)l(ot)i FK([)p FC(s)p
FK(][)p FC(j)c FK(]@)p FC(a:)p FK(r)17 b FD(\000)h FG(fr)l(om)7
b FK([)p FC(j)e FK(])p FC(:)p FG(he)l(al)10 b FK([)p
FC(s)p FK(][)p FC(k)s FK(]@)p FC(a:)p FK(r)22 b FD(\024)h
FK(9)p eop end
%%Page: 103 113
TeXDict begin 103 112 bop 2157 299 a FF(7.4.)53 b(Bounding)26
b(the)f(coun)n(ters)153 b(103)523 506 y Fv(Pr)n(oof.)36
b FK(Assume,)i(to)e(the)i(con)n(trary)-7 b(,)35 b(that)i(the)h
(di\013erence)f(is)g(outside)g FD(f\000)p FK(4)p FC(;)14
b(:)g(:)g(:)e(;)i FK(9)p FD(g)p FK(.)523 606 y(W)-7 b(e)39
b(will)g(reac)n(h)f(the)h(required)f(con)n(tradiction)f(b)n(y)i(sho)n
(wing)e(that)i(the)g(conditions)g(of)523 706 y(claim)27
b(7.10)g(hold,)g(implying)h(that)g FC(a)f FK(ab)r(orts.)606
805 y(Let)40 b FC(b)j FK(=)g(p)p FG(-L)l(ast)p FK(\()p
FC(a:)p FK(r)1314 817 y FA(j)1349 805 y FK(\))d(and)f
FC(c)44 b FK(=)f(p)p FG(-L)l(ast)o FK(\()p FC(a:)p FK(r)2141
817 y FA(k)2182 805 y FK(\).)d(In)g(the)h(\014rst)e(case,)g(assume)g
(the)523 905 y(di\013erence)28 b(is)f(under)h FD(\000)p
FK(4.)e(Then)i(b)n(y)f(claim)h(7.7,)f FC(s)c FK(=)f FG(ss)7
b FK(@)p FC(b)27 b FK(and)741 1088 y FG(he)l(al)10 b
FK([)p FC(s)p FK(][)p FC(k)s FK(]@)p FC(b)22 b FK(=)h
FG(fr)l(om)7 b FK([)p FC(j)e FK(])p FC(:)p FG(he)l(al)k
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FG(sho)l(ot)h FK([)p FC(s)p FK(][)p FC(j)d FK(]@)p FC(a:)p
FK(r)18 b(+)g(5)p FC(:)523 1270 y FK(So)23 b FC(b)g FD(6)p
FK(=)g FD(?)g FK(and)g FC(b:)p FK(s)1117 1282 y FA(k)1182
1270 y FK(read)f(a)h(shot)g(coun)n(ter)g(from)g(user)g
FC(k)j FK(that's)e(at)f(least)g(5)g(greater)f(than)523
1370 y(what)28 b FC(a)f FK(read)g(in)h FC(a:)p FK(r)1183
1382 y FA(k)1224 1370 y FK(.)g(This)f(sho)n(ws)g(the)h(existence)f(of)
1104 1553 y FC(a:)p FK(r)1204 1565 y FA(k)1268 1553 y
FD(\036)22 b FC(w)1414 1565 y FB(1)1452 1553 y FC(:)p
FK(w)1535 1565 y FA(i)1586 1553 y FD(\036)g FC(w)1732
1565 y FB(2)1793 1553 y FD(\036)h FC(w)1940 1565 y FB(3)2001
1553 y FD(\036)f FC(w)2147 1565 y FB(4)2208 1553 y FD(\036)h
FC(w)2355 1565 y FB(5)2392 1553 y FC(:)p FK(w)2475 1565
y FA(j)2533 1553 y FD(\036)g FC(b:)p FK(s)2712 1565 y
FA(k)2753 1553 y FC(;)523 1735 y FK(where)i FC(w)820
1747 y FA(m)883 1735 y FC(;)14 b FK(1)23 b FD(\024)f
FC(m)h FD(\024)g FK(5)i(is)g(the)g FG(\014rst)g FK(non-ab)r(orting)e
(write)i(action)g(b)n(y)g(user)g FC(k)j FK(suc)n(h)c(that)1104
1918 y FG(sho)l(ot)9 b FK([)p FC(s)p FK(][)p FC(j)c FK(]@)p
FC(w)1591 1930 y FA(m)1677 1918 y FK(=)23 b FG(fr)l(om)6
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FK(][)p FC(j)c FK(]@)p FC(a:)p FK(r)18 b(+)g FC(m:)523
2100 y FK(Consequen)n(tly)-7 b(,)27 b(the)h(conditions)f(of)h(claim)f
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FC(w)1810 2295 y FB(4)1871 2283 y FD(\036)f FC(b:)p FK(s)2050
2295 y FA(k)2114 2283 y FD(\036)g FC(b:)p FK(p)2306 2303
y FA(i)2357 2283 y FD(\036)g FC(a:)p FK(r)2544 2295 y
FA(j)2602 2283 y FD(\036)h FC(a:)p FK(t)2789 2295 y FA(k)2830
2283 y FC(:)523 2466 y FK(This)28 b(pro)n(v)n(es)d(the)j(\014rst)g
(inequalit)n(y)f(of)g(the)h(lemma.)606 2565 y(In)g(the)g(other)f(case,)
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b FK([)p FC(k)s FK(])p FC(:)p FG(sho)l(ot)i FK([)p FC(s)p
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2931 y FG(fr)l(om)7 b FK([)p FC(j)e FK(])p FC(:)p FG(he)l(al)k
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b(sho)n(ws)e(the)i(existence)f(of)1372 3263 y FC(w)1431
3275 y FB(7)1492 3263 y FD(\036)22 b FC(w)1638 3275 y
FB(8)1699 3263 y FD(\036)h FC(w)1846 3275 y FB(9)1907
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FK(w)2207 3275 y FA(i)2258 3263 y FD(\036)g FC(a:)p FK(r)2445
3275 y FA(k)2486 3263 y FC(;)523 3445 y FK(where)i FC(w)819
3457 y FA(m)883 3445 y FC(;)14 b FK(7)22 b FD(\024)h
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(W)-7 b(rite)25 b(op)r(eration)f(b)n(y)g(user)g FC(k)k
FK(suc)n(h)523 3545 y(that)1299 3645 y FG(sho)l(ot)8
b FK([)p FC(s)p FK(][)p FC(j)d FK(]@)p FC(w)1785 3657
y FA(m)1871 3645 y FK(=)23 b FG(he)l(al)10 b FK([)p FC(s)p
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y FD(\000)18 b FK(6)23 b(=)f FG(he)l(al)10 b FK([)p FC(s)p
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(sho)n(ws,)f(this)i(can)f(only)f(b)r(e)i(if)g FC(d)c
FD(\036)g FC(w)2949 4171 y FB(7)2987 4159 y FC(:)p FK(r)3042
4171 y FA(j)3077 4159 y FK(,)k(with)f FC(d)523 4259 y
FK(b)r(eing)i FC(N)823 4229 y FB(1)814 4281 y FA(j)887
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FK(p)1973 4562 y FA(i)2024 4541 y FD(\036)g FC(w)2171
4553 y FB(7)2208 4541 y FC(:)p FK(r)2264 4553 y FA(j)2322
4541 y FD(\036)g FC(w)2469 4553 y FB(8)2506 4541 y FC(:)523
4724 y FK(Again)31 b(the)h(conditions)e(of)i(claim)f(7.10)f(hold.)h
(This)g(pro)n(v)n(es)f(the)h(second)g(inequalit)n(y)g(of)523
4824 y(the)d(lemma.)2367 b Fu(2)p eop end
%%Page: 104 114
TeXDict begin 104 113 bop 2157 299 a FF(7.4.)53 b(Bounding)26
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656 y FK(F)e(or)27 b(notational)g(con)n(v)n(enience,)f(w)n(e)h(in)n
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FC(b)36 b FK(are)g(uniquely)523 905 y(de\014ned)28 b(b)n(y)f(the)h
(equations)855 1082 y FD(\000)p FK(4)22 b FD(\024)83
b FC(a)18 b FD(\011)g FC(b)83 b(<)22 b FK(10)321 b FD(^)83
b FC(a)18 b FD(\011)g FC(b)23 b FD(\021)g FC(a)18 b FD(\000)g
FC(b)83 b FK(\(mo)r(d)28 b(14\))855 1182 y FD(\000)p
FK(4)22 b FD(\024)83 b FC(a)18 b FD(\010)g FC(b)83 b(<)22
b FK(10)321 b FD(^)83 b FC(a)18 b FD(\010)g FC(b)23 b
FD(\021)g FC(a)18 b FK(+)g FC(b)83 b FK(\(mo)r(d)28 b(14\))726
1281 y FD(\000)p FK(10)p FC(n)925 1251 y FB(2)984 1281
y FD(\024)83 b FC(a)18 b FD(\014)g FC(b)83 b(<)22 b FK(10)p
FC(n)1617 1251 y FB(2)1672 1281 y FK(+)c FC(n)83 b FD(^)g
FC(a)18 b FD(\014)g FC(b)23 b FD(\021)g FC(a)18 b FD(\000)g
FC(b)83 b FK(\(mo)r(d)28 b(20)p FC(n)2935 1251 y FB(2)2989
1281 y FK(+)18 b FC(n)p FK(\))523 1452 y(By)27 b(Corollary)e(7.14)i
(and)g(a)g(simple)h(reordering)d(of)j(terms,)f(the)h(assignmen)n(t)682
1601 y(select)f FG(max)38 b FK(suc)n(h)27 b(that)h FD(8)p
FC(j)g FK(:)23 b FG(fr)l(om)7 b FK([)p FG(max)j FK(])p
FC(:)p FG(tag)30 b FD(\025)23 b FG(fr)l(om)7 b FK([)p
FC(j)e FK(])p FC(:)p FG(tag)523 1751 y FK(of)28 b(Construction)e(1)i
(is)f(equiv)-5 b(alen)n(t)27 b(to:)682 1900 y(select)g
FG(max)33 b FD(2)23 b FC(A)28 b FK(suc)n(h)g(that)f FD(8)p
FC(j)i FD(2)23 b FC(A)g FK(:)g FG(fr)l(om)7 b FK([)p
FG(max)j FK(])p FC(:)p FG(tag)26 b FD(\000)18 b FG(fr)l(om)7
b FK([)p FC(j)e FK(])p FC(:)p FG(tag)31 b FD(\025)23
b FK(0)682 2000 y(where)k FC(A)c FK(=)g FD(f)p FC(j)k
FD(2)c FC(I)31 b FK(:)23 b FG(alive)7 b FK(\()p FC(j)e
FK(\))p FD(g)523 2149 y FK(whic)n(h,)28 b(b)n(y)f(corollary)e(7.16)h
(and)i(lemma)f(7.17,)g(is)g(in)h(turn)g(equiv)-5 b(alen)n(t)27
b(to)682 2299 y(select)g FG(max)33 b FD(2)23 b FC(A)28
b FK(suc)n(h)g(that)f FD(8)p FC(j)i FD(2)23 b FC(A)g
FK(:)g FG(fr)l(om)7 b FK([)p FG(max)j FK(])p FC(:)p FG(tag)26
b FD(\014)18 b FG(fr)l(om)7 b FK([)p FC(j)e FK(])p FC(:)p
FG(tag)31 b FD(\025)23 b FK(0)682 2398 y(where)k FC(A)c
FK(=)g FD(f)p FC(j)k FD(2)c FC(I)31 b FK(:)c FD(8)p FC(k)f
FD(2)e FC(I)30 b FK(:)23 b FG(fr)l(om)7 b FK([)p FC(k)s
FK(])p FC(:)p FG(sho)l(ot)h FK([)p FC(z)t FK(][)p FC(j)d
FK(])18 b FD(\011)g FG(fr)l(om)7 b FK([)p FC(j)e FK(])p
FC(:)p FG(he)l(al)10 b FK([)p FC(z)t FK(][)p FC(k)s FK(])22
b FC(<)h FK(6)1393 2498 y(where)k FC(z)g FK(=)22 b FG(fr)l(om)7
b FK([)p FC(j)e FK(])p FC(:)p FG(ss)i FD(g)523 2747 y
FK(F)-7 b(urthermore,)24 b(the)h(subtractions)f(referred)f(to)i(in)g
(claim)g(7.6)e(are)h(also)g(equiv)-5 b(alen)n(t)24 b(to)h
FD(\011)p FK(.)606 2847 y(Let)g(Construction)g(2)f(b)r(e)i(the)f
(result)g(of)g(replacing)f(these)h(3)g(program)e(fragmen)n(ts)h(\(se-)
523 2946 y(lection)29 b(of)f FG(max)39 b FK(and)28 b(shot-heal)f
(di\013erences\))i(b)n(y)f(their)h(equiv)-5 b(alen)n(t)28
b(parts.)g(Ob)n(viously)-7 b(,)523 3046 y(Construction)27
b(2)g(is)h(also)e(correct.)606 3146 y(W)-7 b(e)26 b(next)f(consider)f
(our)h(b)r(ounded)h(solution,)f(Construction)f(3,)h(sho)n(wn)f(in)i
(\014gure)f(7.3.)523 3245 y(It)41 b(is)f(iden)n(tical)g(to)h
(Construction)e(2)h(except)h(for)e(the)i(t)n(yp)r(e)g(of)f(tags)g(and)g
(shot/heal)523 3345 y(coun)n(ters,)34 b(and)h(the)h(w)n(a)n(y)e(they)h
(are)f(increased.)h(Note)g(that)g(in)h(the)f(2nd)g(to)h(last)e(line,)
523 3444 y(the)29 b(selection)e(of)i FC(t)f FK(is)g(p)r(ossible)g
(\(and)g(unique\))h(b)r(ecause)f(20)p FC(n)2494 3414
y FB(2)2548 3444 y FK(+)19 b FC(n)28 b FK(is)g(a)g(m)n(ultiple)h(of)f
FC(n)p FK(.)523 3544 y(The)33 b(initial)h(state)e(of)h(Construction)g
(3)f(has)h(0)g(in)g(all)g(\014elds)g(of)g(all)g(shared)f(and)h(static)
523 3644 y(v)-5 b(ariables,)26 b(lik)n(e)i(Constructions)e(1)h(and)h
(2.)606 3743 y(T)-7 b(o)27 b(pro)n(v)n(e)e(Construction)i(3)f(correct,)
g(it)i(su\016ces)e(to)h(sho)n(w)g(that)g(an)n(y)f(of)h(its)h
(histories,)523 3843 y(sa)n(y)33 b FC(h)721 3855 y FB(3)791
3843 y FK(=)g FC(S)940 3855 y FB(0)977 3843 y FC(;)14
b(S)1065 3855 y FB(1)1102 3843 y FC(;)g(S)1190 3855 y
FB(2)1227 3843 y FC(;)g(:)g(:)g(:)p FK(,)34 b(is)g FG(e)l(quivalent)g
FK(to)f(a)h(history)f FC(h)2415 3855 y FB(2)2485 3843
y FK(=)g FC(T)2632 3855 y FB(0)2669 3843 y FC(;)14 b(T)2755
3855 y FB(1)2791 3843 y FC(;)g(T)2877 3855 y FB(2)2914
3843 y FC(;)g(:)g(:)g(:)34 b FK(of)f(Con-)523 3943 y(struction)24
b(2,)g(in)g(the)h(sense)f(that)g(corresp)r(onding)e(states)i
FC(S)29 b FK(and)24 b FC(T)36 b FK(are)23 b(the)i(same,)e(up)i(to)523
4042 y FG(c)l(ongruenc)l(e)30 b FK(of)h(tags)e(and)h(shot/heal)f(coun)n
(ters.)h(That)g(is,)g(when)h FG(tag)2774 4063 y FB(3)2842
4042 y FK(is)f(the)h(v)-5 b(alue)30 b(of)523 4142 y(some)21
b(lo)r(cal)f(or)h(shared)f(v)-5 b(ariable)20 b(of)h(t)n(yp)r(e)g
FG(tagtyp)l(e)28 b FK(in)22 b(state)f FC(S)2459 4154
y FA(r)2496 4142 y FK(,)g(and)g FG(tag)2811 4162 y FB(2)2869
4142 y FK(is)g(the)h(v)-5 b(alue)21 b(of)523 4242 y(that)f(same)g(v)-5
b(ariable)19 b(in)i(state)f FC(T)1535 4254 y FA(r)1571
4242 y FK(,)g(then)h(w)n(e)f(w)n(an)n(t)f FG(tag)2222
4262 y FB(3)2282 4242 y FD(\021)k FG(tag)2485 4262 y
FB(2)2605 4242 y FK(\(mo)r(d)28 b(20)p FC(n)2958 4211
y FB(2)2998 4242 y FK(+)t FC(n)p FK(\).)20 b(Sim-)523
4341 y(ilarly)-7 b(,)25 b(when)h FG(shot)1121 4353 y
FB(3)1184 4341 y FK(is)g(the)g(v)-5 b(alue)25 b(of)h(some)f(lo)r(cal)g
(or)g(shared)g(v)-5 b(ariable)24 b(of)i(t)n(yp)r(e)g
FG(shottyp)l(e)523 4441 y FK(in)32 b(state)f FC(S)882
4453 y FA(r)919 4441 y FK(,)g(and)g FG(shot)1293 4453
y FB(2)1361 4441 y FK(is)g(the)h(v)-5 b(alue)32 b(of)f(that)g(same)g(v)
-5 b(ariable)31 b(in)g(state)g FC(T)2979 4453 y FA(r)3016
4441 y FK(,)g(then)h(w)n(e)523 4540 y(w)n(an)n(t)d FG(shot)882
4552 y FB(3)946 4540 y FD(\021)d FG(shot)1191 4552 y
FB(2)1312 4540 y FK(\(mo)r(d)i(14\).)h(W)-7 b(e)30 b(pro)n(v)n(e)e(the)
i(existence)f(of)h FC(h)2715 4552 y FB(2)2782 4540 y
FK(b)n(y)f(induction)h(on)523 4640 y(the)e(length)g(of)f
FC(h)1062 4652 y FB(3)1099 4640 y FK(.)606 4740 y(If)k
FC(h)740 4752 y FB(3)807 4740 y FK(consists)e(of)i(only)e(the)i
(initial)f(state)g FC(S)2042 4752 y FB(0)2110 4740 y
FK(then)h(it)f(is)g(clearly)f(equiv)-5 b(alen)n(t)30
b(to)g(the)523 4839 y FC(h)571 4851 y FB(2)636 4839 y
FK(consisting)d(only)g(of)h(the)g(initial)g(state)f FC(T)1932
4851 y FB(0)1997 4839 y FK(of)h(Construction)f(2.)g(No)n(w)g(supp)r
(ose)h FC(h)3233 4851 y FB(3)3293 4839 y FK(=)523 4939
y FC(S)574 4951 y FB(0)611 4939 y FC(;)14 b(S)699 4951
y FB(1)736 4939 y FC(;)g(:)g(:)g(:)g(;)g(S)972 4951 y
FA(r)1008 4939 y FC(;)g(S)1096 4951 y FA(r)r FB(+1)1251
4939 y FK(is)33 b(a)g(history)g(of)g(Construction)g(3,)g(and)g(the)h
(one)f(ev)n(en)n(t)g(shorter)523 5039 y(history)22 b
FC(S)849 5051 y FB(0)886 5039 y FC(;)14 b(S)974 5051
y FB(1)1011 5039 y FC(;)g(:)g(:)g(:)g(;)g(S)1247 5051
y FA(r)1306 5039 y FK(is)22 b(equiv)-5 b(alen)n(t)23
b(to)f(a)g(history)g FC(T)2258 5051 y FB(0)2295 5039
y FC(;)14 b(T)2381 5051 y FB(1)2417 5039 y FC(;)g(:)g(:)g(:)g(;)g(T)
2651 5051 y FA(r)2710 5039 y FK(of)22 b(Construction)g(2.)523
5138 y(In)34 b(particular,)e FC(S)1096 5150 y FA(r)1166
5138 y FK(is)h(the)h(same)e(as)h FC(T)1774 5150 y FA(r)1810
5138 y FK(,)h(up)f(to)g(congruence.)f(Let)i FC(T)2761
5150 y FA(r)r FB(+1)2915 5138 y FK(b)r(e)f(the)h(state)523
5238 y(reac)n(hed)f(from)h FC(T)1084 5250 y FA(r)1155
5238 y FK(b)n(y)g(execution)g(of)g(the)h(same)f(pro)r(cedure)f
(statemen)n(t)i(b)n(y)f(the)g(same)523 5337 y(user)d(as)g(in)h(the)g
(ev)n(en)n(t)f FC(S)1331 5349 y FA(r)1397 5337 y FD(!)f
FC(S)1561 5349 y FA(r)r FB(+1)1682 5337 y FK(.)h(One)h(can)f(c)n(hec)n
(k)g(in)g(a)h(straigh)n(tforw)n(ard)c(manner,)p eop end
%%Page: 105 115
TeXDict begin 105 114 bop 2157 299 a FF(7.4.)53 b(Bounding)26
b(the)f(coun)n(ters)153 b(105)523 506 y FK(b)n(y)30 b(considering)f
(all)h(pro)r(cedure)f(statemen)n(ts,)i(that)f FC(T)2260
518 y FA(r)r FB(+1)2411 506 y FK(m)n(ust)g(then)h(b)r(e)g(the)g(same)e
(as)523 606 y FC(T)572 618 y FA(r)r FB(+1)692 606 y FK(,)h(up)h(to)e
(congruence.)g(The)h(test)g(for)f(ab)r(ortion)g(comes)g(out)h(the)g
(same)f(b)r(ecause)h(of)523 706 y(congruence.)k(Similarly)-7
b(,)35 b(the)h(same)f(c)n(hoices)f(of)i FG(max)45 b FK(are)35
b(a)n(v)-5 b(ailable)34 b(in)i FC(T)2961 718 y FA(r)3032
706 y FK(as)f(in)h FC(S)3298 718 y FA(r)3335 706 y FK(.)523
805 y(The)e(incremen)n(t)h(of)f FG(me)6 b FC(:)p FG(sho)l(ot)j
FK([)p FC(s)p FK(][)p FC(j)c FK(])34 b(in)h(the)f(3rd)g(to)g(last)g
(line)h(preserv)n(es)d(congruence,)523 905 y(as)37 b(do)r(es)g(the)h
(selection)f(of)h FC(t)g FK(in)g(the)g(next)f(line.)h(Other)f(statemen)
n(ts)h(are)e(ev)n(en)i(more)523 1005 y(straigh)n(tforw)n(ard.)606
1104 y(This)28 b(completes)f(the)h(induction)g(argumen)n(t)e(and)i(sho)
n(ws)523 1270 y Fv(Theorem)k FH(7.19)41 b FG(A)n(ny)26
b(c)l(omplete)i(pr)l(op)l(er)g(history)g(of)g(Construction)f(3)h(is)f
(line)l(arizable.)523 1677 y FL(7.5)116 b(Complexit)m(y)523
1876 y FK(First)39 b(let's)g(consider)f(the)h(time)h(complexit)n(y)e
(of)h(Construction)f(3.)h(Since)g(all)g(shared)523 1976
y(v)-5 b(ariable)24 b(accesses)g(o)r(ccur)h(in)h(phases,)f(eac)n(h)f
(of)i(whic)n(h)f(consists)g(of)g FC(n)h FK(reads)e(or)g(writes)h(in)523
2076 y(parallel,)e(the)h(time)h(complexit)n(y)e(of)h(b)r(oth)g(the)h
(Read)f(and)f(W)-7 b(rite)25 b(op)r(eration)d(is)i(b)r(ounded)523
2175 y(b)n(y)j(the)h(n)n(um)n(b)r(er)g(of)f(phases,)g(whic)n(h)h(is)f
(clearly)g FC(O)r FK(\(1\).)606 2275 y(Next)33 b(consider)f(the)h
(space)f(complexit)n(y)g(of)h(Construction)f(3,)g(whic)n(h)h(is)g(the)g
(size)f(in)523 2374 y(bits)c(of)f(the)h(t)n(yp)r(e)f
FG(shar)l(e)l(d)10 b FK(.)27 b(This)h(can)f(b)r(e)g(split)h(in)n(to)f
(t)n(w)n(o)g(parts:)g(the)g(data)g(size)g(and)g(the)523
2474 y(con)n(trol)f(size.)606 2574 y(The)j(data)f(size)g(is)g(2)19
b FD(\002)g FG(size)l(of)f FK(\()p Fv(abs)q FK(\).)29
b(This)f(factor)g(2)g(o)n(v)n(erhead)f(can)h(b)r(e)h(traced)f(bac)n(k)
523 2673 y(to)36 b(the)g(use)g(of)g(single)g(reader)e(shared)h(v)-5
b(ariables)35 b(in)h(our)g(construction.)f(In)h(fact,)g(the)523
2773 y(v)n(ersion)31 b(of)i(Construction)e(3)i(using)f(m)n(ulti-reader)
f(v)-5 b(ariables)31 b(can)i(b)r(e)g(easily)e(mo)r(di\014ed)523
2873 y(to)d(do)f(a)n(w)n(a)n(y)e(with)j(the)g FG(pr)l(ev)38
b FK(data)27 b(\014eld)h(\(as)f(sk)n(etc)n(hed)g(in)h(the)g(next)f
(section\).)606 2972 y(The)34 b(con)n(trol)f(size)g(concerns)g(all)g
(the)i(other)e(\014elds)h(in)g(the)g(shared)f(v)-5 b(ariables.)33
b(Note)523 3072 y(\014rst)27 b(that)h(from)g(the)g(v)-5
b(alues)27 b(read)g(from)g FC(R)1905 3084 y FA(j;i)1980
3072 y FK(,)g(user)g FC(i)h FK(nev)n(er)e(uses)h(an)n(y)g(of)h(the)g
(coun)n(ters)523 3171 y FG(he)l(al)10 b FK([1)s FD(\000)s
FG(ss)c FK(][)p FC(k)s FK(],)20 b(where)f FC(k)26 b FD(6)p
FK(=)d FC(i)p FK(.)d(Th)n(us,)f(in)i(addition)e(to)h(the)h(single)e
(coun)n(ter)g FG(he)l(al)10 b FK([1)s FD(\000)s FG(ss)c
FK(][)p FC(i)p FK(],)523 3271 y(only)29 b(a)g(single)h(heal)f(coun)n
(ter)g(set)g(needs)h(to)f(b)r(e)h(stored)f(in)h FC(R)2481
3283 y FA(j;i)2555 3271 y FK(,)g(of)g(whic)n(h)f(the)h(missing)523
3371 y(\014rst)d(index)h(is)g(understo)r(o)r(d)f(to)g(b)r(e)h
FC(R)1713 3383 y FA(j;i)1788 3371 y FC(:)p FG(ss)7 b
FK(.)27 b(This)h(leads)f(to)h(a)f(con)n(trol)f(size)h(of)1010
3553 y FD(d)p FK(log)14 b(20)p FC(n)1302 3519 y FB(2)1357
3553 y FK(+)k FC(n)p FD(e)g FK(+)g(1)g(+)g(\(3)p FC(n)g
FK(+)g(1\))p FD(d)p FK(log)c(14)p FD(e)22 b(\024)g FK(12)p
FC(n)c FK(+)g FC(o)p FK(\()p FC(n)p FK(\))p FC(:)523
3861 y FL(7.6)116 b(Subproblems)523 4060 y FK(The)35
b(\014nal)f(construction)g(presen)n(ts)f(a)h(solution)h(to)f(the)h
(problem)f(of)g(implemen)n(ting)h(a)523 4159 y(m)n(ulti-user)29
b(v)-5 b(ariable)29 b(from)g(single-reader)f(v)-5 b(ariables.)28
b(Man)n(y)i(other)f(pap)r(ers)g(ha)n(v)n(e)f(con-)523
4259 y(sidered)36 b(the)h(in)n(termediate)f(lev)n(el)g(of)h(a)f
(single-writer)f(m)n(ulti-reader)g(v)-5 b(ariable,)36
b(whic)n(h)523 4359 y(splits)26 b(the)h(problem)e(in)n(to)h(t)n(w)n(o)f
(subproblems.)h(W)-7 b(e)26 b(sho)n(w)f(that)i FG(pr)l(oje)l(ctions)g
FK(of)f(our)f(con-)523 4458 y(struction)c(yield)h(solutions)f(to)g
(those)h(t)n(w)n(o)f(subproblems)f(with)j(comp)r(etitiv)n(e)e
(complexit)n(y)523 4558 y(measures.)606 4658 y(The)38
b(\014rst)g(pro)5 b(jection)37 b(is)h(obtained)g(b)n(y)f(collapsing)g
(the)h(ro)n(w)f(of)h(shared)f(v)-5 b(ariables)523 4757
y FC(R)586 4769 y FA(i;)p FB(0)667 4757 y FC(;)14 b(:)g(:)g(:)f(;)h(R)
914 4769 y FA(i;n)p Fw(\000)p FB(1)1115 4757 y FK(in)n(to)27
b(a)h(single)f(m)n(ulti-reader)f(v)-5 b(ariable)27 b
FC(R)2432 4769 y FA(i)2459 4757 y FK(.)h(Eac)n(h)e(lo)r(op)682
4907 y FH(for)h FC(j)h FD(2)c FC(I)35 b FH(do)27 b FC(R)1228
4919 y FA(i;j)1329 4907 y FK(:=)c FG(me)523 5056 y FK(is)28
b(replaced)e(b)n(y)i(the)g(single)f(write)682 5205 y
FC(R)745 5217 y FA(i)795 5205 y FK(:=)c FG(me)p eop end
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TeXDict begin 106 115 bop 2157 299 a FF(7.4.)53 b(Bounding)26
b(the)f(coun)n(ters)153 b(106)p 523 907 2323 4 v 523
1854 4 947 v 551 1002 a FH(t)m(yp)s(e)28 b FC(I)i FK(:)23
b(0)p FC(::n)18 b FD(\000)g FK(1)763 1102 y FG(tagtyp)l(e)35
b FK(:)27 b FD(\000)p FK(10)p FC(n)1299 1071 y FB(2)1335
1102 y FC(::)p FK(10)p FC(n)1515 1071 y FB(2)1570 1102
y FK(+)18 b FC(n)g FD(\000)g FK(1)763 1201 y FG(shottyp)l(e)35
b FK(:)28 b FD(\000)p FK(4)p FC(::)p FK(9)763 1301 y
FG(shar)l(e)l(d)37 b FK(:)28 b FH(record)1187 1400 y
FG(value)7 b FK(,)p FG(pr)l(ev)37 b FK(:)28 b Fv(abs)p
FG(typ)l(e)1187 1500 y(tag)63 b FK(:)28 b FG(tagtyp)l(e)1187
1600 y(ss)34 b FK(:)28 b(0)p FC(::)p FK(1)1187 1699 y
FG(sho)l(ot)8 b FK(,)p FG(he)l(al)38 b FK(:)27 b FH(arra)m(y)t
FK([0)p FC(::)p FK(1][0)p FC(::n)17 b FD(\000)h FK(1])28
b FH(of)36 b FG(shottyp)l(e)1076 1799 y FH(end)p 2843
1854 V 523 1857 2323 4 v 523 1918 2778 4 v 523 4758 4
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eop end
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%%Page: 108 118
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y FK(:=)c FG(me)710 4802 y FH(return)28 b FG(me)6 b FC(:)p
FG(value)551 4902 y FH(end)p 3200 4957 V 523 4960 2681
4 v 1070 5103 a FF(FIGURE)25 b(7.4.)i(Construction)f(4;)g(single)h(to)f
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(Kranakis,)f(P)-7 b(.M.B.)24 b(Vit\023)-42 b(an)n(yi,)25
b FG(A)h(pr)l(o)l(of)i(te)l(chnique)694 2615 y(for)37
b(r)l(e)l(gister)f(atomicity)p FK(,)g(Pro)r(c.)e(8th)g(Conference)g(on)
g(F)-7 b(oundations)35 b(of)f(Soft)n(w)n(are)694 2715
y(T)-7 b(ec)n(hnology)25 b(&)i(Theoretical)f(Computer)g(Science,)h
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FG(Constructing)h(Two-writer)j(A)n(tomic)e(R)l(e)l(gisters)p
FK(,)d(IEEE)g(T)-7 b(ransactions)694 3071 y(on)27 b(Computers,)g(v)n
(ol.)g(37,)g(pp.)h(1506{1514,)23 b(1988.)565 3229 y([3])41
b(J.E.)31 b(Burns)g(and)h(G.L.)g(P)n(eterson,)e FG(Constructing)k
(Multi-r)l(e)l(ader)g(A)n(tomic)g(V)-6 b(alues)694 3328
y(F)g(r)l(om)25 b(Nonatomic)h(V)-6 b(alues)p FK(,)23
b(Pro)r(c.)f(6th)g(A)n(CM)h(Symp)r(osium)g(on)g(Principles)f(of)h(Dis-)
694 3428 y(tributed)28 b(Computing,)f(pp.)h(222{231,)d(1987.)565
3585 y([4])41 b(D.)34 b(Dolev)g(and)f(N.)h(Sha)n(vit,)g
FG(Bounde)l(d)i(Concurr)l(ent)f(Time-Stamp)h(Systems)f(A)n(r)l(e)694
3685 y(Constructible)p FK(,)26 b(Pro)r(c.)f(21th)g(A)n(CM)h(Symp)r
(osium)g(on)f(Theory)g(of)h(Computing,)f(pp.)694 3784
y(454{466,)f(1989.)565 3942 y([5])41 b(M.P)-7 b(.)37
b(Herlih)n(y)-7 b(,)38 b FG(Imp)l(ossibility)j(and)e(Universality)i(R)l
(esults)c(for)j(Wait-F)-6 b(r)l(e)l(e)39 b(Syn-)694 4041
y(chr)l(onization)p FK(,)44 b(Pro)r(c.)d(7th)i(A)n(CM)g(Symp)r(osium)g
(on)f(Principles)f(of)i(Distributed)694 4141 y(Computing,)27
b(1988.)565 4298 y([6])41 b(M.P)-7 b(.)28 b(Herlih)n(y)-7
b(,)27 b(J.)h(Wing,)g FG(Line)l(arizability:)34 b(A)c(Corr)l(e)l
(ctness)g(Condition)h(for)g(Con-)694 4398 y(curr)l(ent)d(Obje)l(cts)p
FK(,)g(A)n(CM)g(TOPLAS)f(v)n(ol.)g(12,)f(No.)i(3,)f(pp.)h(463{492,)c
(1990.)565 4555 y([7])41 b(A.)24 b(Israeli)e(and)h(M.)h(Li,)g
FG(Bounde)l(d)j(Time-Stamps)p FK(,)d(Pro)r(c.)e(28th)h(IEEE)f(Symp)r
(osium)694 4655 y(on)27 b(F)-7 b(oundations)27 b(of)h(Computer)f
(Science,)h(pp.)g(371{382,)c(1987.)565 4812 y([8])41
b(A.)19 b(Israeli)e(and)h(A.)h(Shaham,)f FG(Optimal)k(Multi-Writer)g
(Multi-R)l(e)l(ader)f(A)n(tomic)h(R)l(e)l(g-)694 4912
y(ister)p FK(,)29 b(Pro)r(c.)f(11th)g(A)n(CM)h(Symp)r(osium)h(on)e
(Principles)g(of)h(Distributed)h(Comput-)694 5011 y(ing,)d(pp.)h
(71{82,)d(1992.)565 5169 y([9])41 b(L.M.)c(Kirousis,)f(E.)h(Kranakis,)e
(P)-7 b(.M.B.)37 b(Vit\023)-42 b(an)n(yi,)37 b FG(A)n(tomic)i(Multir)l
(e)l(ader)g(R)l(e)l(gis-)694 5268 y(ter)p FK(,)21 b(Pro)r(c.)g(2nd)h
(In)n(ternational)e(W)-7 b(orkshop)21 b(on)h(Distributed)g(Computing,)g
(Lecture)694 5368 y(Notes)27 b(in)h(Computer)f(Science)h(312,)e(pp.)i
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%%Page: 110 120
TeXDict begin 110 119 bop 2551 299 a FF(7.7.)53 b(Conclusion)154
b(110)523 506 y FK([10])41 b(L.)29 b(Lamp)r(ort,)f FG(On)j(Interpr)l(o)
l(c)l(ess)g(Communic)l(ation)h(Parts)f(I)g(and)h(II)p
FK(,)d(Distributed)694 606 y(Computing,)e(v)n(ol.)g(1,)g(pp.)h(77{101,)
d(1986.)523 769 y([11])41 b(M.)c(Li)f(and)h(P)-7 b(.M.B.)36
b(Vit\023)-42 b(an)n(yi,)37 b FG(A)h(very)h(simple)g(c)l(onstruction)e
(for)i(atomic)g(mul-)694 869 y(tiwriter)h(r)l(e)l(gister)p
FK(,)e(T)-7 b(ec)n(hn.)39 b(Rep)r(ort)f(TR)g(01-87,)e(Aik)n(en)j(Comp.)
f(Lab.,)g(Harv)-5 b(ard)694 969 y(Univ)n(ersit)n(y)e(,)25
b(No)n(v)n(em)n(b)r(er)g(1987.)f(Also,)i(pp.)g(488-505)d(in:)j
FG(Pr)l(o)l(c.)j(International)g(Col-)694 1068 y(lo)l(quium)42
b(on)f(A)n(utomata,)h(L)l(anguages,)h(and)f(Pr)l(o)l(gr)l(amming)p
FK(,)g(Lecture)e(Notes)g(in)694 1168 y(Computer)27 b(Science,)h(V)-7
b(ol.)27 b(372,)g(Springer)f(V)-7 b(erlag,)27 b(1989.)523
1331 y([12])41 b(N.)e(Lync)n(h)g(and)g(M.)g(T)-7 b(uttle,)40
b FG(Hier)l(ar)l(chic)l(al)i(c)l(orr)l(e)l(ctness)e(pr)l(o)l(ofs)h(for)
g(distribute)l(d)694 1431 y(algorithms)p FK(,)25 b(Pro)r(c.)d(6th)h(A)n
(CM)h(Symp)r(osium)f(on)h(Principles)e(of)h(Distributed)h(Com-)694
1531 y(puting,)k(1987.)523 1694 y([13])41 b(R.)19 b(Newman-W)-7
b(olfe,)19 b FG(A)j(Pr)l(oto)l(c)l(ol)g(for)h(Wait-F)-6
b(r)l(e)l(e,)22 b(A)n(tomic,)h(Multi-R)l(e)l(ader)f(Shar)l(e)l(d)694
1794 y(V)-6 b(ariables)p FK(,)29 b(Pro)r(c.)f(6th)g(A)n(CM)g(Symp)r
(osium)h(on)f(Principles)f(of)h(Distributed)h(Com-)694
1893 y(puting,)f(pp.)g(232{248,)c(1987.)523 2057 y([14])41
b(G.L.)36 b(P)n(eterson)e(and)i(J.E.)f(Burns,)g FG(Concurr)l(ent)i(r)l
(e)l(ading)h(while)h(writing)f(II:)f(the)694 2156 y(multiwriter)f(c)l
(ase)p FK(,)g(Pro)r(c.)e(28th)h(IEEE)e(Symp)r(osium)j(on)f(F)-7
b(oundations)34 b(of)h(Com-)694 2256 y(puter)27 b(Science,)h(pp.)g
(383{392,)c(1987.)523 2419 y([15])41 b(G.L.)36 b(P)n(eterson,)e
FG(Concurr)l(ent)j(r)l(e)l(ading)h(while)h(writing)p
FK(,)e(A)n(CM)f(T)-7 b(ransactions)34 b(on)694 2519 y(Programming)25
b(Languages)g(and)i(Systems,)h(v)n(ol.)f(5,)g(No.)h(1,)f(pp.)h(46{55,)d
(1983.)523 2682 y([16])41 b(R.)26 b(Sc)n(ha\013er,)g
FG(On)i(the)h(c)l(orr)l(e)l(ctness)f(of)i(atomic)f(multi-writer)g(r)l
(e)l(gisters)p FK(,)d(T)-7 b(ec)n(hnical)694 2782 y(Rep)r(ort)27
b(MIT/LCS/TM-364,)e(MIT)j(lab.)f(for)g(Computer)h(Science,)f(June)h
(1988.)523 2945 y([17])41 b(A.K.)d(Singh,)g(J.H.)h(Anderson,)e(M.G.)i
(Gouda,)f FG(The)i(Elusive)g(A)n(tomic)g(R)l(e)l(gister)694
3045 y(R)l(evisite)l(d)p FK(,)30 b(Pro)r(c.)e(6th)i(A)n(CM)f(Symp)r
(osium)h(on)f(Principles)g(of)g(Distributed)h(Com-)694
3144 y(puting,)e(pp.)g(206{221,)c(1987.)523 3308 y([18])41
b(J.)27 b(Anderson,)g FG(Comp)l(osite)k(R)l(e)l(gisters)p
FK(,)d(Distributed)g(Computing,)g(to)f(app)r(ear.)523
3471 y([19])41 b(K.)26 b(Vidy)n(asank)-5 b(ar,)24 b FG(Converting)29
b(L)l(amp)l(ort's)g(R)l(e)l(gular)f(R)l(e)l(gister)g(to)g(an)h(atomic)g
(r)l(e)l(g-)694 3571 y(ister)p FK(,)f(Information)e(Pro)r(cessing)g
(Letters,)h(v)n(ol.)g(28,)g(pp.)h(287{290,)c(1988.)523
3734 y([20])41 b(P)-7 b(.M.B.)36 b(Vit\023)-42 b(an)n(yi,)37
b(B.)g(Aw)n(erbuc)n(h,)f FG(A)n(tomic)j(Shar)l(e)l(d)g(R)l(e)l(gister)f
(A)l(c)l(c)l(ess)g(by)g(Asyn-)694 3834 y(chr)l(onous)25
b(Har)l(dwar)l(e)p FK(,)e(Pro)r(c.)f(27th)g(IEEE)e(Symp)r(osium)j(on)f
(F)-7 b(oundations)22 b(of)g(Com-)694 3934 y(puter)27
b(Science,)h(pp.)g(233{243,)c(1986.)i(\(Errata,)g(Ibid.,1987\))523
4097 y([21])41 b(C.)22 b(Dw)n(ork,)g(O.)g(W)-7 b(aarts,)21
b FG(Simple)26 b(and)g(e\016cient)f(b)l(ounde)l(d)h(c)l(oncurr)l(ent)e
(timestamp-)694 4197 y(ing,)45 b(or,)g(Bounde)l(d)h(c)l(oncurr)l(ent)d
(timestamp)i(systems)f(ar)l(e)h(c)l(ompr)l(ehensible!,)j
FK(,)694 4296 y(Pro)r(c.)22 b(24th)g(A)n(CM)h(Symp)r(osium)h(on)e
(Theory)g(of)h(Computing,)g(pp.)h(655{666,)c(1992.)523
4460 y([22])41 b(A.)26 b(Israeli,)g(M.)g(Pinhaso)n(v,)f
FG(A)j(Concurr)l(ent)g(Time-Stamp)h(Scheme)h(which)g(is)f(Lin-)694
4559 y(e)l(ar)h(in)g(Time)h(and)f(Sp)l(ac)l(e)p FK(,)f(Pro)r(c.)d(6th)i
(In)n(ternational)f(W)-7 b(orkshop)27 b(on)g(Distributed)694
4659 y(Algorithms,)g(Lecture)g(Notes)g(in)h(Computer)f(Science)h(647,)e
(pp.)i(95{109,)d(1992.)523 4822 y([23])41 b(R.)22 b(Ga)n(wlic)n(k,)f
(N.)i(Lunc)n(h,)f(N.)h(Sha)n(vit,)f FG(Concurr)l(ent)i(Timestamping)i
(Made)h(Simple)p FK(,)694 4922 y(Pro)r(c.)35 b(1st)g(Israel)g(Symp)r
(osium)h(on)g(Theory)f(of)h(Computing)g(and)f(Systems,)h(pp.)694
5021 y(171{183,)24 b(1992.)523 5185 y([24])41 b(C.)23
b(Dw)n(ork,)f(M.)h(Herlih)n(y)-7 b(,)23 b(S.)g(Plotkin,)f(O)h(W)-7
b(aarts,)22 b FG(Time-L)l(apse)27 b(Snapshots)p FK(,)d(Pro)r(c.)694
5284 y(1st)f(Israel)g(Symp)r(osium)h(on)g(Theory)e(of)i(Computing)g
(and)g(Systems,)g(pp.)g(154{170,)694 5384 y(1992.)p eop
end
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(c)l(ess)g(c)l(ommunic)l(ation)g(and)g(the)g(implementation)h(of)694
606 y(a)30 b(multi-writer)f(atomic)i(r)l(e)l(gister)p
FK(,)d(unpublished)g(man)n(uscript.)523 772 y([26])41
b(J.)25 b(T)-7 b(romp,)25 b FG(How)j(to)f(Construct)g(an)h(A)n(tomic)g
(V)-6 b(ariable)p FK(,)26 b(Chapter)f(6,)g(also)f(in)i(Pro)r(c.)694
872 y(3rd)i(In)n(ternational)g(W)-7 b(orkshop)27 b(on)i(Distributed)h
(Algorithms,)e(Lecture)h(Notes)f(in)694 971 y(Computer)f(Science)h
(392,)e(pp.)i(292{302,)c(1989.)p eop end
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2 v 3357 300 2 75 v 523 557 a Fz(8)523 806 y FI(Binary)52
b(Snapshots)523 2549 y FL(8.1)116 b(In)m(tro)s(duction)523
2748 y FK(Consider)42 b(a)h(concurren)n(t)f(shared)h(memory)f(system.)h
(A)h(snapshot)e(memory)h(ob)5 b(ject)523 2848 y(shared)26
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FC(n)f FK(memory)g(cells,)h(one)f(`o)n(wned')g(b)n(y)h(eac)n(h)523
2947 y(pro)r(cess.)20 b(All)i(pro)r(cesses)d(can)i(indep)r(enden)n(tly)
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b FK(\))22 b(the)523 3047 y(cell)30 b(they)h(o)n(wn,)f(and)g(all)g(pro)
r(cesses)f(can)h(`instan)n(taneously')f(collect)h(\()p
FG(sc)l(an)6 b FK(\))31 b(all)f(v)-5 b(alues)523 3147
y(in)28 b(the)g(v)n(ector)e(in)i(a)f(single)g(op)r(eration.)606
3246 y(The)j(problem)g(of)g(implemen)n(ting)g(a)f(w)n(ait-free)g
(atomic)h(snapshot)f(ob)5 b(ject)30 b(w)n(as)f(inde-)523
3346 y(p)r(enden)n(tly)j(prop)r(osed)e(and)i(solv)n(ed)e(b)n(y)h
(Anderson)g([6,)g(7,)g(8])g(and)h(Afek)g(et)f(al.)g([1].)h(An-)523
3445 y(derson)c(giv)n(es)g(an)h(exp)r(onen)n(tial)f(time)1717
3415 y FB(1)1784 3445 y FK(solution)h(to)g(this)g(problem)g(using)f
(single-writer)523 3545 y(m)n(ulti-reader)h(registers,)g(and)h(also)f
(considers)g(the)i(m)n(ulti-writer)f(case)g(in)g(whic)n(h)h(more)523
3645 y(than)j(one)f(pro)r(cess)g(ma)n(y)g(up)r(date)h(a)g(particular)e
(cell.)i(In)g(his)g(solution)f(for)g(the)i(m)n(ulti-)523
3744 y(writer)40 b(case)g(he)h(uses)f(the)h(single-writer)e(snapshot)h
(ob)5 b(ject)40 b(as)g(a)g(primitiv)n(e,)h(so)f(his)523
3844 y(solution)27 b(do)r(es)g(not)g(rely)g(on)g(m)n(ulti-writer)g(m)n
(ulti-reader)f(registers.)f(Afek)j(et)g(al.)f(giv)n(e)f(a)523
3944 y(p)r(olynomial)c(time)g(implemen)n(tation)h(of)f(a)g
(single-writer)e(atomic)i(snapshot)g(ob)5 b(ject,)22
b(also)523 4043 y(using)31 b(single-writer)f(m)n(ulti-reader)h
(registers.)f(They)h(also)g(consider)f(the)i(m)n(ulti-writer)523
4143 y(case,)27 b(but)h(giv)n(e)f(a)g(solution)g(using)g(m)n
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b(is)h(a)f(p)r(o)n(w)n(erful)g(to)r(ol)g(to)g(construct)g(other)523
4342 y(atomic)26 b(w)n(ait-free)g(ob)5 b(jects,)27 b(for)f(instance)h
(coun)n(ters,)f(logical)f(clo)r(c)n(ks,)h(or)g(b)r(ounded)h(con-)523
4442 y(curren)n(t)h(time-stamp)g(sc)n(hemes.)g(Aspnes)h(and)f(Herlih)n
(y)g([3])h(giv)n(e)e(a)i(general)e(metho)r(d)i(to)523
4541 y(con)n(v)n(ert)c(a)i(sequen)n(tial)f(sp)r(eci\014cation)h(of)f(a)
h(shared)f(memory)g(ob)5 b(ject)26 b(that)i(satis\014es)e(cer-)523
4641 y(tain)31 b(constrain)n(ts)e(to)h(a)h(w)n(ait-free)e(implemen)n
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4741 y(snapshot)35 b(memory)f(ob)5 b(ject)35 b(as)g(a)g(primitiv)n(e.)g
(They)h(also)e(giv)n(e)g(a)h(a)g(p)r(olynomial-time)523
4840 y(implemen)n(tation)28 b(of)f(a)g(w)n(ait-free)g(atomic)g
(snapshot)g(ob)5 b(ject.)606 4940 y(The)26 b(main)f(question)g(remains)
g(whether)g(it)h(is)f(p)r(ossible)g(to)h(deterministically)f(imple-)p
523 5060 499 5 v 627 5142 a Fm(1)661 5173 y FF(In)37
b(the)h(literature)h(on)e(this)h(sub)t(ject)h(the)e(time)h(complexit)n
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(this,)h(b)n(y)f(imp)r(osing)g(certain)g(restrictions)523
805 y(on)j(the)g(applicabilit)n(y)g(of)g(the)h(solutions.)e(In)h([12)o
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(w)n(o)f(scans)h(ev)n(er)f(o)n(v)n(erlap.)f(Dw)n(ork)h(et)h(al.)g([9])
523 1005 y(in)n(tro)r(duce)35 b(the)h(w)n(eak)n(er)e(time-lapse)h
(snapshot)g(ob)5 b(ject,)35 b(and)h(giv)n(e)f(a)g(linear)g(time)h(im-)
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b(Time-lapse)f(snapshots)h(satisfy)g(the)h(same)f(prop)r(erties)523
1204 y(atomic)c(snapshots)g(do,)h(except)g(that)g(the)g(former)f(allo)n
(w)g(concurren)n(t)g(scans)g(to)g(con)n(tra-)523 1303
y(dict)34 b(eac)n(h)f(other.)g(In)g([11)o(],)h(Israeli)e(et)i(al.)f
(presen)n(t)g(linear-time)g(implemen)n(tations)g(for)523
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(or)h(for)h(un)n(balanced)f(systems)h(in)g(whic)n(h)523
1503 y(the)g(n)n(um)n(b)r(er)e(of)i(up)r(daters)f(is)g(substan)n
(tially)f(smaller)g(than)h(the)h(n)n(um)n(b)r(er)f(of)g(scanners,)523
1602 y(or)36 b(vice)g(v)n(ersa.)f(Finally)-7 b(,)37 b(A)n(ttiy)n(a)f
(et)h(al.)f([4])h(in)n(tro)r(duce)f(the)h(lattice)g(agreemen)n(t)e
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(solution)g(to)h(this)f(problem)g(can)g(b)r(e)h(con)n(v)n(erted)e(to)h
(a)523 1802 y(w)n(ait-free)d(atomic)g(snapshot)g(implemen)n(tation.)606
1901 y(In)33 b(this)h(c)n(hapter)e(w)n(e)g(tak)n(e)h(a)f(similar)h
(approac)n(h,)e(and)i(reduce)f(the)i(general)d(atomic)523
2001 y(snapshot)k(problem)f(to)h(a)g(simpler)g(one.)g(W)-7
b(e)36 b(presen)n(t)f(a)f(b)r(ounded,)i(linear)f(time)g(con-)523
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(the)h(general)e(atomic)h(snapshot)g(ob)5 b(ject)523
2200 y(from)29 b(an)h(atomic)f(w)n(ait-free)g FG(binary)38
b FK(snapshot)29 b(ob)5 b(ject)30 b(\(where)f(eac)n(h)g(cell)h(can)f
(con)n(tain)523 2300 y(only)35 b(t)n(w)n(o)g(v)-5 b(alues\))35
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(single-writer)h(regis-)523 2399 y(ters.)29 b(Th)n(us)g(the)g(searc)n
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(ma)n(y)e(b)r(e)523 2499 y(restricted)f(to)g(the)h(binary)f(case.)606
2599 y(W)-7 b(e)29 b(will)f(use)g(a)g(pro)r(of)g(tec)n(hnique)g(prop)r
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(the)523 2698 y(correctness)20 b(of)j(some)f(atomic)f(register)g
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(in)h([13)o(],)f(where)g(his)g(t)n(w)n(o)f(precedence)h(relations)523
2897 y FG(pr)l(e)l(c)l(e)l(des)g FK(and)25 b FG(c)l(an)j(a\013e)l(ct)33
b FK(are)24 b(replaced)g(b)n(y)h(a)g(single)g(in)n(terv)-5
b(al)25 b(order.)f(W)-7 b(e)25 b(\014rst)g(presen)n(t)523
2997 y(the)i(mo)r(del)g(in)f(Section)h(8.2,)f(then)h(w)n(e)f(state)g
(the)h(atomic)f(w)n(ait-free)f(snapshot)h(problem)523
3097 y(in)k(Section)g(8.3.)f(The)h(proto)r(col)f(is)h(presen)n(ted)f
(in)h(Section)g(8.4,)f(and)h(is)g(pro)n(v)n(en)e(correct)523
3196 y(in)g(Section)f(8.5.)523 3504 y FL(8.2)116 b(The)38
b(Mo)s(del)523 3703 y FK(A)k(concurren)n(t)d(shared)i(memory)f(system)h
(is)g(a)g(collection)g(of)g(sequen)n(tial)f(pro)r(cesses)523
3802 y(comm)n(unicating)f(async)n(hronously)e(through)i(shared)g
(memory)f(data)i(structures.)e(A)n(t)523 3902 y(an)n(y)27
b(time)g(a)g(pro)r(cess)f(is)h(executing)g(at)g(most)g(one)g(action.)g
(A)h(pro)r(cess)e(can)h(at)g(an)n(y)f(time)523 4002 y(decide)f(to)h
(start)e(a)h(new)h(action)f(when)g(it)h(is)f(idle,)h(or)e(to)h
(\014nish)h(an)f(ongoing)f(action.)h(The)523 4101 y(start)i(time)h(of)g
(an)f(action)g FC(a)h FK(is)f(denoted)h(b)n(y)f FC(s)p
FK(\()p FC(a)p FK(\))h(and)g(the)g(\014nish)f(time)i(b)n(y)e
FC(f)9 b FK(\()p FC(a)p FK(\).)606 4201 y(W)-7 b(e)37
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(y)h(a)f(tuple)h FD(hA)p FC(;)14 b FD(!i)p FK(,)523 4301
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(ordered)f(b)n(y)i FD(!)f FK(suc)n(h)h(that)g FC(a)f
FK(precedes)g FC(b)p FK(,)523 4400 y FC(a)g FD(!)f FC(b)p
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FC(a)23 b FD(2)h(A)523 4500 y FK(there)32 b(are)f(only)h(a)g(\014nite)g
(n)n(um)n(b)r(er)g(of)g(actions)g FC(b)e FD(2)h(A)h FK(with)h
FD(:)p FK(\()p FC(a)e FD(!)g FC(b)p FK(\).)h(This)g(w)n(a)n(y)f(w)n(e)
523 4599 y(require)24 b(an)h(execution)g(to)g(start)f(at)h(some)g(p)r
(oin)n(t)g(in)h(time,)g(rather)e(than)h(extending)g(in)n(to)523
4699 y(the)d(in\014nite)h(past)f([13)o(,)g(5].)g(With)h(this)g
(de\014nition,)f FD(!)g FK(is)g(a)g(sp)r(ecial)g(kind)g(of)g(partial)f
(order)523 4799 y(called)k(an)g FG(interval)j(or)l(der)36
b FK(\(i.e.)25 b(a)g(transitiv)n(e)g(binary)f(relation)g(suc)n(h)h
(that)h(if)g FC(a)d FD(!)g FC(b)i FK(and)523 4898 y FC(c)31
b FD(!)g FC(d)h FK(then)h FC(a)e FD(!)g FC(d)h FK(or)g
FC(c)e FD(!)h FC(b)p FK(\).)h(No)n(w)g(w)n(e)g(ha)n(v)n(e)f(abstracted)
g(a)n(w)n(a)n(y)g(from)h(the)g(actual)523 4998 y(time)26
b(an)f(action)g(o)r(ccurred,)g(and)g(w)n(e)h(can)f(sp)r(ecify)h(the)g
(b)r(eha)n(viour)e(of)h(actions)g(in)n(v)n(olving)523
5098 y(access)h(to)i(the)g(shared)e(memory)h(in)h(terms)f(of)h(the)g
(in)n(terv)-5 b(al)27 b(order.)606 5197 y(If)j(one)g(wishes)f(to)g
(implemen)n(t)i(a)e(certain)g FG(c)l(omp)l(ound)39 b
FK(shared)29 b(memory)g(ob)5 b(ject,)29 b(one)523 5297
y(\014rst)g(assumes)e(a)i(set)g(of)f FG(primitive)37
b FK(shared)28 b(memory)g(ob)5 b(jects)28 b(used)h(in)g(the)g
(implemen-)p eop end
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(y)g(a)g FG(pr)l(oto)l(c)l(ol)523 606 y FK(whic)n(h)j(in)n(v)n(ok)n(es)
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b(Using)h(the)g(comp)r(ound)g(ob)5 b(ject)523 706 y(will)28
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FD(O)i FK(con)n(tains)d(all)g(op)r(erations)g(in)n(v)n(ok)n(ed)f
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1910 y FK(A)e(snapshot)f(memory)g(ob)5 b(ject)35 b(on)h
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(In)i(the)g(\014rst)g(case)e(it)523 2208 y(p)r(erforms)i(an)g
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(will)523 2607 y(complete)30 b(in)g(an)g(a)g(priori)f(b)r(ounded)h(n)n
(um)n(b)r(er)g(of)g(steps,)g(regardless)e(of)i(the)g(b)r(eha)n(viour)
523 2707 y(of)e(the)g(other)f(pro)r(cesses.)606 2806
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3005 y(time)33 b(the)f(op)r(eration)f(executed)1556 2975
y FB(2)1593 3005 y FK(.)h(This)g(allo)n(ws)f(us)h(to)g(`shrink')f(the)i
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(to)h(return)g(the)g(v)-5 b(alues)523 3205 y(written)29
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(next)i(paragraph)c(formalises)523 3304 y(this.)606 3404
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3504 y(tation)i(execution)f FD(hO)r FC(;)1302 3464 y
FA(o)1277 3504 y FD(!i)h FK(of)g(a)f(snapshot)g(ob)5
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3603 y(that)c FD(O)i FK(includes)e FC(n)f FK(initialising)h(up)r
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523 3703 y(op)r(erations)30 b(in)i FD(O)r FK(.)h(The)e(implemen)n
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(implemen)n(tation)i(of)f(the)h FC(n)f FK(pro)r(cess)f(w)n(ait-)523
4608 y(free)j(atomic)g(snapshot)g(ob)5 b(ject.)38 b(The)g
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(shared)523 4707 y(memory)29 b(ob)5 b(jects)30 b(used)g(b)n(y)g(the)g
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f(op)r(eration)f(on)g(the)523 4807 y(shared)f(memory)h(ob)5
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5 v 627 5208 a Fm(2)661 5240 y FF(Although)23 b(here)h(w)n(e)f(refer)i
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(e)f(app)r(eal,)h(w)n(e)523 5323 y(will)j(actually)f(pro)n(v)n(e)f
(atomicit)n(y)i(b)n(y)d(linearization)k(\(cf.)e([13)r(],)g(and)f(the)g
(previous)h(section\).)p eop end
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718 y FA(i)813 706 y FK([1])g(\(this)h(idea)f(w)n(as)g(indep)r(enden)n
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FC(T)12 b FK(,)30 b FG(val)1816 2478 y FA(i)1843 2466
y FK([0])h(and)g FG(val)2237 2478 y FA(i)2265 2466 y
FK([1],)f(written)i(b)n(y)e(pro)r(cess)g FC(P)3165 2478
y FA(i)3224 2466 y FK(and)731 2565 y(read)35 b(b)n(y)g(all.)h
(Initially)-7 b(,)36 b FG(val)1642 2577 y FA(i)1670 2565
y FK([1])f(ma)n(y)g(b)r(e)i(arbitrary)-7 b(,)34 b(but)i
FG(val)2751 2577 y FA(i)2779 2565 y FK([0])g(m)n(ust)g(b)r(e)g(ini-)731
2665 y(tialised)27 b(to)g(the)h(desired)g(initial)f(v)-5
b(alue)28 b(of)f(cell)h FC(i)f FK(of)h(the)g(snapshot)f(v)n(ector.)648
2831 y FD(\017)41 b FK(1)d(regular)f(register,)g FG(view)1600
2843 y FA(i)1666 2831 y FK(\(an)i FC(n)p FK(-v)-5 b(alue)38
b(v)n(ector)f(with)i(elemen)n(ts)g(of)f(t)n(yp)r(e)h
FC(T)12 b FK(\),)731 2931 y(written)27 b(b)n(y)h(pro)r(cess)e
FC(P)1479 2943 y FA(i)1535 2931 y FK(and)h(read)g(b)n(y)g(all,)h
(initially)f(arbitrary)-7 b(.)648 3097 y FD(\017)41 b
FK(for)30 b(eac)n(h)h FC(j)j FD(2)c(f)p FK(1)p FC(;)14
b(:)g(:)g(:)e(;)i(n)p FD(g)p FK(:)31 b(a)g(safe)g(bit)h
FC(c)2028 3109 y FA(ij)2117 3097 y FK(\(the)g(`complemen)n(t'-bit\),)g
(an)f(atomic)731 3196 y(bit)i FC(s)904 3208 y FA(ij)995
3196 y FK(\(the)h(`start'-bit\))f(and)f(a)h(regular)e(bit)i
FC(e)2289 3208 y FA(ij)2380 3196 y FK(\(the)h(`end'-bit\).)f(All)h
(written)731 3296 y(b)n(y)27 b(pro)r(cess)f FC(P)1188
3308 y FA(i)1216 3296 y FK(,)i(read)f(b)n(y)g(pro)r(cess)f
FC(P)1909 3308 y FA(j)1972 3296 y FK(and)i(initially)g(0.)523
3545 y Fl(8.4.2)99 b(The)34 b(Pr)-5 b(oto)g(c)g(ols)523
3694 y FK(Eac)n(h)31 b(of)h(the)h FC(n)f FK(pro)r(cesses)f
FC(P)1480 3706 y FA(i)1540 3694 y FK(can)h(execute)h(b)r(oth)f(up)r
(dates)h(and)f(scans)f(according)g(to)523 3794 y(the)d(follo)n(wing)f
(proto)r(cols)p 523 3872 1168 4 v 523 4818 4 947 v 551
3966 a FH(Pro)s(cedure)h FG(Up)l(date)1266 3987 y FA(i)1294
3966 y FK(\()p FG(value)7 b FK(\))704 4066 y FC(b)22
b FK(:=)h(1)18 b FD(\000)g FC(b)704 4166 y FK(write)27
b FG(val)1026 4178 y FA(i)1054 4166 y FK([)p FC(b)p FK(])c
FD( )g FG(value)704 4265 y(B-Up)l(date)1049 4286 y FA(i)1076
4265 y FK(\()p FC(b)p FK(\))704 4365 y FH(for)k FC(j)h
FD(2)c(f)p FK(1)p FC(;)14 b(:)g(:)g(:)f(;)h(n)p FD(g)26
b FH(do)847 4464 y FK(write)i FC(s)1099 4476 y FA(ij)1180
4464 y FD( )g FK(\(read)f FC(c)1544 4476 y FA(j)s(i)1602
4464 y FK(\))704 4564 y(write)g FG(view)1082 4576 y FA(i)1133
4564 y FD( )c FG(Sc)l(an)1415 4576 y FA(i)704 4664 y
FH(for)k FC(j)h FD(2)c(f)p FK(1)p FC(;)14 b(:)g(:)g(:)f(;)h(n)p
FD(g)26 b FH(do)847 4763 y FK(write)i FC(e)1099 4775
y FA(ij)1180 4763 y FD( )23 b FC(s)1325 4775 y FA(ij)p
1687 4818 V 523 4821 1168 4 v 1811 3872 1519 4 v 1811
4818 4 947 v 1867 3966 a FH(Pro)s(cedure)28 b FG(Sc)l(an)2501
3978 y FA(i)2019 4066 y FH(for)g FC(j)g FD(2)c(f)p FK(1)p
FC(;)14 b(:)g(:)g(:)e(;)i(n)p FD(g)27 b FH(do)2163 4166
y FK(write)h FC(c)2412 4178 y FA(ij)2493 4166 y FD( )23
b FK(1)p FD(\000)p FK(\(read)j FC(s)2961 4178 y FA(j)s(i)3020
4166 y FK(\))2019 4265 y FC(b)p FK([1)p FC(::n)p FK(])d(:=)f
FG(B-Sc)l(an)2637 4277 y FA(i)2019 4365 y FH(for)28 b
FC(j)g FD(2)c(f)p FK(1)p FC(;)14 b(:)g(:)g(:)e(;)i(n)p
FD(g)27 b FH(do)2163 4464 y FK(read)g FC(v)s FK([)p FC(j)5
b FK(])23 b FD( )h FG(val)2716 4476 y FA(j)2751 4464
y FK([)p FC(b)p FK([)p FC(j)5 b FK(]])2163 4564 y FH(if)28
b FC(c)2283 4576 y FA(ij)2369 4564 y FK(=)f(\(read)g
FC(s)2717 4576 y FA(j)s(i)2775 4564 y FK(\))h(=)g(\(read)f
FC(e)3184 4576 y FA(j)s(i)3242 4564 y FK(\))2163 4664
y FH(then)32 b(return)c FK(\(read)f FG(view)3058 4676
y FA(j)3093 4664 y FK(\))2019 4763 y FH(return)i FC(v)s
FK([1)p FC(::n)p FK(])p 3327 4818 V 1811 4821 1519 4
v 606 4942 a(The)f(Up)r(date-proto)r(col)f(uses)h(lo)r(cal)g(v)-5
b(ariables)26 b FC(j)34 b FK(\(ranging)26 b(o)n(v)n(er)h
FD(f)p FK(1)p FC(;)14 b(:)g(:)g(:)e(;)i(n)p FD(g)p FK(\),)28
b(and)g FC(b)p FK(,)523 5041 y(a)34 b(static)h(bit)g(v)-5
b(ariable)34 b(initially)g(0,)h(whic)n(h)f(retains)g(its)h(v)-5
b(alue)35 b(in)g(b)r(et)n(w)n(een)f(successiv)n(e)523
5141 y(in)n(v)n(o)r(cations)27 b(of)i(the)g(proto)r(col.)f(The)h
(Scan-proto)r(col)e(uses)h(lo)r(cal)h(v)-5 b(ariables)27
b FC(b)i FK(\(an)g FC(n)p FK(-bit)523 5241 y(v)n(ector\),)d
FC(j)32 b FK(\(ranging)26 b(o)n(v)n(er)g FD(f)p FK(1)p
FC(;)14 b(:)g(:)g(:)e(;)i(n)p FD(g)p FK(\),)27 b(and)g
FC(v)k FK(\(an)c FC(n)p FK(-v)-5 b(alue)26 b(v)n(ector)g(with)i(elemen)
n(ts)f(of)523 5340 y(t)n(yp)r(e)h FC(T)12 b FK(\).)p
eop end
%%Page: 116 126
TeXDict begin 116 125 bop 2483 299 a FF(8.4.)53 b(The)26
b(Solution)153 b(116)606 506 y FK(A)32 b(few)f(w)n(ords)f(on)h(the)h
(programming)d(notation)i(are)f(in)i(order.)e(Some)h(assignmen)n(ts)523
606 y(in)n(v)n(olv)n(e)23 b(b)r(oth)i(a)g(write)f(and)h(a)f(read)g(or)f
(Scan.)i(These)f(are)g(to)h(b)r(e)g(executed)f(sequen)n(tially)-7
b(,)523 706 y(the)30 b(read/Scan)e(\014rst)i(and)g(then)g(the)g(write.)
g(E.g.)f(`write)h FC(s)d FD( )f FK(\(read)k FC(c)p FK(\)')g(is)g
(shorthand)523 805 y(for)38 b(`)h(read)e FC(t)k FD( )h
FC(c)p FK(;)c(write)g FC(s)j FD( )h FC(t)p FK('.)c(This)h(should)f(not)
g(to)h(b)r(e)g(confused)f(with)h(read-)523 905 y(mo)r(dify-write)g(op)r
(erations)e(that)j(execute)f(atomically)-7 b(.)38 b(W)-7
b(e)39 b(assume)f(that)i(the)f(v)-5 b(alue)523 1005 y(of)34
b(a)g(shared)f(register)g(written)i(b)n(y)f(a)g(pro)r(cess)f(also)g(b)r
(elongs)h(to)g(that)g(pro)r(cess's)f(lo)r(cal)523 1104
y(state.)d(This)f(means)h(that)g(the)g(v)-5 b(alue)29
b(of)h(for)f(instance)h(the)g(shared)e(v)-5 b(ariable)29
b FC(c)3055 1116 y FA(ij)3143 1104 y FK(in)h(the)523
1204 y(Scan-proto)r(col)c(need)j(not)f(b)r(e)h(explicitly)f(read.)g
(The)g(return)g(statemen)n(ts)g(in)h(the)f(Scan-)523
1303 y(proto)r(col)d(serv)n(e)g(to)h(return)g(the)h(indicated)f(v)-5
b(alue)26 b(to)g(the)h(caller,)e(and)h(to)g(terminate)g(the)523
1403 y(proto)r(col)g(immediately)-7 b(.)606 1503 y(The)33
b(for)g(lo)r(ops)g(are)f(indexed)i(o)n(v)n(er)d(a)i(set)h(to)f(mak)n(e)
g(clear)f(that)i(the)f FC(n)h FK(lo)r(op)f(b)r(o)r(dies)523
1602 y(ma)n(y)d(b)r(e)h(in)n(terlea)n(v)n(ed)f(arbitrarily)-7
b(.)29 b(Since)i(the)g(registers)e(accessed)h(in)h(the)g(lo)r(op)g(b)r
(o)r(dies)523 1702 y(are)25 b(all)h(disjoin)n(t,)h(suc)n(h)f(a)g(for)f
(statemen)n(t)i(can)f(also)f(b)r(e)h(in)n(terpreted)g(as)g(a)g
(do-in-parallel)523 1802 y(construct.)21 b(Th)n(us)h(the)g(parallel)e
(time)i(complexit)n(y)f([2])h(of)f(snapshots)g(equals)g(the)h(parallel)
523 1901 y(time)28 b(complexit)n(y)f(of)h(binary)f(snapshots)f(\(up)i
(to)g(a)f(constan)n(t)g(factor\).)523 2208 y FL(8.5)116
b(Pro)s(of)37 b(of)h(Correctness)523 2408 y FK(T)-7 b(o)32
b(pro)n(v)n(e)f(correctness)g(w)n(e)h(assume)g(the)h(usual)f
(correctness)f(conditions)h(on)h(the)g(read)523 2507
y(write)25 b(registers)e(that)j(w)n(e)f(use)g(in)g(our)g(implemen)n
(tation.)g(W)-7 b(e)25 b(also)f(assume)h(the)g(correct-)523
2607 y(ness)j(of)h(the)g(atomic)f(binary)f(snapshot)h(ob)5
b(ject)29 b(used)f(b)n(y)g(our)g(implemen)n(tation.)h(I.e.)f(in)523
2707 y(an)d(execution)g FD(hA)p FC(;)14 b FD(!i)27 b
FK(w)n(e)e(assume)g(there)g(exists)g(a)g(total)g(order)g
FD(\))g FK(extending)h FD(!)f FK(suc)n(h)523 2806 y(that)f(ev)n(ery)f
(binary)g(scan)g FC(B)t(S)28 b FK(returns)c(for)f(bit)h
FC(i)g FK(the)g(v)-5 b(alue)23 b(written)h(b)n(y)g(the)g(last)g(binary)
523 2906 y(up)r(date)k FC(B)t(U)926 2918 y FA(i)981 2906
y FK(executed)g(b)n(y)f FC(P)1493 2918 y FA(i)1549 2906
y FK(preceding)g FC(B)t(S)32 b FK(in)c FD(\))p FK(.)606
3005 y(W)-7 b(e)30 b(write)g FC(U)39 b FK(for)29 b FG(Up)l(date)37
b FK(and)30 b FC(S)35 b FK(for)29 b FG(Sc)l(an)6 b FK(.)31
b(F)-7 b(or)29 b(op)r(eration)g FC(O)g FD(2)f(f)p FC(U;)14
b(S;)g(B)t(U;)g(B)t(S)5 b FD(g)p FK(,)523 3105 y FC(O)588
3075 y FA(x)586 3127 y(i)669 3105 y FK(denotes)39 b(the)g
FC(x)p FK(-th)g(execution)f(of)h FC(O)i FK(b)n(y)d(pro)r(cess)g
FC(P)2399 3117 y FA(i)2427 3105 y FK(,)h(including)f(scans)g
FC(S)3139 3117 y FA(i)3205 3105 y FK(that)523 3205 y(are)32
b(in)n(v)n(ok)n(ed)f(b)n(y)i(some)f(up)r(date)h FC(U)1658
3165 y FA(y)1649 3228 y(i)1698 3205 y FK(.)g(These)f(scans)g(are)g
(sometimes)g(written)h(as)f FG(US)3295 3165 y FA(y)3295
3228 y(i)3335 3205 y FK(.)523 3304 y(Note)h(that)g FC(B)t(S)1037
3274 y FA(x)1032 3326 y(i)1112 3304 y FK(is)g(in)n(v)n(ok)n(ed)e(b)n(y)
i FC(S)1685 3274 y FA(x)1680 3326 y(i)1726 3304 y FK(,)h(and)e
FC(B)t(U)2082 3274 y FA(x)2073 3326 y(i)2157 3304 y FK(is)h(in)n(v)n
(ok)n(ed)e(b)n(y)i FC(U)2740 3274 y FA(x)2731 3326 y(i)2782
3304 y FK(.)g FD(O)i FK(con)n(tains)d(all)523 3404 y(in)n(v)n(o)r
(cations)20 b FC(S)1007 3374 y FA(x)1002 3426 y(i)1069
3404 y FK(and)h FC(U)1290 3374 y FA(x)1281 3426 y(i)1353
3404 y FK(for)g FC(i)i FD(2)g(f)p FK(1)p FC(;)14 b(:)g(:)g(:)f(;)h(n)p
FD(g)20 b FK(and)h FC(x)j FD(\025)e FK(0.)f(Note)h(that)f(this)h(also)e
(includes)523 3504 y(up)r(dates)26 b FC(U)899 3473 y
FB(0)890 3525 y FA(i)963 3504 y FK(that)g(wrote)g(the)g(initial)h(v)-5
b(alues)25 b(for)h(the)h(cells)f FC(i)p FK(,)g(and)g(scans)f
FG(US)3015 3467 y FA(x)3015 3524 y(i)3083 3504 y FK(in)n(v)n(ok)n(ed)
523 3603 y(b)n(y)i(up)r(dates)h FC(U)1016 3573 y FA(x)1007
3625 y(i)1058 3603 y FK(.)606 3703 y(If)23 b(scan)g FC(S)921
3673 y FA(x)916 3724 y(i)985 3703 y FK(sees)g FC(c)1184
3715 y FA(ij)1265 3703 y FK(=)g(\(read)f FC(s)1604 3715
y FA(j)s(i)1662 3703 y FK(\))i(=)e(\(read)h FC(e)2057
3715 y FA(j)s(i)2115 3703 y FK(\),)g(then)h(pro)r(cess)d
FC(P)2715 3715 y FA(j)2774 3703 y FK(\(or)h(some)g(up)r(date)523
3802 y FC(U)580 3814 y FA(j)615 3802 y FK(\))31 b(is)g(said)g(to)g
FG(interfer)l(e)38 b FK(with)32 b FC(S)1637 3772 y FA(x)1632
3824 y(i)1679 3802 y FK(.)f(A)h(scan)e(is)h FG(dir)l(e)l(ct)40
b FK(if)32 b(no)f(pro)r(cess)f(in)n(terferes)g(with)523
3902 y(it.)f FC(S)686 3872 y FA(x)681 3924 y(i)755 3902
y FG(c)l(ontains)35 b FC(S)1145 3862 y FA(y)1140 3925
y(j)1213 3902 y FK(i\013)28 b FC(s)p FK(\()p FC(S)1439
3872 y FA(x)1434 3924 y(i)1481 3902 y FK(\))c FC(<)f(s)p
FK(\()p FC(S)1752 3862 y FA(y)1747 3925 y(j)1792 3902
y FK(\))h FC(<)f(f)9 b FK(\()p FC(S)2074 3862 y FA(y)2069
3925 y(j)2114 3902 y FK(\))24 b FC(<)f(f)9 b FK(\()p
FC(S)2396 3872 y FA(x)2391 3924 y(i)2437 3902 y FK(\).)29
b(The)f(next)g(lemma)g(sho)n(ws)523 4002 y(that)g(direct)f(scans)g
(will)h(return)f(correct)f(v)-5 b(alues.)523 4151 y Fv(Lemma)32
b FH(8.1)40 b FG(Assume)27 b FC(P)1357 4163 y FA(j)1420
4151 y FG(do)l(es)i(not)e(interfer)l(e)h(with)g(some)g(sc)l(an)g
FC(S)2707 4121 y FA(x)2702 4173 y(i)2749 4151 y FG(,)g(and)g(let)g
FC(S)3132 4121 y FA(x)3127 4173 y(i)3201 4151 y FG(sc)l(an)523
4261 y(the)e(value)h FC(b)p FK([)p FC(j)5 b FK(])25 b
FG(up)l(date)l(d)i(by)f(some)h FC(U)1691 4221 y FA(y)1682
4284 y(j)1730 4261 y FG(,)g(i.e.)g FC(B)t(U)2055 4221
y FA(y)2046 4284 y(j)2118 4261 y FD(\))c FC(B)t(S)2347
4230 y FA(x)2342 4282 y(i)2412 4261 y FD(\))g FC(B)t(U)2651
4221 y FA(y)r FB(+1)2642 4284 y FA(j)2775 4261 y FG(.)j(Then)h(the)f
(value)523 4360 y(val)633 4372 y FA(j)668 4360 y FK([)p
FC(b)p FK([)p FC(j)5 b FK(]])30 b FG(r)l(e)l(ad)g(by)g
FC(S)1208 4330 y FA(x)1203 4382 y(i)1280 4360 y FG(was)g(written)f
(ther)l(e)h(by)g FC(U)2107 4320 y FA(y)2098 4383 y(j)2147
4360 y FG(.)523 4510 y Fv(Pr)n(oof.)19 b FK(Assume)h(scan)f
FC(S)1362 4479 y FA(x)1357 4531 y(i)1423 4510 y FK(do)r(es)g(not)h(see)
f FC(c)1904 4522 y FA(ij)1982 4510 y FK(=)h(\(read)f
FC(s)2315 4522 y FA(j)s(i)2373 4510 y FK(\))h(=)f(\(read)g
FC(e)2757 4522 y FA(j)s(i)2816 4510 y FK(\))h(and)f(that)h
FC(B)t(S)3316 4479 y FA(x)3311 4531 y(i)523 4619 y FK(scanned)27
b(the)h(v)-5 b(alue)28 b FC(b)p FK([)p FC(j)5 b FK(])27
b(for)g(cell)h FC(j)k FK(up)r(dated)c(b)n(y)g FC(B)t(U)2258
4579 y FA(y)2249 4642 y(j)2298 4619 y FK(,)f(i.e.)h FC(B)t(U)2615
4579 y FA(y)2606 4642 y(j)2678 4619 y FD(\))23 b FC(B)t(S)2907
4589 y FA(x)2902 4641 y(i)2972 4619 y FD(\))g FC(B)t(U)3211
4579 y FA(y)r FB(+1)3202 4642 y FA(j)3335 4619 y FK(.)606
4719 y(The)i(write)g(of)f FG(val)1186 4731 y FA(j)1221
4719 y FK([)p FC(b)p FK(])g(b)n(y)h FC(U)1506 4679 y
FA(y)1497 4742 y(j)1570 4719 y FK(precedes)f FC(B)t(U)2033
4679 y FA(y)2024 4742 y(j)2098 4719 y FK(in)h FD(!)p
FK(,)g(and)g(the)g(read)f(of)h FG(val)3006 4731 y FA(j)3041
4719 y FK([)p FC(b)p FK(])g(b)n(y)f FC(S)3316 4688 y
FA(x)3311 4740 y(i)523 4818 y FK(follo)n(ws)k FC(B)t(S)920
4788 y FA(x)915 4840 y(i)991 4818 y FK(in)h FD(!)p FK(.)g(Since)g
FC(B)t(U)1575 4778 y FA(y)1566 4841 y(j)1640 4818 y FD(\))d
FC(B)t(S)1872 4788 y FA(x)1867 4840 y(i)1913 4818 y FK(,)k(w)n(e)e(ha)n
(v)n(e)g FD(:)p FK(\()p FC(B)t(S)2492 4788 y FA(x)2487
4840 y(i)2560 4818 y FD(!)d FC(B)t(U)2801 4778 y FA(y)2792
4841 y(j)2841 4818 y FK(\),)k(so)g(the)g(write)523 4918
y(of)d FG(val)726 4930 y FA(j)761 4918 y FK([)p FC(b)p
FK(])g(b)n(y)g FC(U)1049 4878 y FA(y)1040 4941 y(j)1114
4918 y FK(precedes)f(the)i(read)e(of)g(it)i(b)n(y)e FC(S)2113
4888 y FA(x)2108 4939 y(i)2155 4918 y FK(.)h(So)f(if)i
FC(S)2448 4888 y FA(x)2443 4939 y(i)2515 4918 y FK(do)r(es)f(not)g
(read)f(the)h(v)-5 b(alue)523 5017 y(written)26 b(b)n(y)g
FC(U)992 4978 y FA(y)983 5041 y(j)1058 5017 y FK(it)g(m)n(ust)g(b)r(e)h
(concurren)n(t)d(with)j(or)e(o)r(ccur)g(after)h(a)g(write)f(to)h
FG(val)3060 5029 y FA(j)3095 5017 y FK([)p FC(b)p FK(])f(b)n(y)h(a)523
5138 y(later)f(up)r(date)h FC(U)1058 5107 y FA(z)1049
5159 y(j)1121 5138 y FK(Note)g(that)g(this)f(later)g(up)r(date)h
(cannot)f(b)r(e)h FC(U)2573 5098 y FA(y)r FB(+1)2564
5161 y FA(j)2697 5138 y FK(,)f(since)h(this)f(up)r(date)523
5237 y(will)j(write)f(to)h FG(val)1104 5249 y FA(j)1139
5237 y FK([1)18 b FD(\000)g FC(b)p FK(].)606 5337 y(Supp)r(ose)33
b(the)g(read)e(of)i FG(val)1484 5349 y FA(j)1519 5337
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%%Page: 117 127
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b Fu(2)p eop end
%%Page: 118 128
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%%Page: 119 129
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FG(Comp)l(osite)k(r)l(e)l(gisters)p FK(,)d(Pro)r(c.)e(9th)h(A)n(CM)h
(Symp)r(osium)g(on)f(Prin-)694 5011 y(ciples)j(of)h(Distributed)g
(Computing,)g(pp.)g(15{29,)d(1990.)565 5169 y([9])41
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FC(x)2810 1216 y FA(i)2873 1204 y FK(p)r(oin)n(ts)g(ev)n(enly)523
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(e)h(generalized.)523 2889 y(One)35 b(w)n(ould)f(b)r(e)i(a)f(sc)n(heme)
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FC(X)3307 3001 y FA(i)3335 2989 y FD(j)523 3088 y FK(equal)24
b FC(n)p FK(,)h(but)g(in)g(general)e(the)i(condition)1864
3026 y Fn(P)1952 3113 y FA(i)p Fw(2)p FA(I)2072 3088
y FK(1)p FC(=)p FD(j)p FC(X)2248 3100 y FA(i)2275 3088
y FD(j)e(\024)f FK(1)i(will)h(not)g(b)r(e)g(su\016cien)n(t)g(an)n(y)523
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(the)g(study)g(of)g(general)f(sc)n(hemes)g(where)h FC(i)f
FK(can)h(c)n(ho)r(ose)523 3387 y(its)29 b(lab)r(el)h(so)e(as)h(to)g
(satisfy)g(some)f(constrain)n(t)g(on)h(a)g(function)h(of)f(all)g(the)g
(lab)r(els,)g(where)523 3487 y(this)f(function)g(migh)n(t)g(represen)n
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(o-)523 3421 y(ordigt,)27 b(onafhank)n(elijk)f(v)-5 b(an)28
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(complexiteit)h(v)-5 b(an)35 b(staartdelen)g(is)g(v)-5
b(an)36 b(de)f(order)f(het)523 4517 y(pro)r(dukt)e(v)-5
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(cijfers)h(in)g(teller.)g(Een)g(complex-)523 4617 y(iteits)e(maat)e
(zegt)h(altijd)h(ho)r(e)f(het)g(gebruik)f(v)-5 b(an)29
b(middelen)h(to)r(eneem)n(t)f(met)h(de)f(gro)r(otte)523
4717 y(v)-5 b(an)28 b(de)f(probleem)g(instan)n(tie.)606
4816 y(Naast)f(geheugen)g(en)h(tijd)g(k)n(omen)f(o)r(ok)g(minder)h(b)r
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4916 y(energiegebruik)f(in)i(electronisc)n(he)e(circuits.)606
5015 y(Zelfs)e(de)h(gro)r(otte)e(v)-5 b(an)25 b(de)g(oplossing)e(k)-5
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5115 y(ingev)-5 b(al)24 b(die)i(oplossing)d(niet)i(uniek)g(is.)g(Zo)g
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5215 y(leem)37 b(b)r(estudeerd:)f(gegev)n(en)f(een)h(v)n(erzameling)f
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b(w)2654 5227 y FB(2)2691 5215 y FC(;)g(:)g(:)g(:)g(;)g(w)2935
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5326 y FA(i)1472 5314 y FK(deel)h(v)-5 b(an)36 b(uit)h(mak)n(en.)e(Zo)h
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end
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2300 y(in)i(de)h(kleuren)e(wit)i(en)f(zw)n(art,)f(w)n(aarbij)g(de)h
(hele)h(witte)f(omgeving)f(v)-5 b(an)27 b(de)g(startp)r(ositie)523
2399 y(\(die)h(dus)g(b)r(egrensd)f(is)g(do)r(or)g(zw)n(arte)f(stukk)n
(en\))i(zw)n(art)e(gev)n(erfd)h(mo)r(et)h(w)n(orden.)606
2499 y(Ho)r(ofdstuk)41 b(4)f(gaat)g(v)n(erv)n(olgens)d(in)k(op)g(de)f
(sc)n(hak)n(el-energie)e(v)-5 b(an)41 b(circuits)f(die)h(de)523
2599 y(binaire)28 b(\(2)g(w)n(aardig;)f(0)h(of)g(1\))h(of-functie)g
(\(en)f(generalisaties)f(daarv)-5 b(an\))27 b(b)r(erek)n(enen)h(op)523
2698 y FC(n)d FK(inputs.)g(Do)r(or)g(een)g(kleine)f(redundan)n(tie)h
(in)g(de)g(ho)r(ev)n(eelheid)g(sc)n(hak)n(el-elemen)n(ten)e(k)-5
b(an)523 2798 y(de)23 b(ho)r(ev)n(eelheid)f(b)r(edrading)h(die)g(omsc)n
(hak)n(elt)e(bij)j(v)n(erandering)c(v)-5 b(an)23 b(de)g(inputs)g(met)h
(een)523 2897 y(factor)j FC(l)i FK(gereduceerd)d(w)n(orden,)h(w)n
(aarbij)f FC(l)j FK(het)f(aan)n(tal)e(cijfers)i(v)-5
b(an)27 b FC(n)h FK(is.)606 2997 y(In)35 b(Ho)r(ofdstuk)g(5)f(w)n(ordt)
g(een)h(nieu)n(w)g(so)r(ort)e(computer)i(v)n(o)r(orgesteld,)e(w)n(aarv)
-5 b(an)32 b(het)523 3097 y(geheugen)18 b(b)r(estaat)h(uit)h(`cellen')f
(die)g(elk)-5 b(aar)18 b(aan)g(kunnen)i(wijzen,)f(elk)n(e)g(cel)g
(wijst)g(bijv)n(o)r(or-)523 3196 y(b)r(eeld)h(naar)e(3)h(andere,)f(met)
i(een)f(ro)r(de,)g(gro)r(ene,)f(en)i(blau)n(w)n(e)e(wijzer.)h(Op)g(elk)
h(momen)n(t)f(v)-5 b(an)523 3296 y(de)31 b(b)r(erek)n(ening)f(is)h(er)f
(een)h(sp)r(eciale)f(cel)h(v)-5 b(an)31 b(w)n(aaruit)e(de)i(andere)f
(ge-adresseerd)e(kun-)523 3396 y(nen)j(w)n(orden.)g(Niet)g(alleen)g
(enk)n(ele)g(cellen)g(kunnen)g(ge-adresseeerd)d(w)n(orden,)i(o)r(ok)h
(k)-5 b(an)523 3495 y(v)g(an)n(uit)26 b(een)h(cel)f(alle)g(cellen)g
(die)h(er)e(naar)g(wijzen,)i(b)r(ev)n(o)r(orb)r(eeld)f(met)g(de)h(gro)r
(ene)e(wijzer,)523 3595 y(tegelijk)20 b(ge-adresseerd)d(w)n(orden.)i
(Op)h(een)g(v)n(erzameling)e(ge-adresseerde)f(cellen)j(kunnen)523
3694 y(allerlei)k(op)r(eraties)g(w)n(orden)g(uitgev)n(o)r(erd,)g(zoals)
f(het)j(cre)n(\177)-39 b(eren)22 b(v)-5 b(an)25 b(nieu)n(w)n(e)g
(cellen,)g(of)f(het)523 3794 y(omleggen)k(v)-5 b(an)29
b(wijzers.)g(Dit)h(blijkt)g(een)f(bijzonder)g(krac)n(h)n(tige)f(so)r
(ort)g(computer)h(op)f(te)523 3894 y(lev)n(eren,)23 b(die)h(bijv)n(o)r
(orb)r(eeld)f(in)h(een)g(relatief)g(klein)g(aan)n(tal)e(stapp)r(en)i
(het)h(sc)n(haaksp)r(el)d(zou)523 3993 y(kunnen)32 b(oplossen)e(\(ho)r
(ew)n(el)i(het)g(aan)n(tal)e(cellen)i(w)n(at)f(in)h(die)g(b)r(erek)n
(ening)f(aangemaakt)523 4093 y(w)n(ordt)c(w)n(el)g(v)n(ersc)n(hrikk)n
(elijk)e(gro)r(ot)i(w)n(ordt.\))606 4193 y(In)i(de)h(laatste)e(4)h(ho)r
(ofdstukk)n(en)f(k)n(omen)h(comm)n(unicatie)g(proto)r(collen)f(aan)g(b)
r(o)r(d,)i(die)523 4292 y(v)n(ersc)n(hillende)21 b(pro)r(cessen)g(de)i
(indruk)f(gev)n(en)f(dat)i(ze)f(gezamenlijk)d(\023)-39
b(e)n(\023)g(en)21 b(geheugen)h(delen.)523 4392 y(W)-7
b(at)34 b(de)h(een)f(erin)f(sc)n(hrijft)i(k)-5 b(an)34
b(een)g(ander)f(dan)h(lezen.)g(Het)h(b)r(egin)n(t)f(heel)g(een)n(v)n
(oudig)523 4491 y(met)25 b(een)f(enk)n(el)f(bit)i(\(2-w)n(aardig)c
(geheugenelemen)n(t\))j(dat)g(do)r(or)f(slec)n(h)n(ts)h(een)g(pro)r
(ces)f(k)-5 b(an)523 4591 y(w)n(orden)24 b(gesc)n(hrev)n(en)e(en)j(do)r
(or)f(een)g(ander)g(gelezen.)g(V)-7 b(an)n(uit)25 b(deze)g(b)r(ou)n
(wstenen)f(kunnen)523 4691 y(geheugens)h(met)i(v)n(eel)e(meer)h
(mogelijkheden)g(w)n(orden)f(geb)r(ou)n(wd:)g(meer)h(w)n(aardes,)e
(meer)523 4790 y(lezers,)g(meer)h(sc)n(hrijv)n(ers,)f(enz.)h(Het)h
(uniek)n(e)f(v)-5 b(an)25 b(deze)g(proto)r(collen)f(is)h(dat)h(ze)f
FG(wachtvrij)523 4890 y FK(zijn,)35 b(w)n(at)e(wil)i(zeggen)e(dat)h
(het)g(ene)h(pro)r(ces)e(no)r(oit)h(op)g(het)g(ander)g(ho)r(eft)g(te)h
(w)n(ac)n(h)n(ten)523 4990 y(om)30 b(zijn)h(geheugen-op)r(eratie)d(te)i
(kunnen)h(v)n(olto)r(oien.)e(Er)g(k)-5 b(an)30 b(rustig)f(tegelijk)n
(ertijd)h(in)523 5089 y(gelezen)d(en)h(gesc)n(hrev)n(en)d(w)n(orden,)i
(zonder)g(dat)g(er)g(onzin)h(uit)g(k)n(om)n(t.)p eop
end
%%Page: 127 137
TeXDict begin 127 136 bop 3357 223 2 75 v 3283 225 150
2 v 3357 300 2 75 v 523 543 a FI(Curriculum)52 b(Vitae)523
2516 y FK(Johannes)26 b(Theo)r(dorus)h(T)-7 b(romp)523
2715 y(13)27 b(mei)h(1966:)108 b(geb)r(oren)27 b(te)h(Alkmaar)523
2914 y(1978{1984:)166 b(A)n(theneum)1090 3014 y(Han)28
b(F)-7 b(ortman)27 b(College)g(te)g(Heerh)n(ugo)n(w)n(aard)523
3213 y(1984{1989)189 b(Do)r(ctoraal)27 b(Informatica)1090
3313 y(Univ)n(ersiteit)h(v)-5 b(an)27 b(Amsterdam)523
3512 y(1989{1993)189 b(Onderzo)r(ek)n(er)26 b(in)i(opleiding)1090
3611 y(in)g(het)g(pro)5 b(ject)27 b(\\Algorithms)g(and)g(Complexit)n
(y")1090 3711 y(v)-5 b(an)28 b(het)g(CWI)g(te)g(Amsterdam)p
eop end
%%Trailer

userdict /end-hook known{end-hook}if
%%EOF