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1673 y Fj(k)r Fg(+1)375 1753 y Fj(i)p Fg(=1)541 1723 y Fk(X)624 1738 y Fj(i)685 1723 y Fk(>)28 b Fn(\()p Fk(n)22 b Fm(\000)g Fn(1\))p Fk(\030)1148 1613 y Fh(o)1213 1723 y Fk(;)44 b Fn(where)g Fk(X)1666 1738 y Fj(i)1726 1660 y Fe(L)1715 1716 y Fn(=)1809 1723 y Fk(X)8 b Fn(,)45 b(for)f Fk(i)h Fn(=)h(1)p Fk(;)17 b(:)g(:)g(:)d(;)j(k)s Fn(,)45 b(and)f(the)g Fk(X)3207 1738 y Fj(i)3239 1723 y Fn('s)g(are)f(indep)s(en-)0 1882 y(den)m(t.)i(Then)34 b(b)m(y)f(Mark)m(o)m(v's)g(and)h(Jensen's)h (inequalities,)d(for)i(all)e Fk(n)h Fn(large)g(enough,)389 2306 y Fk(b)432 2264 y Fj(k)497 2306 y Fi(P)592 2135 y Fh(\()673 2181 y Fj(k)r Fg(+1)673 2211 y Fh(X)680 2423 y Fj(i)p Fg(=1)834 2306 y Fk(X)917 2321 y Fj(i)977 2306 y Fk(>)c Fn(\()p Fk(n)21 b Fm(\000)i Fn(1\))p Fk(\030)1441 2135 y Fh(\))1548 2306 y Fm(\024)1665 2172 y Fk(b)1708 2136 y Fj(k)1773 2181 y Fi(E)1865 2032 y Fh(\032)1940 2062 y(\020)1999 2097 y(P)2104 2122 y Fj(k)r Fg(+1)2104 2202 y Fj(i)p Fg(=1)2271 2172 y Fk(X)2354 2187 y Fj(i)2386 2062 y Fh(\021)2446 2082 y Fj(\022)2491 2032 y Fh(\033)p 1665 2283 902 4 v 1875 2374 a Fn(\(\()p Fk(n)f Fm(\000)g Fn(1\))p Fk(\030)5 b Fn(\))2311 2345 y Fj(\022)1548 2631 y Fm(\024)1665 2558 y Fk(b)1708 2521 y Fj(k)1756 2558 y Fn(\()p Fk(k)25 b Fn(+)e(1\))2061 2521 y Fj(\022)2122 2567 y Fi(E)2214 2477 y Fh(\010)2272 2558 y Fk(X)2363 2521 y Fj(\022)2408 2477 y Fh(\011)p 1665 2608 802 4 v 1825 2700 a Fn(\(\()p Fk(n)f Fm(\000)g Fn(1\))p Fk(\030)5 b Fn(\))2261 2671 y Fj(\022)1548 2937 y Fm(\024)1665 2863 y Fk(b)1708 2827 y Fe(b)p Fj(d)13 b Fg(log)1902 2845 y Fa(b)1939 2827 y Fg(\()p Fj(n)p Fe(\000)p Fg(1\))p Fe(c)2210 2863 y Fn(\()p Fm(b)p Fk(d)k Fn(log)2490 2887 y Fj(b)2530 2863 y Fn(\()p Fk(n)22 b Fm(\000)h Fn(1\))p Fm(c)f Fn(+)g(1\))3094 2813 y Fj(\022)3156 2872 y Fi(E)3247 2782 y Fh(\010)3306 2863 y Fk(X)3397 2827 y Fj(\022)3441 2782 y Fh(\011)p 1665 2914 1835 4 v 2342 3005 a Fn(\(\()p Fk(n)f Fm(\000)i Fn(1\))p Fk(\030)5 b Fn(\))2778 2976 y Fj(\022)1548 3228 y Fm(\024)1665 3154 y Fn(\(2)p Fk(d)p Fn(\))1845 3118 y Fj(\022)1905 3163 y Fi(E)1997 3074 y Fh(\010)2055 3154 y Fk(X)2146 3118 y Fj(\022)2191 3074 y Fh(\011)p 1665 3205 585 4 v 1910 3297 a Fk(\030)1959 3268 y Fj(\022)2273 3161 y Fn(\(log)2440 3185 y Fj(b)2480 3161 y Fn(\()p Fk(n)22 b Fm(\000)h Fn(1\)\))2829 3125 y Fj(\022)p 2273 3205 601 4 v 2344 3297 a Fn(\()p Fk(n)f Fm(\000)g Fn(1\))2653 3268 y Fj(\022)r Fe(\000)p Fj(d)1548 3520 y Fm(\024)1665 3446 y Fn(\(2)p Fk(d)p Fn(\))1845 3410 y Fj(\022)1905 3455 y Fi(E)1997 3365 y Fh(\010)2055 3446 y Fk(X)2146 3410 y Fj(\022)2191 3365 y Fh(\011)p 1665 3497 585 4 v 1910 3588 a Fk(\030)1959 3559 y Fj(\022)2425 3452 y Fn(1)p 2273 3497 354 4 v 2273 3588 a(\()p Fk(n)g Fm(\000)g Fn(1\))2582 3559 y Fg(2)2638 3520 y Fk(:)p 3840 3745 60 5 v 3840 3795 5 50 v 3895 3795 V 3840 3800 60 5 v 0 4013 a Fo(Pr)n(oposition)48 b(2.)55 b Ff(Let)44 b Fk(T)1040 4028 y Fj(n)1094 4013 y Ff(,)i Fk(N)1248 4028 y Fj(k)1297 4013 y Ff(,)g Fk(k)s Ff(,)f Fk(\022)h Ff(and)e Fk(d)f Ff(as)h(in)f(the)h(previous)g(prop)s(osition.)74 b(Then,)46 b(for)e(all)0 4172 y Fk(\030)32 b(>)c Fn(0)p Ff(,)33 b(there)h(is)f Fk(')28 b Fm(2)g Fn(\(0)p Fk(;)17 b Fn(1\))p Ff(,)31 b(suc)m(h)k(that)745 4447 y Fi(P)17 b Fm(f)o Fk(M)986 4462 y Fj(n;)p Fg(1)1131 4447 y Fk(>)28 b Fn(\()p Fk(n)22 b Fm(\000)h Fn(1\)\()p Fk(\013)e Fn(+)i Fk(\030)5 b Fn(\))p Fk(;)17 b(N)1983 4462 y Fj(k)2058 4447 y Fm(\024)28 b Fn(\()p Fk(n)22 b Fm(\000)g Fn(1\))p Fk(\030)5 b(=)p Fn(2)p Fm(g)26 b(\024)i Fk(')2867 4406 y Fg(\()p Fj(n)p Fe(\000)p Fg(1\))3080 4376 y Fa(d)3128 4447 y Fk(;)0 4723 y Ff(for)33 b(all)g Fk(n)g Ff(large)g(enough.)0 4941 y Fo(Pr)n(oof.)48 b Fn(Denote)28 b(b)m(y)g Fk(v)908 4956 y Fg(1)953 4941 y Fk(;)17 b(:)g(:)g(:)d(;)j(v)1223 4961 y Fj(b)1258 4941 y Fa(k)37 b Fn(all)27 b(no)s(des)i(in)e Fk(T)1909 4956 y Fj(n)1991 4941 y Fn(at)g(depth)i Fk(k)s Fn(.)42 b(De\014ne)29 b Fk(B)2893 4905 y Fj(i)2888 4969 y(n)p Fe(\000)p Fj(k)3076 4941 y Fn(to)e(b)s(e)h(the)g(minim)m(um)0 5100 y(of)33 b(all)g Fk(V)310 5115 y Fj(v)350 5125 y Fa(i)381 5115 y Fj(;w)503 5100 y Fn(suc)m(h)h(that)f Fk(w)k Fn(is)c(a)g(leaf)g(descendan)m(t)j(of)d Fk(v)2093 5115 y Fj(i)2127 5100 y Fn(.)44 b(Then,)33 b(b)m(y)h(indep)s(endence)i (of)d(the)h Fk(B)3608 5064 y Fj(i)3603 5128 y(n)p Fe(\000)p Fj(k)3763 5100 y Fn('s,)1900 5339 y(10)p eop %%Page: 11 11 11 10 bop 15 375 a Fi(P)110 285 y Fh(\010)168 366 y Fk(M)265 381 y Fj(n;)p Fg(1)410 366 y Fk(>)28 b Fn(\()p Fk(n)22 b Fm(\000)g Fn(1\)\()p Fk(\013)g Fn(+)g Fk(\030)5 b Fn(\))p Fk(;)32 b(N)1276 381 y Fj(k)1353 366 y Fm(\024)c Fn(\()p Fk(n)22 b Fm(\000)g Fn(1\))p Fk(\030)5 b(=)p Fn(2)1916 285 y Fh(\011)195 555 y Fm(\024)29 b Fi(P)396 444 y Fh(n)462 555 y Fk(B)543 513 y Fg(1)538 579 y Fj(n)p Fe(\000)p Fj(k)720 555 y Fn(+)23 b(\()p Fk(n)e Fm(\000)i Fn(1\))p Fk(\030)5 b(=)p Fn(2)26 b Fk(>)i Fn(\()p Fk(n)22 b Fm(\000)h Fn(1\)\()p Fk(\013)e Fn(+)i Fk(\030)5 b Fn(\))p Fk(;)17 b(:)g(:)g(:)c(;)k(B)2333 513 y Fj(b)2368 483 y Fa(k)2328 579 y Fj(n)p Fe(\000)p Fj(k)2509 555 y Fn(+)23 b(\()p Fk(n)f Fm(\000)g Fn(1\))p Fk(\030)5 b(=)p Fn(2)26 b Fk(>)i Fn(\()p Fk(n)22 b Fm(\000)h Fn(1\)\()p Fk(\013)e Fn(+)i Fk(\030)5 b Fn(\))3820 444 y Fh(o)195 774 y Fn(=)29 b(\()o Fi(P)17 b Fm(f)p Fk(B)560 789 y Fj(n)p Fe(\000)p Fj(k)748 774 y Fk(>)28 b Fn(\()p Fk(n)22 b Fm(\000)g Fn(1\)\()p Fk(\013)g Fn(+)g Fk(\030)5 b(=)p Fn(2\))p Fm(g)o Fn(\))1662 724 y Fj(b)1697 694 y Fa(k)195 963 y Fm(\024)29 b Fn(\()o Fi(P)17 b Fm(f)p Fk(B)560 978 y Fj(n)p Fe(\000)p Fj(k)748 963 y Fk(>)28 b Fn(\()p Fk(n)22 b Fm(\000)g Fk(k)s Fn(\)\()p Fk(\013)g Fn(+)h Fk(\030)5 b(=)p Fn(2\))p Fm(g)n Fn(\))1667 912 y Fj(b)1702 882 y Fa(k)1767 963 y Fk(;)0 1229 y Fn(where)28 b Fk(B)358 1244 y Fj(n)p Fe(\000)p Fj(k)545 1229 y Fn(is)f(the)g (random)f(v)-6 b(ariable)27 b(de\014ned)i(in)e(Theorem)f(3.)42 b(Substituting)27 b Fk(k)j Fn(b)m(y)e Fm(b)p Fk(d)17 b Fn(log)3532 1253 y Fj(b)3572 1229 y Fn(\()p Fk(n)10 b Fm(\000)g Fn(1\))p Fm(c)0 1388 y Fn(and)34 b(using)f(Theorem)g(3)g(w) m(e)h(get)f(that)352 1654 y Fi(P)447 1573 y Fh(\010)505 1654 y Fk(B)581 1672 y Fj(n)p Fe(\000b)p Fj(d)14 b Fg(log)887 1691 y Fa(b)923 1672 y Fg(\()p Fj(n)p Fe(\000)p Fg(1\))p Fe(c)1206 1654 y Fk(>)28 b Fn(\()p Fk(n)22 b Fm(\000)g(b)p Fk(d)17 b Fn(log)1773 1678 y Fj(b)1813 1654 y Fn(\()p Fk(n)22 b Fm(\000)g Fn(1\))p Fm(c)p Fn(\)\()p Fk(\013)g Fn(+)g Fk(\030)5 b(=)p Fn(2\))2617 1573 y Fh(\011)2702 1654 y Fm(\000)-16 b(!)27 b Fn(0)p Fk(;)116 b Fn(as)33 b Fk(n)27 b Fm(!)h(1)p Fk(:)0 1920 y Fn(Th)m(us)34 b(w)m(e)g(can)g (\014nd)g Fk(')28 b Fm(2)g Fn(\(0)p Fk(;)17 b Fn(1\))31 b(suc)m(h)k(that)676 2187 y Fi(P)771 2106 y Fh(\010)829 2187 y Fk(B)905 2205 y Fj(n)p Fe(\000b)p Fj(d)14 b Fg(log)1211 2223 y Fa(b)1247 2205 y Fg(\()p Fj(n)p Fe(\000)p Fg(1\))p Fe(c)1530 2187 y Fk(>)28 b Fn(\()p Fk(n)22 b Fm(\000)h(b)p Fk(d)17 b Fn(log)2097 2211 y Fj(b)2137 2187 y Fn(\()p Fk(n)22 b Fm(\000)g Fn(1\))p Fm(c)p Fn(\)\()p Fk(\013)g Fn(+)h Fk(\030)5 b(=)p Fn(2\))2942 2106 y Fh(\011)3026 2187 y Fm(\024)28 b Fk(';)0 2453 y Fn(for)33 b(all)g Fk(n)g Fn(large)g(enough.)45 b(Hence,)142 2719 y Fi(P)237 2639 y Fh(\010)295 2719 y Fk(M)392 2734 y Fj(n;)p Fg(1)537 2719 y Fk(>)28 b Fn(\()p Fk(n)22 b Fm(\000)h Fn(1\))o(\()p Fk(\013)f Fn(+)h Fk(\030)5 b Fn(\))p Fk(;)32 b(N)1404 2734 y Fj(k)1480 2719 y Fm(\024)d Fn(\()p Fk(n)21 b Fm(\000)i Fn(1\))p Fk(\030)5 b(=)p Fn(2)2044 2639 y Fh(\011)979 2919 y Fm(\024)1084 2838 y Fh(\000)1130 2919 y Fi(P)1225 2838 y Fh(\010)1283 2919 y Fk(B)1359 2937 y Fj(n)p Fe(\000b)p Fj(d)13 b Fg(log)1665 2955 y Fa(b)1701 2937 y Fg(\()p Fj(n)p Fe(\000)p Fg(1\))p Fe(c)1984 2919 y Fk(>)28 b Fn(\()p Fk(n)22 b Fm(\000)g(b)p Fk(d)17 b Fn(log)2551 2943 y Fj(b)2591 2919 y Fn(\()p Fk(n)k Fm(\000)i Fn(1\))p Fm(c)p Fn(\)\()p Fk(\013)e Fn(+)i Fk(\030)5 b(=)p Fn(2\))3395 2838 y Fh(\011)o(\001)3497 2859 y Fg(\()p Fj(n)p Fe(\000)p Fg(1\))3710 2829 y Fa(d)979 3107 y Fm(\024)28 b Fk(')1149 3066 y Fg(\()p Fj(n)p Fe(\000)p Fg(1\))1362 3036 y Fa(d)1410 3107 y Fk(;)0 3374 y Fn(for)33 b(all)g Fk(n)g Fn(large)g(enough.)p 3840 3324 60 5 v 3840 3374 5 50 v 3895 3374 V 3840 3379 60 5 v 199 3590 a(The)h(follo)m(wing)f(corollary)f(is)h(immediate)e (from)h(the)i(t)m(w)m(o)f(previous)h(prop)s(ositions.)0 3805 y Fo(Cor)n(ollar)-7 b(y)41 b(1.)55 b Ff(Assume)1094 3814 y Fi(E)1185 3725 y Fh(\010)1244 3805 y Fk(X)1335 3769 y Fj(\022)1379 3725 y Fh(\011)1468 3805 y Fk(<)32 b Fm(1)j Ff(for)g(some)g Fk(\022)e(>)f Fn(2)p Ff(.)50 b(Pic)m(k)35 b Fk(d)c Fm(2)g Fn(\(0)p Fk(;)17 b(\022)25 b Fm(\000)f Fn(2\))p Ff(.)49 b(F)-8 b(or)35 b(all)g Fk(\030)g(>)d Fn(0)p Ff(,)0 3964 y(there)i(are)f Fk(\020)i(>)28 b Fn(0)p Ff(,)33 b Fk(')28 b Fm(2)g Fn(\(0)p Fk(;)17 b Fn(1\))p Ff(,)31 b(and)j Fk(n)1439 3979 y Fg(0)1516 3964 y Ff(suc)m(h)h(that)544 4243 y Fi(P)17 b Fm(f)o Fk(M)785 4258 y Fj(n;)p Fg(1)930 4243 y Fk(>)28 b Fn(\()p Fk(n)22 b Fm(\000)h Fn(1\)\()p Fk(\013)e Fn(+)i Fk(\030)5 b Fn(\))p Fm(g)26 b(\024)2002 4176 y Fk(\020)p 1850 4220 354 4 v 1850 4312 a Fn(\()p Fk(n)c Fm(\000)h Fn(1\))2160 4283 y Fg(2)2238 4243 y Fn(+)g Fk(')2403 4202 y Fg(\()p Fj(n)p Fe(\000)p Fg(1\))2616 4172 y Fa(d)2663 4243 y Fk(;)50 b Ff(for)33 b(all)g Fk(n)27 b Fm(\025)h Fk(n)3284 4258 y Fg(0)3329 4243 y Fk(:)0 4725 y Fm(x)p Fi(4.)39 b(Pro)s(of)g(of)g(main)e(theorem)199 4941 y Fn(W)-8 b(e)41 b(\014rst)f(pro)m(v)m(e)g(one)h(half)f(of)g(part) g(A)g(of)g(Theorem)f(1.)64 b(The)41 b(second)g(half)f(of)h(part)e(A)h (follo)m(ws)0 5100 y(from)33 b(part)g(B,)f(and)i(will)e(b)s(e)i(pro)m (v)m(ed)g(in)f(Theorem)g(6)g(b)s(elo)m(w.)1900 5339 y(11)p eop %%Page: 12 12 12 11 bop 0 100 a Fo(Theorem)48 b(5.)55 b Ff(Let)43 b Fk(X)50 b Ff(b)s(e)43 b(a)f(regular)h(random)e(v)-6 b(ariable)43 b(with)2538 109 y Fi(E)2630 19 y Fh(\010)2688 100 y Fk(X)2779 63 y Fj(\022)2823 19 y Fh(\011)2925 100 y Fk(<)h Fm(1)e Ff(for)h(some)f Fk(\022)k(>)e Fn(2)p Ff(.)0 261 y(Then)34 b(for)g(ev)m(ery)f Fk(")28 b(>)g Fn(0)1359 431 y(lim)1324 491 y Fj(n)p Fe(!1)1549 431 y Fi(P)1644 320 y Fh(n)1710 431 y Fk(N)1801 390 y Fg(1)p Fj(=n)1964 431 y Fk(>)g(\014)g Fn(+)22 b Fk(")2298 320 y Fh(o)2393 431 y Fn(=)28 b(0)p Fk(:)0 901 y Fo(Pr)n(oof.)81 b Fn(Let)43 b Fk(T)670 916 y Fj(n)767 901 y Fn(b)s(e)h(the)f(random)g Fk(b)p Fn(-ary)f(tree)h(as)g (de\014ned)i(in)e(Section)h(1.)73 b(It)43 b(is)g(clear)g(that)g(the)0 1062 y(n)m(um)m(b)s(er)38 b(descendan)m(ts)i(of)e Fk(v)1089 1077 y Fj(k)r(l)1201 1062 y Fn(visited)f(b)m(y)h(the)g(algorithm)d(is)i (smaller)g(than)g(or)h(equal)f(to)g Fk(bZ)3621 1077 y Fj(k)r(l)3720 1062 y Fn(+)26 b(1,)0 1223 y(b)s(ecause)41 b(if)e(the)h(v)-6 b(alue)40 b(of)f(a)h(no)s(de)g(is)f(less)h(than)g (the)g(minimal)c(leaf)k(v)-6 b(alue)40 b(seen)g(th)m(us)h(far,)f(all)f (its)g Fk(b)0 1384 y Fn(c)m(hildren)34 b(will)f(b)s(e)g(visited.)44 b(It)33 b(follo)m(ws)g(that)1307 1706 y Fk(N)38 b Fm(\024)29 b Fk(b)1591 1535 y Fh( )1717 1581 y Fj(n)1669 1611 y Fh(X)1669 1826 y Fj(k)r Fg(=1)1834 1581 y Fj(b)p Fe(\000)p Fg(1)1830 1611 y Fh(X)1839 1826 y Fj(l)p Fg(=1)1991 1706 y Fk(Z)2059 1721 y Fj(k)r(l)2156 1706 y Fn(+)22 b Fk(n)2315 1535 y Fh(!)2416 1706 y Fn(+)h(1)p Fk(:)0 2044 y Fn(Th)m(us,)641 2333 y Fi(P)736 2223 y Fh(n)802 2333 y Fk(N)893 2292 y Fg(1)p Fj(=n)1055 2333 y Fk(>)29 b(\014)e Fn(+)c Fk(")1390 2223 y Fh(o)1484 2333 y Fn(=)29 b Fi(P)17 b Fm(f)o Fk(N)39 b(>)28 b Fn(\()p Fk(\014)g Fn(+)22 b Fk(")p Fn(\))2265 2292 y Fj(n)2320 2333 y Fm(g)1484 2621 y(\024)29 b Fi(P)1685 2451 y Fh(\()1765 2621 y Fk(b)1872 2496 y Fj(n)1825 2526 y Fh(X)1825 2741 y Fj(k)r Fg(=1)1989 2496 y Fj(b)p Fe(\000)p Fg(1)1986 2526 y Fh(X)1994 2741 y Fj(l)p Fg(=1)2146 2621 y Fk(Z)2214 2636 y Fj(k)r(l)2311 2621 y Fn(+)23 b Fk(bn)e Fn(+)i(1)k Fk(>)i Fn(\()p Fk(\014)e Fn(+)c Fk(")p Fn(\))3125 2580 y Fj(n)3179 2451 y Fh(\))1484 2974 y Fm(\024)1637 2849 y Fj(n)1590 2879 y Fh(X)1590 3093 y Fj(k)r Fg(=1)1755 2849 y Fj(b)p Fe(\000)p Fg(1)1751 2879 y Fh(X)1760 3093 y Fj(l)p Fg(=1)1912 2974 y Fi(P)2007 2833 y Fh(\032)2081 2974 y Fk(b)17 b(Z)2209 2989 y Fj(k)r(l)2311 2974 y Fk(>)2509 2906 y Fn(\()p Fk(\014)27 b Fn(+)c Fk(")p Fn(\))2816 2870 y Fj(n)p 2428 2951 524 4 v 2428 3042 a Fn(\()p Fk(b)f Fm(\000)g Fn(1\))p Fk(n)g Fn(+)g(1)2963 2833 y Fh(\033)3038 2974 y Fk(;)0 3317 y Fn(if)33 b(\()p Fk(\014)27 b Fn(+)22 b Fk(")p Fn(\))397 3281 y Fj(n)452 3317 y Fk(=)p Fn(\(\()p Fk(b)e Fm(\000)i Fn(1\))p Fk(n)f Fn(+)h(1\))28 b Fm(\025)g Fk(bn)21 b Fn(+)h(1,)32 b(whic)m(h)i(is)f(true)g(for)h(all)e Fk(n)h Fn(large)f(enough.)45 b(No)m(w,)32 b(notice)h(that)0 3478 y Fk(Z)68 3493 y Fj(k)r(l)170 3478 y Fm(\024)28 b Fn(1)22 b(+)h Fk(b)f Fn(+)h Fm(\001)17 b(\001)g(\001)j Fn(+)j Fk(b)893 3442 y Fj(n)p Fe(\000)p Fj(k)1081 3478 y Fm(\024)28 b Fk(b)1229 3442 y Fj(n)p Fe(\000)p Fj(k)r Fg(+1)1490 3478 y Fn(,)33 b(so)g(that)g(for)g Fk(k)e Fm(\025)d(d)p Fn(\(1)22 b Fm(\000)h Fk(\016)t Fn(\))p Fk(n)p Fm(e)p Fn(,)1100 3784 y Fk(bZ)1211 3799 y Fj(k)r(l)1313 3784 y Fm(\024)28 b Fk(b)1461 3743 y Fj(n)p Fe(\000)p Fj(k)r Fg(+1)1750 3784 y Fm(\024)g Fk(b)1898 3743 y Fj(\016)s(n)p Fg(+1)2120 3784 y Fm(\024)2318 3716 y Fn(\()p Fk(\014)f Fn(+)c Fk(")p Fn(\))2625 3680 y Fj(n)p 2237 3761 V 2237 3852 a Fn(\()p Fk(b)f Fm(\000)g Fn(1\))p Fk(n)g Fn(+)g(1)2772 3784 y Fk(;)0 4085 y Fn(where)34 b Fk(\016)j Fn(is)d(tak)m(en)965 4255 y(0)27 b Fk(<)h(\016)k Fm(\024)c Fn(log)1457 4280 y Fj(b)1497 4255 y Fn(\()p Fk(\014)f Fn(+)c Fk(")p Fn(\))f Fm(\000)1938 4188 y Fn(log)2067 4212 y Fj(b)2123 4188 y Fn(\(\()p Fk(b)g Fm(\000)g Fn(1\))p Fk(n)g Fn(+)g(1\))g(+)h(1)p 1938 4233 958 4 v 2387 4324 a Fk(n)2908 4255 y(:)0 4495 y Fn(Observ)m(e)39 b(that)e(the)h(previous)g(can)g(b)s(e)g(done)h(for)e (all)g Fk(n)h Fn(large)f(enough.)58 b(The)38 b(previous)g(observ)-6 b(ation)0 4656 y(implies)32 b(that)393 4886 y Fj(n)346 4915 y Fh(X)345 5130 y Fj(k)r Fg(=1)510 4886 y Fj(b)p Fe(\000)p Fg(1)507 4915 y Fh(X)515 5130 y Fj(l)p Fg(=1)667 5010 y Fi(P)762 4870 y Fh(\032)837 5010 y Fk(b)17 b(Z)965 5025 y Fj(k)r(l)1066 5010 y Fk(>)1264 4943 y Fn(\()p Fk(\014)28 b Fn(+)22 b Fk(")p Fn(\))1571 4907 y Fj(n)p 1184 4987 524 4 v 1184 5078 a Fn(\()p Fk(b)f Fm(\000)i Fn(1\))p Fk(n)e Fn(+)i(1)1719 4870 y Fh(\033)1821 5010 y Fm(\024)1926 4877 y Fe(b)p Fg(\(1)p Fe(\000)p Fj(\016)s Fg(\))p Fj(n)p Fe(c)2017 4915 y Fh(X)2016 5130 y Fj(k)r Fg(=1)2272 4886 y Fj(b)p Fe(\000)p Fg(1)2268 4915 y Fh(X)2277 5130 y Fj(l)p Fg(=1)2429 5010 y Fi(P)2524 4870 y Fh(\032)2598 5010 y Fk(b)17 b(Z)2726 5025 y Fj(k)r(l)2828 5010 y Fk(>)3026 4943 y Fn(\()p Fk(\014)27 b Fn(+)c Fk(")p Fn(\))3333 4907 y Fj(n)p 2945 4987 V 2945 5078 a Fn(\()p Fk(b)f Fm(\000)g Fn(1\))p Fk(n)g Fn(+)g(1)3480 4870 y Fh(\033)3773 5010 y Fn(\()p Fm(\003)p Fn(\))1900 5339 y(12)p eop %%Page: 13 13 13 12 bop 0 100 a Fn(for)33 b(all)g Fk(n)g Fn(large)g(enough.)45 b(Next,)32 b(for)h(all)g Fk(\030)e(>)e Fn(0,)316 322 y Fi(P)411 182 y Fh(\032)485 322 y Fk(b)17 b(Z)613 337 y Fj(k)r(l)715 322 y Fk(>)913 255 y Fn(\()p Fk(\014)27 b Fn(+)c Fk(")p Fn(\))1220 219 y Fj(n)p 832 300 524 4 v 832 391 a Fn(\()p Fk(b)f Fm(\000)g Fn(1\))p Fk(n)g Fn(+)g(1)1367 182 y Fh(\033)715 601 y Fm(\024)28 b Fi(P)17 b Fm(f)p Fk(M)1062 616 y Fj(n;k)1211 601 y Fk(>)28 b Fn(\()p Fk(n)22 b Fm(\000)h Fk(k)s Fn(\)\()p Fk(\013)e Fn(+)i Fk(\030)5 b Fn(\))p Fm(g)15 b Fn(+)i Fi(P)2197 461 y Fh(\032)2272 601 y Fk(b)g(Z)2400 616 y Fj(k)r(l)2491 601 y Fn(\()o Fk(\013)23 b Fn(+)f Fk(\030)5 b Fn(\))27 b Fk(>)3028 534 y Fn(\()p Fk(\014)g Fn(+)c Fk(")p Fn(\))3335 498 y Fj(n)p 2947 579 V 2947 670 a Fn(\()p Fk(b)e Fm(\000)i Fn(1\))p Fk(n)f Fn(+)g(1)3482 461 y Fh(\033)3557 601 y Fk(;)0 830 y Fn(where)47 b(the)g(random)e(v)-6 b(ariable)46 b Fk(Z)1319 845 y Fj(k)r(l)1411 830 y Fn(\()o Fk(\013)23 b Fn(+)f Fk(\030)5 b Fn(\))45 b(represen)m(ts)k(the)d(n)m(um)m(b)s(er)h (of)g(descendan)m(ts)i Fk(w)g Fn(of)d Fk(v)3797 845 y Fj(k)r(l)3872 830 y Fn(,)0 985 y(suc)m(h)d(that)e Fk(V)515 1000 y Fj(u)562 1011 y Fa(k)q Fb(\000)p Fd(1)694 1000 y Fj(;w)823 985 y Fn(is)h(not)f(larger)g(than)h(\()p Fk(\013)27 b Fn(+)h Fk(\030)5 b Fn(\)\()p Fk(n)27 b Fm(\000)h Fk(k)s Fn(\).)68 b(Corollary)40 b(1)h(implies)g(that)g(there)h(are)0 1141 y Fk(\020)54 b(>)48 b Fn(0,)f Fk(')g Fm(2)g Fn(\(0)p Fk(;)17 b Fn(1\),)46 b(and)g Fk(n)1136 1156 y Fg(0)1180 1141 y Fn(,)i(suc)m(h)e(that)e Fi(P)17 b Fm(f)p Fk(M)1961 1156 y Fj(n;k)2110 1141 y Fk(>)29 b Fn(\()p Fk(n)21 b Fm(\000)i Fk(k)s Fn(\)\()p Fk(\013)f Fn(+)g Fk(\030)5 b Fn(\))p Fm(g)44 b Fn(is)h(smaller)e(or)i(equal)f(to)0 1296 y Fk(\020)7 b(=)p Fn(\()p Fk(n)21 b Fm(\000)i Fk(k)s Fn(\))415 1260 y Fg(2)482 1296 y Fn(+)f Fk(')646 1260 y Fg(\()p Fj(n)p Fe(\000)p Fj(k)r Fg(\))863 1230 y Fa(d)911 1296 y Fn(,)33 b(for)g Fk(n)28 b Fm(\025)g Fk(n)1377 1311 y Fg(0)1422 1296 y Fn(,)k(uniformly)h(o)m(v)m(er)g(1)27 b Fm(\024)i Fk(k)h Fm(\024)e Fn(\(1)22 b Fm(\000)h Fk(\016)t Fn(\))p Fk(n)p Fn(.)43 b(Th)m(us,)34 b(for)g Fk(n)27 b Fm(\025)h Fk(n)3634 1311 y Fg(0)3679 1296 y Fn(,)50 1443 y Fe(b)p Fg(\(1)p Fe(\000)p Fj(\016)s Fg(\))p Fj(n)p Fe(c)140 1481 y Fh(X)140 1695 y Fj(k)r Fg(=1)395 1451 y Fj(b)p Fe(\000)p Fg(1)391 1481 y Fh(X)400 1695 y Fj(l)p Fg(=1)552 1576 y Fi(P)17 b Fm(f)p Fk(M)794 1591 y Fj(n;k)943 1576 y Fk(>)28 b Fn(\()p Fk(n)22 b Fm(\000)g Fk(k)s Fn(\)\()p Fk(\013)g Fn(+)h Fk(\030)5 b Fn(\))p Fm(g)26 b(\024)2052 1451 y Fj(n)p Fe(\000)p Fg(1)2055 1481 y Fh(X)1856 1699 y Fj(k)r Fg(=)p Fj(n)p Fe(\000b)p Fg(\(1)p Fe(\000)p Fj(\016)s Fg(\))p Fj(n)p Fe(c)2415 1576 y Fk(b)2475 1435 y Fh(\022)2584 1508 y Fk(\020)p 2560 1553 100 4 v 2560 1644 a(k)2615 1615 y Fg(2)2693 1576 y Fn(+)d Fk(')2858 1535 y Fj(k)2902 1504 y Fa(d)2949 1435 y Fh(\023)1751 1903 y Fm(\024)2166 1836 y Fk(bn\020)p 1868 1880 750 4 v 1868 1972 a Fn(\()p Fk(n)f Fm(\000)h(b)p Fn(\(1)f Fm(\000)g Fk(\016)t Fn(\))p Fk(n)p Fm(c)p Fn(\))2573 1943 y Fg(2)2652 1903 y Fn(+)g Fk(b')2859 1862 y Fg(\()p Fj(n)p Fe(\000b)p Fg(\(1)p Fe(\000)p Fj(\016)s Fg(\))p Fj(n)p Fe(c)p Fg(\))3358 1832 y Fa(d)3405 1903 y Fm(b)p Fn(\(1)g Fm(\000)h Fk(\016)t Fn(\))p Fk(n)p Fm(c)1751 2173 y(\024)1898 2105 y Fk(b\020)p 1868 2150 153 4 v 1868 2241 a(\016)1916 2212 y Fg(2)1961 2241 y Fk(n)2055 2173 y Fn(+)g Fk(b)p Fn(\(1)e Fm(\000)i Fk(\016)t Fn(\))p Fk(')2560 2132 y Fg(\()p Fj(\016)s(n)p Fg(\))2710 2102 y Fa(d)2757 2173 y Fk(n)1751 2358 y Fm(\000)-16 b(!)27 b Fn(0)199 b(as)34 b Fk(n)27 b Fm(!)g(1)p Fk(:)0 2563 y Fn(No)m(w,)33 b(for)g(the)h(other)f(part)g(of)h(the)f(sum)g(in)g(\()p Fm(\003)p Fn(\),)g(note)g(that)303 2791 y Fi(P)398 2651 y Fh(\032)473 2791 y Fk(b)17 b(Z)601 2806 y Fj(k)r(l)692 2791 y Fn(\()o Fk(\013)23 b Fn(+)f Fk(\030)5 b Fn(\))27 b Fk(>)1229 2724 y Fn(\()p Fk(\014)g Fn(+)c Fk(")p Fn(\))1536 2688 y Fj(n)p 1148 2768 524 4 v 1148 2860 a Fn(\()p Fk(b)f Fm(\000)g Fn(1\))p Fk(n)g Fn(+)g(1)1683 2651 y Fh(\033)1785 2791 y Fm(\024)1903 2733 y Fi(E)1994 2724 y Fm(f)p Fk(b)17 b(Z)2172 2739 y Fj(k)r(l)2263 2724 y Fn(\()p Fk(\013)22 b Fn(+)g Fk(\030)5 b Fn(\))o Fm(g)p 1903 2768 722 4 v 2083 2860 a Fn(\()p Fk(\014)27 b Fn(+)c Fk(")p Fn(\))2390 2831 y Fj(n)2653 2791 y Fn(\()o(\()p Fk(b)f Fm(\000)h Fn(1\))p Fk(n)e Fn(+)i(1\))1785 3144 y(=)1903 3049 y Fi(E)1994 2930 y Fh(n)2061 3040 y Fk(b)2121 2966 y Fh(P)2225 2990 y Fj(n)p Fe(\000)p Fj(k)2225 3070 y(j)t Fg(=1)2403 3040 y Fk(Z)2478 2992 y Fj(j)2471 3070 y(k)r(l)2562 3040 y Fn(\()p Fk(\013)f Fn(+)g Fk(\030)5 b Fn(\))2873 2930 y Fh(o)2956 3040 y Fn(\(\()p Fk(b)22 b Fm(\000)g Fn(1\))p Fk(n)g Fn(+)g(1\))p 1903 3121 1655 4 v 2549 3212 a(\()p Fk(\014)28 b Fn(+)22 b Fk(")p Fn(\))2856 3183 y Fj(n)3569 3144 y Fk(;)0 3390 y Fn(where)33 b Fk(Z)362 3342 y Fj(j)355 3420 y(k)r(l)447 3390 y Fn(\()o Fk(\013)23 b Fn(+)f Fk(\030)5 b Fn(\))32 b(is)g(the)h(n)m(um)m(b)s(er)g(of)g(descendan)m(ts)i Fk(w)g Fn(of)e Fk(v)2354 3405 y Fj(k)r(l)2462 3390 y Fn(suc)m(h)h(that)e Fk(V)2959 3405 y Fj(u)3006 3416 y Fa(k)q Fb(\000)p Fd(1)3137 3405 y Fj(;w)3258 3390 y Fn(is)g(smaller)g (than)0 3546 y(or)h(equal)g(to)g(\()p Fk(\013)22 b Fn(+)g Fk(\030)5 b Fn(\)\()p Fk(n)21 b Fm(\000)i Fk(k)s Fn(\),)32 b(and)i(the)f(distance)h(from)f Fk(w)j Fn(to)d Fk(v)2459 3561 y Fj(k)r(l)2567 3546 y Fn(is)g Fk(j)6 b Fn(.)44 b(No)m(w,)33 b(for)g Fk(\022)d(>)e Fn(0,)28 3750 y Fi(E)120 3670 y Fh(\010)178 3750 y Fk(Z)253 3702 y Fj(j)246 3780 y(k)r(l)337 3750 y Fn(\()p Fk(\013)22 b Fn(+)h Fk(\030)5 b Fn(\))650 3670 y Fh(\011)734 3750 y Fm(\024)29 b Fk(b)883 3709 y Fj(j)941 3750 y Fi(P)17 b Fm(f)p Fk(X)1169 3765 y Fg(1)1235 3750 y Fn(+)23 b Fm(\001)17 b(\001)g(\001)d Fk(X)1550 3765 y Fj(j)1614 3750 y Fn(+)23 b Fk(X)1797 3765 y Fj(j)t Fg(+1)1967 3750 y Fm(\024)28 b Fn(\()p Fk(n)22 b Fm(\000)g Fk(k)s Fn(\)\()p Fk(\013)g Fn(+)h Fk(\030)5 b Fn(\))p Fm(g)468 3936 y(\024)28 b Fk(b)616 3895 y Fj(j)674 3945 y Fi(E)766 3825 y Fh(n)832 3936 y Fk(e)878 3895 y Fj(\022)r Fg(\(\()p Fj(n)p Fe(\000)p Fj(k)r Fg(\)\()p Fj(\013)p Fg(+)p Fj(\030)r Fg(\))p Fe(\000)p Fj(X)1506 3905 y Fd(1)1548 3895 y Fe(\000\001\001\001)o(\000)p Fj(X)1808 3905 y Fa(j)s Fd(+1)1931 3895 y Fg(\))1967 3825 y Fh(o)2084 3936 y Fn(\(b)m(y)33 b(Mark)m(o)m(v's)g(inequalit)m (y\))468 4226 y(=)28 b Fk(b)616 4184 y Fj(j)658 4226 y Fk(e)704 4184 y Fj(\022)r Fg(\()p Fj(n)p Fe(\000)p Fj(k)r Fg(\)\()p Fj(\013)p Fg(+)p Fj(\030)r Fg(\))1197 4096 y Fj(j)t Fg(+1)1203 4131 y Fh(Y)1202 4343 y Fj(i)p Fg(=1)1352 4235 y Fi(E)1444 4145 y Fh(\010)1502 4226 y Fk(e)1548 4184 y Fe(\000)p Fj(X)1675 4194 y Fa(i)1707 4184 y Fj(\022)1752 4145 y Fh(\011)1860 4226 y Fn(\(b)m(y)33 b(the)h(indep)s(endence)i(of)e(the)f Fk(X)3194 4241 y Fj(i)3227 4226 y Fn('s\))468 4504 y(=)28 b Fk(b)616 4463 y Fj(j)658 4504 y Fk(e)704 4463 y Fj(\022)r Fg(\()p Fj(n)p Fe(\000)p Fj(k)r Fg(\)\()p Fj(\013)p Fg(+)p Fj(\030)r Fg(\))1197 4423 y Fh(\000)1243 4513 y Fi(E)1335 4423 y Fh(\010)1393 4504 y Fk(e)1439 4463 y Fe(\000)p Fj(X)5 b(\022)1617 4423 y Fh(\011)q(\001)1721 4439 y Fj(j)t Fg(+1)1914 4504 y Fn(\(b)s(ecause)34 b(the)g Fk(X)2576 4519 y Fj(i)2608 4504 y Fn('s)g(are)f(iden)m(tically)g(distributed\)) 468 4745 y Fm(\024)573 4604 y Fh(\022)646 4745 y Fk(b)706 4754 y Fi(E)797 4604 y Fh(\032)872 4745 y Fk(e)918 4692 y Fj(\022)958 4618 y Fh(\000)1016 4656 y Fd(\()p Fa(n)p Fb(\000)p Fa(k)q Fd(\)\()p Fa(\013)p Fd(+)p Fa(\030)q Fd(\))p 1016 4676 376 4 v 1145 4716 a Fa(j)s Fd(+1)1403 4692 y Fe(\000)p Fj(X)1535 4618 y Fh(\001)1587 4604 y(\033\023)1735 4625 y Fj(j)t Fg(+1)468 5045 y Fn(=)573 4905 y Fh(\022)646 5045 y Fk(\026)723 4905 y Fh(\022)808 4978 y Fn(\()p Fk(n)22 b Fm(\000)g Fk(k)s Fn(\)\()p Fk(\013)g Fn(+)h Fk(\030)5 b Fn(\))p 808 5022 626 4 v 1012 5114 a Fk(j)28 b Fn(+)22 b(1)1446 4905 y Fh(\023\023)1592 4925 y Fj(j)t Fg(+1)1752 5045 y Fk(;)1900 5339 y Fn(13)p eop %%Page: 14 14 14 13 bop 0 100 a Fn(where)34 b(for)g(the)f(last)g(equalit)m(y)f(to)h (hold)g(w)m(e)h(to)s(ok)e Fk(\022)e Fn(=)e Fk(\022)2129 63 y Fe(\003)2208 100 y Fn(suc)m(h)34 b(that)896 304 y Fk(b)956 313 y Fi(E)1048 164 y Fh(\032)1122 304 y Fk(e)1168 251 y Fj(\022)1208 221 y Fb(\003)1251 178 y Fh(\000)1308 215 y Fd(\()p Fa(n)p Fb(\000)p Fa(k)q Fd(\)\()p Fa(\013)p Fd(+)p Fa(\030)q Fd(\))p 1308 236 376 4 v 1437 275 a Fa(j)s Fd(+1)1695 251 y Fe(\000)p Fj(X)1827 178 y Fh(\001)1879 164 y(\033)1981 304 y Fn(=)28 b Fk(\026)2163 164 y Fh(\022)2248 237 y Fn(\()p Fk(n)22 b Fm(\000)h Fk(k)s Fn(\)\()p Fk(\013)e Fn(+)i Fk(\030)5 b Fn(\))p 2248 281 626 4 v 2452 373 a Fk(j)28 b Fn(+)23 b(1)2886 164 y Fh(\023)2976 304 y Fk(:)0 515 y Fn(Th)m(us,)34 b(since)g Fk(\026)17 b Fn(\()p Fk(a)p Fn(\))32 b(is)h(an)h(increasing)f(function)i(of)e Fk(a)27 b Fm(\025)i Fn(0,)65 763 y Fi(P)160 623 y Fh(\032)234 763 y Fk(b)17 b(Z)362 778 y Fj(k)r(l)453 763 y Fn(\()p Fk(\013)22 b Fn(+)h Fk(\030)5 b Fn(\))26 b Fk(>)990 696 y Fn(\()p Fk(\014)i Fn(+)22 b Fk(")p Fn(\))1297 660 y Fj(n)p 909 741 524 4 v 909 832 a Fn(\()p Fk(b)g Fm(\000)g Fn(1\))p Fk(n)g Fn(+)h(1)1444 623 y Fh(\033)1547 763 y Fm(\024)1664 696 y Fk(b)p Fn(\(\()p Fk(b)e Fm(\000)i Fn(1\))p Fk(n)e Fn(+)i(1\))p 1664 741 644 4 v 1805 832 a(\()p Fk(\014)k Fn(+)c Fk(")p Fn(\))2112 803 y Fj(n)2336 639 y(n)p Fe(\000)p Fj(k)2342 669 y Fh(X)2345 881 y Fj(j)t Fg(=1)2508 772 y Fi(E)2600 653 y Fh(n)2666 763 y Fk(Z)2741 716 y Fj(j)2734 794 y(k)r(l)2826 763 y Fn(\()o Fk(\013)g Fn(+)f Fk(\030)5 b Fn(\))3137 653 y Fh(o)1547 1127 y Fm(\024)1664 1060 y Fk(b)p Fn(\(\()p Fk(b)21 b Fm(\000)i Fn(1\))p Fk(n)e Fn(+)i(1\))p 1664 1104 V 1805 1196 a(\()p Fk(\014)k Fn(+)c Fk(")p Fn(\))2112 1167 y Fj(n)2336 1003 y(n)p Fe(\000)p Fj(k)2342 1033 y Fh(X)2345 1245 y Fj(j)t Fg(=1)2508 987 y Fh(\022)2581 1127 y Fk(\026)2658 987 y Fh(\022)2743 1060 y Fn(\()p Fk(n)f Fm(\000)h Fk(k)s Fn(\)\()p Fk(\013)e Fn(+)i Fk(\030)5 b Fn(\))p 2743 1104 626 4 v 2947 1196 a Fk(j)28 b Fn(+)22 b(1)3381 987 y Fh(\023\023)3528 1007 y Fj(j)t Fg(+1)1547 1491 y Fn(=)1664 1424 y Fk(b)p Fn(\(\()p Fk(b)f Fm(\000)i Fn(1\))p Fk(n)e Fn(+)i(1\))p 1664 1468 644 4 v 1805 1559 a(\()p Fk(\014)k Fn(+)c Fk(")p Fn(\))2112 1531 y Fj(n)2336 1367 y(n)p Fe(\000)p Fj(k)2342 1397 y Fh(X)2345 1609 y Fj(j)t Fg(=1)2508 1351 y Fh(\022)2581 1491 y Fk(\026)2659 1415 y Fa(j)s Fd(+1)p 2653 1433 129 4 v 2653 1472 a Fa(n)p Fd(+1)2816 1351 y Fh(\022)2901 1424 y Fn(\()p Fk(n)f Fn(+)g(1\)\()p Fk(\013)g Fn(+)h Fk(\030)5 b Fn(\))p 2901 1468 621 4 v 3102 1559 a Fk(j)28 b Fn(+)23 b(1)3534 1351 y Fh(\023\023)3680 1371 y Fj(n)p Fg(+1)1547 1867 y Fm(\024)1664 1799 y Fk(b)p Fn(\(\()p Fk(b)e Fm(\000)i Fn(1\))p Fk(n)e Fn(+)i(1\))p 1664 1844 644 4 v 1805 1935 a(\()p Fk(\014)k Fn(+)c Fk(")p Fn(\))2112 1906 y Fj(n)2336 1742 y(n)p Fe(\000)p Fj(k)2342 1772 y Fh(X)2345 1984 y Fj(j)t Fg(=1)2508 1696 y Fh( )2644 1867 y Fn(sup)2587 1955 y Fj(x)p Fe(2)p Fg(\(0)p Fj(;)p Fg(1\))2868 1867 y Fk(\026)2928 1826 y Fj(x)2995 1726 y Fh(\022)3080 1799 y Fk(\013)f Fn(+)h Fk(\030)p 3080 1844 235 4 v 3168 1935 a(x)3326 1726 y Fh(\023)3399 1696 y(!)3478 1717 y Fj(n)p Fg(+1)1547 2242 y Fm(\024)1664 2175 y Fk(nb)p Fn(\(\()p Fk(b)e Fm(\000)h Fn(1\))p Fk(n)g Fn(+)h(1\))p 1664 2219 704 4 v 1835 2311 a(\()p Fk(\014)k Fn(+)c Fk(")p Fn(\))2142 2282 y Fj(n)2396 2072 y Fh( )2532 2242 y Fn(sup)2475 2331 y Fj(x)p Fe(2)p Fg(\(0)p Fj(;)p Fg(1\))2755 2242 y Fk(\026)2815 2201 y Fj(x)2882 2102 y Fh(\022)2967 2175 y Fk(\013)g Fn(+)f Fk(\030)p 2967 2219 235 4 v 3056 2311 a(x)3213 2102 y Fh(\023)3287 2072 y(!)3366 2093 y Fj(n)p Fg(+1)3537 2242 y Fk(:)199 2477 y Fn(W)-8 b(e)34 b(no)m(w)f(sum)g(all)g(the)g(terms)g(and)h(get)180 2600 y Fe(b)p Fg(\(1)p Fe(\000)p Fj(\016)s Fg(\))p Fj(n)p Fe(c)271 2638 y Fh(X)270 2852 y Fj(k)r Fg(=1)525 2608 y Fj(b)p Fe(\000)p Fg(1)522 2638 y Fh(X)530 2852 y Fj(l)p Fg(=1)682 2733 y Fi(P)777 2592 y Fh(\032)852 2733 y Fk(b)17 b(Z)980 2748 y Fj(k)r(l)1070 2733 y Fn(\()p Fk(\013)22 b Fn(+)h Fk(\030)5 b Fn(\))27 b Fk(>)1608 2665 y Fn(\()p Fk(\014)g Fn(+)c Fk(")p Fn(\))1915 2629 y Fj(n)p 1527 2710 524 4 v 1527 2801 a Fn(\()p Fk(b)e Fm(\000)i Fn(1\))p Fk(n)f Fn(+)g(1)2062 2592 y Fh(\033)1410 3097 y Fm(\024)1527 3030 y Fk(n)1587 2993 y Fg(2)1631 3030 y Fk(b)p Fn(\()p Fk(b)g Fm(\000)g Fn(1\)\(\()p Fk(b)f Fm(\000)i Fn(1\))p Fk(n)e Fn(+)i(1\))p 1527 3074 1040 4 v 1866 3165 a(\()p Fk(\014)k Fn(+)c Fk(")p Fn(\))2173 3137 y Fj(n)2595 2927 y Fh( )2731 3097 y Fn(sup)2674 3185 y Fj(x)p Fe(2)p Fg(\(0)p Fj(;)p Fg(1\))2955 3097 y Fk(\026)3015 3056 y Fj(x)3082 2956 y Fh(\022)3167 3030 y Fk(\013)f Fn(+)h Fk(\030)p 3167 3074 235 4 v 3255 3165 a(x)3413 2956 y Fh(\023)3486 2927 y(!)3565 2947 y Fj(n)p Fg(+1)1410 3456 y Fm(\024)1612 3389 y Fk(b)1655 3352 y Fg(3)1699 3389 y Fk(n)1759 3352 y Fg(3)p 1527 3433 362 4 v 1527 3524 a Fn(\()p Fk(\014)k Fn(+)c Fk(")p Fn(\))1834 3496 y Fj(n)1917 3286 y Fh(")2032 3456 y Fn(sup)1975 3544 y Fj(x)p Fe(2)p Fg(\(0)p Fj(;)p Fg(1\))2256 3456 y Fk(\026)2316 3415 y Fj(x)2383 3316 y Fh(\022)2468 3389 y Fk(\013)f Fn(+)h Fk(\030)p 2468 3433 235 4 v 2556 3524 a(x)2714 3316 y Fh(\023)2787 3286 y(#)2845 3306 y Fj(n)p Fg(+1)3017 3456 y Fk(:)0 3703 y Fn(Notice)33 b(that)h(if)g Fk(\030)k Fn(w)m(ere)d(equal)f(to)f(0,)h (then)h(the)f(quan)m(tit)m(y)f(in)h(the)h(square)f(brac)m(k)m(ets)h(w)m (ould)f(b)s(e)h Fk(\014)3717 3666 y Fj(n)p Fg(+1)3872 3703 y Fn(.)0 3857 y(Lemma)d(1)h(implies)f(that)h(for)g Fk(")28 b(>)g Fn(0,)33 b(there)h(exists)f Fk(\030)2014 3821 y Fe(\003)2087 3857 y Fk(>)28 b Fn(0)33 b(suc)m(h)i(that)912 4062 y(sup)855 4150 y Fj(x)p Fe(2)p Fg(\(0)p Fj(;)p Fg(1\))1135 4062 y Fk(\026)1195 4021 y Fj(x)1262 3921 y Fh(\022)1347 3994 y Fk(\013)23 b Fn(+)f Fk(\030)1582 3958 y Fe(\003)p 1347 4039 280 4 v 1459 4130 a Fk(x)1639 3921 y Fh(\023)1740 4062 y Fm(\024)86 b Fn(sup)1845 4150 y Fj(x)p Fe(2)p Fg(\(0)p Fj(;)p Fg(1\))2126 4062 y Fk(\026)2186 4021 y Fj(x)2253 3951 y Fh(\020)2324 3994 y Fk(\013)p 2324 4039 65 4 v 2328 4130 a(x)2401 3951 y Fh(\021)2477 4062 y Fn(+)2568 3994 y Fk(")p 2566 4039 50 4 v 2566 4130 a Fn(2)2656 4062 y(=)28 b Fk(\014)f Fn(+)2958 3994 y Fk(")p 2956 4039 V 2956 4130 a Fn(2)3018 4062 y Fk(:)0 4288 y Fn(Therefore,)77 4504 y Fk(b)120 4468 y Fg(3)164 4504 y Fk(n)224 4468 y Fg(3)p -8 4549 362 4 v -8 4640 a Fn(\()p Fk(\014)g Fn(+)c Fk(")p Fn(\))299 4611 y Fj(n)382 4401 y Fh( )518 4571 y Fn(sup)461 4660 y Fj(x)p Fe(2)p Fg(\(0)p Fj(;)p Fg(1\))742 4571 y Fk(\026)802 4530 y Fj(x)868 4431 y Fh(\022)954 4504 y Fk(\013)f Fn(+)h Fk(\030)1189 4468 y Fe(\003)p 954 4549 280 4 v 1065 4640 a Fk(x)1246 4431 y Fh(\023)1319 4401 y(!)1398 4422 y Fj(n)p Fg(+1)1581 4571 y Fm(\024)28 b Fk(b)1729 4530 y Fg(3)1773 4571 y Fk(n)1833 4530 y Fg(3)1894 4371 y Fh(0)1894 4551 y(@)1993 4468 y Fn(sup)2143 4492 y Fj(x)p Fe(2)p Fg(\(0)p Fj(;)p Fg(1\))2429 4468 y Fk(\026)2489 4432 y Fj(x)2556 4357 y Fh(\020)2627 4423 y Fj(\013)p Fg(+)p Fj(\030)2778 4393 y Fb(\003)p 2627 4445 194 4 v 2701 4502 a Fj(x)2833 4357 y Fh(\021)p 1993 4549 899 4 v 2328 4640 a Fk(\014)g Fn(+)22 b Fk(")2904 4371 y Fh(1)2904 4551 y(A)2991 4378 y Fj(n)3119 4571 y Fn(sup)3062 4660 y Fj(x)p Fe(2)p Fg(\(0)p Fj(;)p Fg(1\))3343 4571 y Fk(\026)3403 4530 y Fj(x)3470 4431 y Fh(\022)3555 4504 y Fk(\013)g Fn(+)h Fk(\030)3790 4468 y Fe(\003)p 3555 4549 280 4 v 3666 4640 a Fk(x)3847 4431 y Fh(\023)1581 4916 y Fm(\024)28 b Fk(b)1729 4874 y Fg(3)1773 4916 y Fk(n)1833 4874 y Fg(3)1894 4775 y Fh(\022)1980 4848 y Fk(\014)f Fn(+)c Fk("=)p Fn(2)p 1980 4893 330 4 v 2029 4984 a Fk(\014)28 b Fn(+)23 b Fk(")2321 4775 y Fh(\023)2394 4796 y Fj(n)2465 4916 y Fn(\()p Fk(\014)28 b Fn(+)22 b Fk("=)p Fn(2\))1577 5100 y Fm(\000)-16 b(!)1553 5160 y Fj(n)p Fe(!1)1761 5100 y Fn(0)p Fk(:)p 3840 5050 60 5 v 3840 5100 5 50 v 3895 5100 V 3840 5105 60 5 v 1900 5339 a Fn(14)p eop %%Page: 15 15 15 14 bop 199 100 a Fn(T)-8 b(o)32 b(\014nish)i(the)e(pro)s(of)h(of)g (Theorem)f(1,)g(w)m(e)h(sho)m(w)g(the)g(univ)m(ersal)g(lo)m(w)m(er)f(b) s(ound)i(of)e(part)h(B.)e(Note)0 263 y(that)24 b(an)m(y)g(algorithm)f (m)m(ust)h(visit)f(all)h(no)s(des)h Fk(v)j Fn(in)d Fk(T)1923 278 y Fj(n)2002 263 y Fn(with)f(v)-6 b(alue)24 b(strictly)g(less)h (than)f Fk(B)3275 278 y Fj(n)3329 263 y Fn(,)i(the)f(minimal)0 426 y(leaf)33 b(v)-6 b(alue)34 b(in)f Fk(T)617 441 y Fj(n)671 426 y Fn(.)44 b(Th)m(us,)34 b(it)f(su\016ces)i(to)e(pro)m(v)m (e)g(the)h(follo)m(wing.)0 672 y Fo(Theorem)k(6.)55 b Ff(Let)34 b Fk(X)i Fm(\025)28 b Fn(0)34 b Ff(b)s(e)g(a)f(regular)g (random)g(v)-6 b(ariable.)45 b(Let)34 b Fk(N)2722 636 y Fe(0)2783 672 y Ff(b)s(e)g(the)g(n)m(um)m(b)s(er)g(of)g(no)s(des)g Fk(v)0 836 y Ff(in)f Fk(T)174 851 y Fj(n)262 836 y Ff(with)g(v)-6 b(alue)33 b Fk(V)804 851 y Fj(v)879 836 y Fk(<)28 b(B)1060 851 y Fj(n)1114 836 y Ff(.)44 b(Then,)34 b(for)g(ev)m(ery)f Fk(\017)28 b(>)g Fn(0)p Ff(,)1351 1142 y Fn(lim)1316 1201 y Fj(n)p Fe(!1)1541 1142 y Fi(P)1636 1031 y Fh(n)1702 1142 y Fk(N)1793 1101 y Fe(0)p Fg(1)p Fj(=n)1978 1142 y Fk(<)g(\014)g Fm(\000)23 b Fk(\017)2307 1031 y Fh(o)2401 1142 y Fn(=)29 b(0)p Fk(:)33 1448 y Fo(Pr)n(oof.)50 b Fn(W)-8 b(e)34 b(use)g(the)f(notation)g(from)f(Theorem)h(2,)g(and)h (note)f(that)1451 1784 y Fk(N)1542 1743 y Fe(0)1597 1784 y Fm(\025)1750 1660 y Fj(n)1703 1690 y Fh(X)1706 1902 y Fj(j)t Fg(=0)1863 1784 y Fk(Z)1938 1743 y Fg(\()p Fj(j)t Fg(\))2043 1784 y Fn(\()p Fk(B)2158 1799 y Fj(n)2233 1784 y Fm(\000)23 b Fn(1\))p Fk(:)0 2153 y Fn(De\014ne)35 b Fk(\013)p Fn(,)e Fk(\014)38 b Fn(and)c Fk(\015)k Fn(as)c(in)f (section)h(2,)e(and)i(set)g Fk(k)c Fn(=)e Fm(b)p Fk(\015)6 b(n)p Fm(c)p Fn(.)44 b(Th)m(us,)34 b(for)f(an)m(y)h Fk(\030)d Fm(2)d Fn(\(0)p Fk(;)17 b(\013)p Fn(\),)1160 2459 y Fk(N)1251 2418 y Fe(0)1307 2459 y Fm(\025)28 b Fk(Z)1487 2418 y Fg(\()p Fj(k)r Fg(\))1615 2459 y Fn(\(\()p Fk(\013)22 b Fm(\000)g Fk(\030)5 b Fn(\))p Fk(n)p Fn(\))16 b Fk(I)2125 2477 y Fg([)p Fj(B)2208 2487 y Fa(n)2256 2477 y Fe(\000)p Fg(1)p Fj(>)p Fg(\()p Fj(\013)p Fe(\000)p Fj(\030)r Fg(\))p Fj(n)p Fg(])2712 2459 y Fk(:)0 2765 y Fn(By)33 b(Theorem)h(3,)g(with)g (probabilit)m(y)g(one,)h Fk(B)1718 2780 y Fj(n)1794 2765 y Fm(\000)24 b Fn(1)29 b Fk(>)h Fn(\()p Fk(\013)23 b Fm(\000)g Fk(\030)5 b Fn(\))p Fk(n)33 b Fn(for)i(all)e Fk(n)h Fn(large)g(enough.)48 b(Th)m(us,)36 b(w)m(e)0 2928 y(are)d(done)h(if)g(w)m(e)f(can)h(sho)m(w)g(that)f(giv)m(en)g Fk(\017)28 b(>)g Fn(0,)33 b(w)m(e)g(can)h(\014nd)g Fk(\030)e(>)c Fn(0)33 b(suc)m(h)i(that)1014 3245 y(lim)979 3304 y Fj(n)p Fe(!1)1204 3245 y Fi(P)1299 3104 y Fh(\032)1374 3134 y(\020)1433 3245 y Fk(Z)1508 3203 y Fg(\()p Fj(k)r Fg(\))1636 3245 y Fn(\(\()p Fk(\013)22 b Fm(\000)g Fk(\030)5 b Fn(\))p Fk(n)p Fn(\))2085 3134 y Fh(\021)2144 3155 y Fg(1)p Fj(=n)2307 3245 y Fk(<)28 b(\014)g Fm(\000)22 b Fk(\017)2635 3104 y Fh(\033)2738 3245 y Fn(=)28 b(0)p Fk(:)0 3561 y Fn(T)-8 b(o)33 b(this)g(e\013ect,)h(observ)m(e)g(that)765 3825 y Fh(\020)825 3935 y Fk(Z)900 3894 y Fg(\()p Fj(k)r Fg(\))1028 3935 y Fn(\()o(\()p Fk(\013)22 b Fm(\000)h Fk(\030)5 b Fn(\))p Fk(n)p Fn(\))1476 3825 y Fh(\021)1536 3845 y Fg(1)p Fj(=n)1698 3935 y Fn(=)1803 3765 y Fh(")1862 3795 y(\022)1935 3935 y Fk(Z)2010 3894 y Fg(\()p Fj(k)r Fg(\))2138 3795 y Fh(\022)2223 3868 y Fn(\()p Fk(\013)22 b Fm(\000)h Fk(\030)5 b Fn(\))p Fk(n)p 2223 3912 372 4 v 2381 4004 a(k)2607 3935 y(k)2662 3795 y Fh(\023\023)2808 3815 y Fj(k)r(=n)2947 3765 y Fh(#)3005 3785 y Fg(1)p Fj(=k)1698 4306 y Fm(\025)1803 4136 y Fh(")1862 4165 y(\022)1935 4306 y Fk(Z)2010 4265 y Fg(\()p Fj(k)r Fg(\))2138 4165 y Fh(\022)2223 4239 y Fk(\013)22 b Fm(\000)h Fk(\030)p 2223 4283 235 4 v 2312 4374 a(\015)2469 4306 y(k)2524 4165 y Fh(\023)q(\023)2671 4186 y Fj(\015)t Fe(\000)2796 4159 y Fd(1)p 2791 4170 44 4 v 2791 4210 a Fa(n)2852 4136 y Fh(#)2910 4152 y Fg(1)p Fj(=k)1698 4620 y Fm(!)1826 4480 y Fh(\024)1878 4620 y Fk(\026)1955 4480 y Fh(\022)2040 4553 y Fk(\013)f Fm(\000)h Fk(\030)p 2040 4597 235 4 v 2129 4689 a(\015)2286 4480 y Fh(\023\025)2412 4500 y Fj(\015)0 4937 y Fn(b)m(y)40 b(Theorem)g(2.)64 b(By)39 b(the)i(con)m(tin)m(uit)m(y)f(of)g Fk(\026)p Fn(,)i(the)e(lo)m(w)m(er)g (b)s(ound)h(tends)g(to)f Fk(\014)46 b Fn(as)40 b Fk(\030)j Fm(#)c Fn(0,)i(since)g Fk(\014)k Fn(=)0 5100 y Fk(\026)60 5064 y Fj(\015)111 5100 y Fn(\()p Fk(\013=\015)6 b Fn(\).)p 3840 5050 60 5 v 3840 5100 5 50 v 3895 5100 V 3840 5105 60 5 v 1900 5339 a(15)p eop %%Page: 16 16 16 15 bop 1115 125 1670 4 v 1115 145 4 20 v 1418 145 V 1948 145 V 2251 145 V 2781 145 V 1115 265 4 120 v 1276 230 a Fk(b)p 1418 265 V 340 w(\032)p 1948 265 V 397 w(b)p 2251 265 V 341 w(\032)p 2781 265 V 1115 285 4 20 v 1418 285 V 1948 285 V 2251 285 V 2781 285 V 1115 289 1670 4 v 1115 408 4 120 v 1269 373 a Fn(2)p 1418 408 V 203 w(.334648)p 1948 408 V 251 w(3)p 2251 408 V 203 w(.257101)p 2781 408 V 1115 528 V 1269 493 a(4)p 1418 528 V 203 w(.220361)p 1948 528 V 251 w(5)p 2251 528 V 203 w(.198027)p 2781 528 V 1115 647 V 1269 612 a(6)p 1418 647 V 203 w(.182672)p 1948 647 V 251 w(7)p 2251 647 V 203 w(.171302)p 2781 647 V 1115 767 V 1269 732 a(8)p 1418 767 V 203 w(.162452)p 1948 767 V 251 w(9)p 2251 767 V 203 w(.155311)p 2781 767 V 1115 886 V 1219 852 a(10)p 1418 886 V 203 w(.149393)p 1948 886 V 202 w(11)p 2251 886 V 202 w(.144383)p 2781 886 V 1115 1006 V 1219 971 a(12)p 1418 1006 V 203 w(.140071)p 1948 1006 V 202 w(13)p 2251 1006 V 202 w(.136306)p 2781 1006 V 1115 1126 V 1219 1091 a(14)p 1418 1126 V 203 w(.132983)p 1948 1126 V 202 w(15)p 2251 1126 V 202 w(.130019)p 2781 1126 V 1115 1245 V 1219 1210 a(16)p 1418 1245 V 203 w(.127354)p 1948 1245 V 202 w(17)p 2251 1245 V 202 w(.124941)p 2781 1245 V 1115 1365 V 1219 1330 a(18)p 1418 1365 V 203 w(.122741)p 1948 1365 V 202 w(19)p 2251 1365 V 202 w(.120725)p 2781 1365 V 1115 1484 V 1219 1449 a(20)p 1418 1484 V 203 w(.118868)p 1948 1484 V 202 w(21)p 2251 1484 V 202 w(.117149)p 2781 1484 V 1115 1504 4 20 v 1418 1504 V 1948 1504 V 2251 1504 V 2781 1504 V 1115 1508 1670 4 v 618 1785 a Fo(T)-9 b(able)38 b(1.)94 b Fn(V)-8 b(alues)33 b(of)g Fk(b)g Fn(vs.)44 b Fk(\032)33 b Fn(for)g(the)h(exp)s(onen)m(tial)f (distribution.)0 2157 y Fm(x)p Fi(5.)39 b(Some)f(examples.)199 2389 y Fn(In)j(this)g(section)g(w)m(e)h(will)d(presen)m(t)k(some)d (examples)g(of)h Fk(\026)p Fn(-functions)h(and)f Fk(\014)5 b Fn(-v)-6 b(alues)43 b(for)e(some)0 2551 y(w)m(ell-kno)m(wn)34 b(distributions.)0 2784 y Fo(Example.)43 b Fn(Exp)s(onen)m(tial)33 b(Distribution)p 508 2799 1089 4 v(.)43 b(If)32 b Fk(X)38 b Fn(is)32 b(exp)s(onen)m(tially)e(distributed,)i Fk(m)p Fn(\()p Fk(\022)s Fn(\))c(=)g(1)p Fk(=)p Fn(\()p Fk(\022)20 b Fn(+)f(1\),)0 2945 y(and)1470 3127 y Fi(E)1562 3007 y Fh(n)1628 3118 y Fk(e)1674 3077 y Fj(\022)r Fg(\()p Fj(a)p Fe(\000)p Fj(X)5 b Fg(\))1958 3007 y Fh(o)2052 3118 y Fn(=)2213 3051 y Fk(e)2259 3014 y Fj(\022)r(a)p 2169 3095 222 4 v 2169 3186 a Fk(\022)25 b Fn(+)e(1)2402 3118 y Fk(;)0 3361 y Fn(for)33 b Fk(\022)e Fm(\025)d Fn(0.)44 b(Also,)927 3554 y Fk(\026)17 b Fn(\()p Fk(a)p Fn(\))27 b(=)h Fk(b)f Fn(inf)1327 3619 y Fj(\022)r Fe(\025)p Fg(0)1485 3413 y Fh(\032)1615 3486 y Fk(e)1661 3450 y Fj(\022)r(a)p 1572 3531 V 1572 3622 a Fk(\022)d Fn(+)f(1)1805 3413 y Fh(\033)1907 3554 y Fn(=)2012 3413 y Fh(\032)2103 3500 y Fk(bae)2245 3464 y Fg(1)p Fe(\000)p Fj(a)2395 3500 y Fk(;)100 b(a)27 b Fm(\024)h Fn(1;)2103 3619 y Fk(b)249 b(;)100 b Fn(otherwise,)0 3854 y(b)s(ecause)42 b(log)521 3744 y Fh(\020)592 3815 y Fj(be)665 3785 y Fa(\022)r(a)p 592 3831 152 4 v 648 3889 a Fj(\022)756 3744 y Fh(\021)856 3854 y Fn(is)e(minim)m(um)e(for)i Fk(\022)i Fn(=)1794 3815 y Fg(1)p 1793 3831 44 4 v 1793 3889 a Fj(a)1875 3854 y Fm(\000)27 b Fn(1.)65 b(The)40 b(v)-6 b(alue)41 b Fk(\013)f Fn(is)g(de\014ned)i(to)e(b)s(e)h(the)f (unique)0 4016 y(solution)33 b(of)1689 4189 y Fk(\013be)1842 4148 y Fg(1)p Fe(\000)p Fj(\013)2029 4189 y Fn(=)28 b(1)p Fk(:)1561 b Fn(\(2\))0 4432 y(Observ)m(e)35 b(that)e(as)g Fk(b)27 b Fm(!)h(1)p Fn(,)k Fk(\013)c Fm(\030)g Fn(1)p Fk(=)p Fn(2)p Fk(b)p Fn(.)43 b(Note)33 b(that)970 4724 y(sup)913 4812 y Fj(x)p Fe(2)p Fg(\(0)p Fj(;)p Fg(1\))1194 4724 y Fk(\026)1254 4683 y Fj(x)1321 4614 y Fh(\020)1392 4657 y Fk(\013)p 1392 4701 65 4 v 1396 4793 a(x)1468 4614 y Fh(\021)1556 4724 y Fn(=)91 b(sup)1661 4812 y Fj(x)p Fe(2)p Fg(\()p Fj(\013;)p Fg(1\))1954 4614 y Fh(n)2032 4657 y Fk(\013)p 2032 4701 V 2036 4793 a(x)2108 4724 y(be)2197 4683 y Fg(1)p Fe(\000)2311 4656 y Fa(\013)p 2311 4667 46 4 v 2314 4707 a(x)2374 4614 y Fh(o)2440 4634 y Fj(x)2512 4724 y Fm(_)86 b Fn(sup)2601 4812 y Fj(x)p Fe(2)p Fg(\(0)p Fj(;\013)p Fg(\))2894 4724 y Fk(b)2937 4683 y Fj(x)1556 5024 y Fn(=)91 b(sup)1661 5112 y Fj(x)p Fe(2)p Fg(\()p Fj(\013;)p Fg(1\))1954 4884 y Fh(\032)2040 4957 y Fk(e)2086 4921 y Fj(\013x)p 2040 5001 149 4 v 2061 5093 a Fk(x)2118 5064 y Fj(x)2201 5024 y Fk(e)2247 4983 y Fe(\000)p Fj(\013)2366 4884 y Fh(\033)2463 5024 y Fm(_)22 b Fk(b)2594 4983 y Fj(\013)2651 5024 y Fk(;)1900 5339 y Fn(16)p eop %%Page: 17 17 17 16 bop 1115 4 1670 4 v 1115 24 4 20 v 1418 24 V 1948 24 V 2251 24 V 2781 24 V 1115 143 4 120 v 1276 109 a Fk(b)p 1418 143 V 340 w(\032)p 1948 143 V 397 w(b)p 2251 143 V 341 w(\032)p 2781 143 V 1115 163 4 20 v 1418 163 V 1948 163 V 2251 163 V 2781 163 V 1115 167 1670 4 v 1115 287 4 120 v 1269 252 a Fn(2)p 1418 287 V 203 w(.590941)p 1948 287 V 251 w(3)p 2251 287 V 203 w(.455860)p 2781 287 V 1115 406 V 1269 372 a(4)p 1418 406 V 203 w(.372014)p 1948 406 V 251 w(5)p 2251 406 V 203 w(.311999)p 2781 406 V 1115 526 V 1269 491 a(6)p 1418 526 V 203 w(.265677)p 1948 526 V 251 w(7)p 2251 526 V 203 w(.228184)p 2781 526 V 1115 646 V 1269 611 a(8)p 1418 646 V 203 w(.196821)p 1948 646 V 251 w(9)p 2251 646 V 203 w(.169940)p 2781 646 V 1115 765 V 1219 730 a(10)p 1418 765 V 203 w(.146465)p 1948 765 V 202 w(11)p 2251 765 V 202 w(.125653)p 2781 765 V 1115 885 V 1219 850 a(12)p 1418 885 V 203 w(.106973)p 1948 885 V 202 w(13)p 2251 885 V 202 w(.090028)p 2781 885 V 1115 1004 V 1219 969 a(14)p 1418 1004 V 203 w(.074516)p 1948 1004 V 202 w(15)p 2251 1004 V 202 w(.061973)p 2781 1004 V 1115 1124 V 1219 1089 a(16)p 1418 1124 V 203 w(.046876)p 1948 1124 V 202 w(17)p 2251 1124 V 202 w(.034384)p 2781 1124 V 1115 1243 V 1219 1208 a(18)p 1418 1243 V 203 w(.022566)p 1948 1243 V 202 w(19)p 2251 1243 V 202 w(.011245)p 2781 1243 V 1115 1263 4 20 v 1418 1263 V 1948 1263 V 2251 1263 V 2781 1263 V 1115 1267 1670 4 v 559 1529 a Fo(T)-9 b(able)38 b(2.)94 b Fn(V)-8 b(alues)33 b(of)g Fk(b)g Fn(vs.)44 b Fk(\032)33 b Fn(for)g(the)h(Bernoulli)17 b(\(.05\))31 b(distribution.)0 1751 y(b)s(ecause)i Fk(\013b)27 b Fn(=)h Fk(e)652 1715 y Fj(\013)p Fe(\000)p Fg(1)811 1751 y Fk(:)j Fn(The)h(v)-6 b(alue)31 b(of)h Fk(x)f Fn(whic)m(h)h (maximizes)c(the)k(\014rst)f(quan)m(tit)m(y)g(is)g Fk(e)3207 1715 y Fj(\013)p Fe(\000)p Fg(1)3397 1751 y Fn(\(whic)m(h)h(is)f(in)0 1911 y(\()p Fk(\013;)17 b Fn(1\)\).)42 b(By)33 b(manipulating)f(\(2\))g (w)m(e)i(get)1288 2241 y Fk(\014)f Fn(=)1483 2071 y Fh(\()1563 2241 y Fk(e)1609 2200 y Fe(\000)p Fj(\013)1885 2174 y Fk(e)1931 2138 y Fj(\013e)2021 2108 y Fa(\013)p Fb(\000)p Fd(1)p 1740 2218 570 4 v 1740 2312 a Fk(e)1786 2284 y Fj(\013e)1876 2264 y Fa(\013)p Fb(\000)p Fd(1)2020 2312 y Fk(e)2066 2284 y Fe(\000)p Fj(e)2166 2264 y Fa(\013)p Fb(\000)p Fd(1)2321 2071 y Fh(\))2424 2241 y Fm(_)22 b Fk(b)2555 2200 y Fj(\013)1377 2520 y Fn(=)29 b(max)1685 2410 y Fh(n)1751 2520 y Fk(e)1797 2479 y Fj(e)1835 2449 y Fa(\013)p Fb(\000)p Fd(1)1974 2479 y Fe(\000)p Fj(\013)2093 2520 y Fk(;)17 b(b)2181 2479 y Fj(\013)2236 2410 y Fh(o)1377 2710 y Fn(=)29 b Fk(e)1529 2669 y Fj(\013)p Fg(\()p Fj(b)p Fe(\000)p Fg(1\))1785 2710 y Fk(;)0 2991 y Fn(b)s(ecause)h Fk(e)410 2955 y Fj(\013)p Fe(\000)p Fg(1)597 2991 y Fn(=)e Fk(\013b)p Fn(.)42 b(Note)29 b(that)f Fk(e)1373 2955 y Fj(\013)p Fg(\()p Fj(b)p Fe(\000)p Fg(1\))1657 2991 y Fk(<)g(e)i Fn(for)f(an)m(y)f(v)-6 b(alue)29 b(of)g Fk(b)p Fn(,)g(and)g Fk(\014)34 b Fm(!)3037 2920 y(p)p 3120 2920 47 4 v 71 x Fk(e)29 b Fn(as)g Fk(b)e Fm(!)h(1)p Fn(.)42 b(Also,)0 3151 y Fk(\032)27 b Fn(=)h(log)313 3175 y Fj(b)370 3151 y Fk(\014)33 b Fm(\030)28 b Fn(1)p Fk(=)p Fn(\(2)17 b(log)f Fk(b)p Fn(\),)33 b(as)g Fk(b)27 b Fm(!)h(1)p Fn(.)0 3373 y Fo(Example.)165 b Fn(Bernoulli)33 b(distribution)g(with)g(parameter)f Fk(p)c Fm(2)g Fn(\(0)p Fk(;)17 b Fn(1\))p 630 3389 2071 4 v -1 w(.)79 b(Let)45 b Fk(X)52 b Fn(b)s(e)45 b(a)g(Bernoulli\()p Fk(p)p Fn(\))0 3532 y(distributed)34 b(random)f(v)-6 b(ariable,)32 b(then)1341 3823 y Fi(E)1433 3733 y Fh(\010)1491 3814 y Fk(e)1537 3773 y Fe(\000)p Fj(\022)r(X)1716 3733 y Fh(\011)1801 3814 y Fn(=)c Fk(p)23 b Fn(+)f(\(1)g Fm(\000)h Fk(p)p Fn(\))p Fk(e)2424 3773 y Fe(\000)p Fj(\022)2531 3814 y Fk(;)0 4095 y Fn(and)925 4376 y Fk(\026)17 b Fn(\()o Fk(a)p Fn(\))27 b(=)i Fk(b)d Fn(inf)1325 4441 y Fj(\022)r Fe(\025)p Fg(0)1483 4385 y Fi(E)1575 4266 y Fh(n)1641 4376 y Fk(e)1687 4335 y Fj(\022)r Fg(\()p Fj(a)p Fe(\000)p Fj(X)5 b Fg(\))1971 4266 y Fh(o)1159 4594 y Fn(=)29 b Fk(b)d Fn(inf)1325 4659 y Fj(\022)r Fe(\025)p Fg(0)1483 4594 y Fk(e)1529 4552 y Fj(\022)r(a)1617 4594 y Fn(\()p Fk(p)c Fn(+)h(\(1)f Fm(\000)g Fk(p)p Fn(\))p Fk(e)2173 4552 y Fj(\022)2219 4594 y Fn(\))1159 4956 y(=)1265 4721 y Fh(8)1265 4811 y(>)1265 4841 y(<)1265 5020 y(>)1265 5050 y(:)1370 4861 y Fk(b)1430 4750 y Fh(\020)1553 4816 y Fj(p)p 1501 4838 146 4 v 1501 4895 a Fg(1)p Fe(\000)p Fj(a)1658 4750 y Fh(\021)1717 4771 y Fg(1)p Fe(\000)p Fj(a)1884 4750 y Fh(\020)1955 4816 y Fg(1)p Fe(\000)p Fj(p)p 1955 4838 144 4 v 2005 4895 a(a)2110 4750 y Fh(\021)2170 4771 y Fj(a)2218 4861 y Fk(;)99 b Fn(0)28 b Fk(<)g(a)f Fm(\024)i Fn(1)22 b Fm(\000)g Fk(p)p Fn(,)1370 5070 y Fk(b)789 b(;)115 b(a)28 b Fm(\025)g Fn(1)22 b Fm(\000)h Fk(p)p Fn(.)1900 5339 y(17)p eop %%Page: 18 18 18 17 bop 1115 4 1670 4 v 1115 24 4 20 v 1418 24 V 1948 24 V 2251 24 V 2781 24 V 1115 143 4 120 v 1276 109 a Fk(b)p 1418 143 V 340 w(\032)p 1948 143 V 397 w(b)p 2251 143 V 341 w(\032)p 2781 143 V 1115 163 4 20 v 1418 163 V 1948 163 V 2251 163 V 2781 163 V 1115 167 1670 4 v 1115 287 4 120 v 1269 252 a Fn(2)p 1418 287 V 203 w(.522782)p 1948 287 V 251 w(3)p 2251 287 V 203 w(.445452)p 2781 287 V 1115 406 V 1269 372 a(4)p 1418 406 V 203 w(.405182)p 1948 406 V 251 w(5)p 2251 406 V 203 w(.379173)p 2781 406 V 1115 526 V 1269 491 a(6)p 1418 526 V 203 w(.360477)p 1948 526 V 251 w(7)p 2251 526 V 203 w(.346137)p 2781 526 V 1115 646 V 1269 611 a(8)p 1418 646 V 203 w(.334648)p 1948 646 V 251 w(9)p 2251 646 V 203 w(.325150)p 2781 646 V 1115 765 V 1219 730 a(10)p 1418 765 V 203 w(.317110)p 1948 765 V 202 w(11)p 2251 765 V 202 w(.310176)p 2781 765 V 1115 885 V 1219 850 a(12)p 1418 885 V 203 w(.304107)p 1948 885 V 202 w(13)p 2251 885 V 202 w(.298731)p 2781 885 V 1115 1004 V 1219 969 a(14)p 1418 1004 V 203 w(.293919)p 1948 1004 V 202 w(15)p 2251 1004 V 202 w(.289575)p 2781 1004 V 1115 1124 V 1219 1089 a(16)p 1418 1124 V 203 w(.285625)p 1948 1124 V 202 w(17)p 2251 1124 V 202 w(.282010)p 2781 1124 V 1115 1243 V 1219 1208 a(18)p 1418 1243 V 203 w(.278682)p 1948 1243 V 202 w(19)p 2251 1243 V 202 w(.275604)p 2781 1243 V 1115 1363 V 1219 1328 a(20)p 1418 1363 V 203 w(.272745)p 1948 1363 V 202 w(21)p 2251 1363 V 202 w(.270078)p 2781 1363 V 1115 1383 4 20 v 1418 1383 V 1948 1383 V 2251 1383 V 2781 1383 V 1115 1387 1670 4 v 625 1615 a Fo(T)-9 b(able)38 b(3.)93 b Fn(V)-8 b(alues)33 b(of)h Fk(b)f Fn(vs.)44 b Fk(\032)32 b Fn(for)i(the)f(Gamma)17 b(\(3\))30 b(distribution.)0 1820 y(So)37 b(as)g(to)g(de\014ne)h Fk(\013)p Fn(,)g(w)m(e)f(m)m(ust)g(assume)g(that)f Fk(bp)e(<)h Fn(1.)54 b(Th)m(us)39 b Fk(\013)e Fn(is)g(de\014ned)h(as)f(the)h (solution)e(of)i(the)0 1976 y(equation)1300 2170 y Fk(b)1360 2030 y Fh(\022)1537 2103 y Fk(p)p 1444 2147 236 4 v 1444 2239 a Fn(1)22 b Fm(\000)h Fk(\013)1692 2030 y Fh(\023)1765 2050 y Fg(1)p Fe(\000)p Fj(\013)1941 2030 y Fh(\022)2026 2103 y Fn(1)f Fm(\000)h Fk(p)p 2026 2147 222 4 v 2105 2239 a(\013)2260 2030 y Fh(\023)2333 2050 y Fj(\013)2418 2170 y Fn(=)28 b(1)p Fk(:)0 2439 y Fo(Example.)93 b Fn(Gamma)31 b(distribution)i(with)g(parameter)g Fk(r)p 558 2455 1672 4 v 3 w Fn(.)44 b(W)-8 b(e)33 b(\014rst)h(compute)1160 2693 y Fk(\026)17 b Fn(\()p Fk(a)p Fn(\))27 b(=)h Fk(b)f Fn(inf)1560 2759 y Fj(\022)r Fe(\025)p Fg(0)1718 2558 y Fh(Z)1817 2583 y Fe(1)1773 2784 y Fg(0)1930 2626 y Fk(x)1987 2590 y Fj(r)r Fe(\000)p Fg(1)2133 2626 y Fk(e)2179 2590 y Fe(\000)p Fj(x)2292 2626 y Fk(e)2338 2590 y Fj(\022)r Fg(\()p Fj(a)p Fe(\000)p Fj(x)p Fg(\))p 1930 2671 667 4 v 2170 2762 a Fn(\000\()p Fk(r)s Fn(\))2609 2693 y Fk(dx)1395 2979 y Fn(=)h Fk(b)f Fn(inf)1560 3044 y Fj(\022)r Fe(\025)p Fg(0)1834 2911 y Fk(e)1880 2875 y Fj(\022)r(a)p 1730 2956 343 4 v 1730 3047 a Fn(\(1)22 b(+)g Fk(\022)s Fn(\))2029 3018 y Fj(r)1395 3289 y Fn(=)28 b Fk(b)1555 3222 y(e)1601 3179 y Fj(a)1644 3186 y Fn(\()1697 3152 y Fa(r)p 1695 3163 39 4 v 1695 3203 a(a)1745 3179 y Fe(\000)p Fg(1)1847 3186 y Fn(\))p 1555 3266 337 4 v 1622 3288 a Fh(\000)1681 3329 y Fj(r)p 1679 3346 44 4 v 1679 3403 a(a)1734 3288 y Fh(\001)1780 3308 y Fj(r)1925 3289 y Fm(_)23 b Fk(b)1395 3598 y Fn(=)1500 3458 y Fh(\032)1591 3533 y Fk(be)1680 3497 y Fj(r)r Fe(\000)p Fj(a)1846 3453 y Fh(\000)1904 3494 y Fj(a)p 1904 3510 V 1906 3568 a(r)1959 3453 y Fh(\001)2005 3473 y Fj(r)2066 3533 y Fk(;)99 b Fn(if)33 b(0)28 b Fm(\024)g Fk(a)f Fm(\024)h Fk(r)s Fn(;)1591 3668 y Fk(b)415 b(;)116 b(a)27 b Fm(\025)h Fk(r)s Fn(.)0 3856 y(b)s(ecause)37 b Fk(\022)s(a)24 b Fm(\000)g Fk(r)c Fn(log\(1)k(+)g Fk(\022)s Fn(\))36 b(is)g(minimal)d(when)k Fk(a)32 b Fn(=)2145 3817 y Fj(r)p 2094 3833 141 4 v 2094 3891 a Fg(1+)p Fj(\022)2247 3856 y Fn(.)53 b(Th)m(us)37 b Fk(\013)f Fn(is)g(de\014ned)i(as)e(the)h(solution)e(to)0 4013 y(the)f(equation:)1587 4169 y Fk(be)1676 4128 y Fj(r)r Fe(\000)p Fj(\013)1851 4059 y Fh(\020)1923 4102 y Fk(\013)p 1923 4146 65 4 v 1931 4238 a(r)1999 4059 y Fh(\021)2058 4079 y Fj(r)2130 4169 y Fn(=)28 b(1)p Fk(:)0 4373 y Fn(Finally)-8 b(,)968 4529 y Fk(\014)34 b Fn(=)85 b(sup)1163 4618 y Fj(x)p Fe(2)p Fg(\(0)p Fj(;)p Fg(1\))1444 4529 y Fk(\026)1504 4488 y Fj(x)1570 4419 y Fh(\020)1642 4462 y Fk(\013)p 1642 4506 V 1646 4598 a(x)1718 4419 y Fh(\021)1805 4529 y Fn(=)g(sup)1910 4618 y Fj(x)p Fe(2)p Fg(\(0)p Fj(;)p Fg(1\))2191 4529 y Fk(b)2234 4488 y Fj(x)2284 4529 y Fk(e)2330 4488 y Fj(r)r(x)p Fe(\000)p Fj(\013)2550 4419 y Fh(\020)2642 4462 y Fk(\013)p 2622 4506 105 4 v 2622 4598 a(r)s(x)2738 4419 y Fh(\021)2798 4439 y Fj(r)r(x)2904 4529 y Fk(:)0 4790 y Fn(Note)33 b(that)g Fk(\026)517 4754 y Fj(x)583 4790 y Fn(\()p Fk(\013=x)p Fn(\))g(is)g(maximal)d(at)j Fk(x)27 b Fn(=)h Fk(b)1726 4754 y Fg(1)p Fj(=r)1851 4790 y Fk(=r)36 b Fn(and)d(therefore)1185 5065 y Fk(\014)g Fn(=)1391 4997 y Fk(b)1434 4961 y Fj(b)1469 4931 y Fd(1)p Fa(=r)1580 4961 y Fj(\013=r)1716 4997 y Fk(e)1762 4961 y Fj(\013)p Fg(\()p Fj(b)1880 4931 y Fd(1)p Fa(=r)1991 4961 y Fe(\000)p Fg(1\))p 1391 5042 739 4 v 1598 5140 a Fk(b)1641 5111 y Fj(b)1676 5091 y Fd(1)p Fa(=r)1786 5111 y Fj(\013=r)2169 5065 y Fn(=)28 b Fk(e)2320 5024 y Fj(\013)p Fg(\()p Fj(b)2438 4994 y Fd(1)p Fa(=r)2549 5024 y Fe(\000)p Fg(1\))2688 5065 y Fk(:)1900 5339 y Fn(18)p eop %%Page: 19 19 19 18 bop 0 100 a Fm(x)p Fi(6.)39 b(References.)0 324 y Fo(Asmussen,)e(S.,)g(and)h(Hering,)f(H.)c Ff(Branc)m(hing)h(Pro)s (cesses)p Fn(.)45 b(Birkh\177)-50 b(auser,)33 b(Boston,)f(1983.)0 549 y Fo(A)-9 b(threy)i(a,)38 b(K.)f(B.,)g(and)i(Ney,)e(P.)g(E.)c Ff(Branc)m(hing)h(Pro)s(cesses)p Fn(.)45 b(Springer-V)-8 b(erlag,)33 b(Berlin,)f(1972.)0 773 y Fo(Biggins,)47 b(J.)g(D.)40 b Fn(Cherno\013)8 b('s)42 b(theorem)e(in)i(the)f(branc)m (hing)h(random)f(w)m(alk.)67 b Ff(Journal)41 b(of)h(Applied)0 933 y(Probabilit)m(y)32 b(14)41 b Fn(\(1977\),)31 b(630{636.)0 1158 y Fo(Br)n(o)n(wn,)40 b(C.)h(A.,)f(and)g(Purdom,)i(P.)e(W.)35 b Fn(An)h(a)m(v)m(erage)f(time)f(analysis)h(of)h(bac)m(ktrac)m(king.)51 b Ff(SIAM)0 1318 y(Journal)33 b(of)h(Computing)e(10)41 b Fn(\(1981\),)32 b(583{593.)0 1542 y Fo(Hammersley,)f(J.)e(M.)c Fn(P)m(ostulates)h(for)f(subadditiv)m(e)h(pro)s(cesses.)43 b Ff(Annals)26 b(of)f(Probabilit)m(y)f(2)34 b Fn(\(1974\),)0 1703 y(652{680.)0 1927 y Fo(Harris,)j(T.)h(E.)33 b Ff(The)h(Theory)f (of)h(Branc)m(hing)f(Pro)s(cesses)p Fn(.)45 b(Springer-V)-8 b(erlag,)33 b(Berlin,)g(1963.)0 2151 y Fo(Karp,)h(R.)f(M.,)g(and)h (Pearl,)f(J.)d Fn(Searc)m(hing)g(for)f(an)h(optimal)d(path)i(in)g(a)g (tree)h(with)e(random)h(costs.)0 2312 y Ff(Arti\014cial)j(In)m (telligence)i(21)41 b Fn(\(1983\),)31 b(99{117.)0 2536 y Fo(Kingman,)59 b(J.)54 b(F.)g(C.)49 b Fn(The)g(\014rst-birth)h (problem)e(for)h(an)g(age-dep)s(endan)m(t)i(branc)m(hing)f(pro)s(cess.) 0 2696 y Ff(Annals)33 b(of)h(Probabilit)m(y)e(3)p Fn(,)h(5)g(\(1975\),) e(341{345.)0 2921 y Fo(Kumar,)50 b(V.)42 b Fn(Searc)m(h,)k(branc)m(h)e (and)f(b)s(ound,)i(in:)62 b Ff(Encyclop)s(edia)43 b(of)f(Arti\014cial)g (In)m(telligence)h(\(2nd)0 3081 y(edition\))p Fn(.)h(Ed.)66 b(S.)33 b(C.)g(Shapiro,)g(pp.)h(1468{1472.)42 b(Wiley-In)m(terscience,) 35 b(1992.)0 3305 y Fo(McDiarmid,)j(C.)f(J.)h(H.)c Fn(Probabilistic)e (analysis)h(of)g(tree)h(searc)m(h,)g(in:)44 b Ff(Disorder)34 b(in)g(Ph)m(ysical)f(Sys-)0 3466 y(tems)p Fn(.)41 b(Ed.)55 b(D.)27 b(J.)g(A.)g(W)-8 b(elsh)28 b(and)g(G.)f(R.)g(Grimmett,)f(pp.)i (249{260.)41 b(Oxford)28 b(Science)h(Publications,)0 3626 y(1990.)0 3850 y Fo(McDiarmid,)f(C.)f(J.)g(H.,)h(and)f(Pr)n(o)n(v) -9 b(an,)29 b(G.)d(M.)g(A.)c Fn(An)h(exp)s(ected-cost)g(analysis)f(of)h (bac)m(ktrac)m(king)0 4010 y(and)47 b(non-bac)m(ktrac)m(king)g (algorithms.)82 b Ff(IJCAI-91:)70 b(Pro)s(ceedings)47 b(of)f(the)h(Tw)m(elfth)g(In)m(ternational)0 4171 y(Conference)35 b(on)f(Arti\014cial)e(In)m(telligence)39 b Fn(\(1991\),)32 b(172{177.)43 b(Morgan)33 b(Kaufmann)h(Publishing,)f(San)0 4331 y(Mateo,)g(CA,.)0 4555 y Fo(Pearl,)k(J.)32 b Ff(Heuristics:)44 b(In)m(telligen)m(t)33 b(Searc)m(h)g(Strategies)g(for)f(Computer)g (Problem)g(Solving)p Fn(.)43 b(Addi-)0 4715 y(son-W)-8 b(esley)g(,)33 b(Reading,)h(Mass,)g(1984.)0 4940 y Fo(Purdom,)39 b(P.)f(W.)33 b Fn(Searc)m(h)i(rearrangemen)m(t)f(bac)m(ktrac)m(king)f (and)h(p)s(olynomial)d(a)m(v)m(erage)j(time.)44 b Ff(Arti-)0 5100 y(\014cial)33 b(In)m(telligence)h(21)41 b Fn(\(1983\),)31 b(117{133.)1900 5339 y(19)p eop %%Page: 20 20 20 19 bop 0 100 a Fo(Reingold,)44 b(E.)f(M.,)g(Niever)n(gel)-7 b(t,)43 b(J.,)i(and)e(Deo,)h(N.)38 b Ff(Com)m(binatorial)e(Algorithms:) 52 b(Theory)0 259 y(and)34 b(Practice)p Fn(.)44 b(Pren)m(tice)33 b(Hall,)f(Englew)m(o)s(o)s(d)h(Cli\013s,)g(N.J.,)f(1977.)0 478 y Fo(Smith,)43 b(D.)e(R.)36 b Fn(Random)h(trees)g(and)h(the)f (analysis)f(of)h(branc)m(h)i(and)e(b)s(ound)h(pro)s(cedures.)57 b Ff(Journal)0 638 y(of)33 b(the)h(A)m(CM,)f(31)41 b Fn(\(1984\),)32 b(163{188.)0 857 y Fo(Stone,)41 b(H.)g(S.,)g(and)g(Sip) -7 b(ala,)42 b(P.)36 b Fn(The)h(a)m(v)m(erage)f(complexit)m(y)f(of)h (depth-\014rst)i(searc)m(h)g(with)e(bac)m(k-)0 1016 y(trac)m(king)d (and)g(cuto\013.)45 b Ff(IBM)33 b(Journal)g(of)h(Researc)m(h)h(and)e (Dev)m(elopmen)m(t)h(30)40 b Fn(\(1986\),)32 b(242{258.)0 1235 y Fo(W)-12 b(ah,)38 b(B.)f(W.,)g(and)h(Yu,)g(C.)f(F.)c Fn(Sto)s(c)m(hastic)h(mo)s(deling)d(of)j(branc)m(h-and-b)s(ound)i (algorithms)31 b(with)0 1395 y(b)s(est-\014rst)j(searc)m(h.)45 b Ff(IEEE)33 b(T)-8 b(ransactions)33 b(of)h(Soft)m(w)m(are)g (Engineering)g(SE-11)41 b Fn(\(1985\),)31 b(922{934.)0 1614 y Fo(Zhang,)i(W.,)g(and)g(K)m(orf,)h(R.)e(E.)c Fn(An)g(a)m(v)m (erage-case)h(analysis)f(of)h(branc)m(h-and-b)s(ound)i(with)d(appli-)0 1773 y(cations.)51 b Ff(Pro)s(ceedings)36 b(of)g(the)g(10th)g(National) e(Conference)k(on)d(AI|AAAI-92)42 b Fn(\(1992\),)35 b(1{6.)50 b(San)0 1933 y(Jose,)33 b(CA.)1900 5339 y(20)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF