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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"

"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">

<head>

<title>Dictionaries — Pense Python 2e documentation</title>

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URL_ROOT: './',

VERSION: '2e',

COLLAPSE_INDEX: false,

FILE_SUFFIX: '.html',

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</head>

<h1>Dictionaries<a class="headerlink" href="#dictionaries" title="Permalink to this headline">¶</a></h1>

This chapter presents another built-in type called a dictionary.

Dictionaries are one of Python’s best features; they are the building

blocks of many efficient and elegant algorithms.

<h2>A dictionary is a mapping<a class="headerlink" href="#a-dictionary-is-a-mapping" title="Permalink to this headline">¶</a></h2>

A dictionary is like a list, but more general. In a list, the

indices have to be integers; in a dictionary they can be (almost) any

type.

A dictionary contains a collection of indices, which are called

keys, and a collection of values. Each key is associated with a

single value. The association of a key and a value is called a

key-value pair or sometimes an item.

In mathematical language, a dictionary represents a mapping from

keys to values, so you can also say that each key “maps to” a value. As

an example, we’ll build a dictionary that maps from English to Spanish

words, so the keys and the values are all strings.

The function dict creates a new dictionary with no items. Because dict

is the name of a built-in function, you should avoid using it as a

variable name.

<div class="highlight-python"><div class="highlight"><pre>>>> eng2sp = dict()

>>> eng2sp

{}

</pre></div>

</div>

The squiggly-brackets, <code class="docutils literal">{}</code>, represent an empty dictionary. To add

items to the dictionary, you can use square brackets:

<div class="highlight-python"><div class="highlight"><pre>>>> eng2sp['one'] = 'uno'

</pre></div>

</div>

This line creates an item that maps from the key <code class="docutils literal">'one'</code> to the value

<code class="docutils literal">'uno'</code>. If we print the dictionary again, we see a key-value pair

with a colon between the key and value:

<div class="highlight-python"><div class="highlight"><pre>>>> eng2sp

{'one': 'uno'}

</pre></div>

</div>

This output format is also an input format. For example, you can create

a new dictionary with three items:

<div class="highlight-python"><div class="highlight"><pre>>>> eng2sp = {'one': 'uno', 'two': 'dos', 'three': 'tres'}

</pre></div>

</div>

But if you print eng2sp, you might be surprised:

<div class="highlight-python"><div class="highlight"><pre>>>> eng2sp

{'one': 'uno', 'three': 'tres', 'two': 'dos'}

</pre></div>

</div>

The order of the key-value pairs might not be the same. If you type the

same example on your computer, you might get a different result. In

general, the order of items in a dictionary is unpredictable.

But that’s not a problem because the elements of a dictionary are never

indexed with integer indices. Instead, you use the keys to look up the

corresponding values:

<div class="highlight-python"><div class="highlight"><pre>>>> eng2sp['two']

'dos'

</pre></div>

</div>

The key <code class="docutils literal">'two'</code> always maps to the value <code class="docutils literal">'dos'</code> so the order of the

items doesn’t matter.

If the key isn’t in the dictionary, you get an exception:

<div class="highlight-python"><div class="highlight"><pre>>>> eng2sp['four']

KeyError: 'four'

</pre></div>

</div>

The len function works on dictionaries; it returns the number of

key-value pairs:

<div class="highlight-python"><div class="highlight"><pre>>>> len(eng2sp)

3

</pre></div>

</div>

The in operator works on dictionaries, too; it tells you whether

something appears as a key in the dictionary (appearing as a value is

not good enough).

<div class="highlight-python"><div class="highlight"><pre>>>> 'one' in eng2sp

True

>>> 'uno' in eng2sp

False

</pre></div>

</div>

To see whether something appears as a value in a dictionary, you can use

the method values, which returns a collection of values, and then use

the in operator:

<div class="highlight-python"><div class="highlight"><pre>>>> vals = eng2sp.values()

>>> 'uno' in vals

True

</pre></div>

</div>

The in operator uses different algorithms for lists and dictionaries.

For lists, it searches the elements of the list in order, as in

Section [find]. As the list gets longer, the search time gets longer in

direct proportion.

For dictionaries, Python uses an algorithm called a hashtable that

has a remarkable property: the in operator takes about the same amount

of time no matter how many items are in the dictionary. I explain how

that’s possible in Section [hashtable], but the explanation might not

make sense until you’ve read a few more chapters.

</div>

<h2>Dictionary as a collection of counters<a class="headerlink" href="#dictionary-as-a-collection-of-counters" title="Permalink to this headline">¶</a></h2>

Suppose you are given a string and you want to count how many times each

letter appears. There are several ways you could do it:

<li>You could create 26 variables, one for each letter of the alphabet.

Then you could traverse the string and, for each character, increment

the corresponding counter, probably using a chained conditional.</li>

<li>You could create a list with 26 elements. Then you could convert each

character to a number (using the built-in function ord), use the

number as an index into the list, and increment the appropriate

counter.</li>

<li>You could create a dictionary with characters as keys and counters as

the corresponding values. The first time you see a character, you

would add an item to the dictionary. After that you would increment

the value of an existing item.</li>

</ol>

Each of these options performs the same computation, but each of them

implements that computation in a different way.

An implementation is a way of performing a computation; some

implementations are better than others. For example, an advantage of the

dictionary implementation is that we don’t have to know ahead of time

which letters appear in the string and we only have to make room for the

letters that do appear.

Here is what the code might look like:

<div class="highlight-python"><div class="highlight"><pre>def histogram(s):

d = dict()

for c in s:

if c not in d:

d[c] = 1

else:

d[c] += 1

return d

</pre></div>

</div>

The name of the function is histogram, which is a statistical term for a

collection of counters (or frequencies).

The first line of the function creates an empty dictionary. The for loop

traverses the string. Each time through the loop, if the character c is

not in the dictionary, we create a new item with key c and the initial

value 1 (since we have seen this letter once). If c is already in the

dictionary we increment d[c].

Here’s how it works:

<div class="highlight-python"><div class="highlight"><pre>>>> h = histogram('brontosaurus')

>>> h

{'a': 1, 'b': 1, 'o': 2, 'n': 1, 's': 2, 'r': 2, 'u': 2, 't': 1}

</pre></div>

</div>

The histogram indicates that the letters <code class="docutils literal">'a'</code> and <code class="docutils literal">'b'</code> appear

once; <code class="docutils literal">'o'</code> appears twice, and so on.

Dictionaries have a method called get that takes a key and a default

value. If the key appears in the dictionary, get returns the

corresponding value; otherwise it returns the default value. For

example:

>>> h

{'a': 1}

>>> h.get('a', 0)

1

>>> h.get('b', 0)

0

</pre></div>

</div>

As an exercise, use get to write histogram more concisely. You should be

able to eliminate the if statement.

</div>

<h2>Looping and dictionaries<a class="headerlink" href="#looping-and-dictionaries" title="Permalink to this headline">¶</a></h2>

If you use a dictionary in a for statement, it traverses the keys of the

dictionary. For example, <code class="docutils literal">print_hist</code> prints each key and the

corresponding value:

<div class="highlight-python"><div class="highlight"><pre>def print_hist(h):

for c in h:

print(c, h[c])

</pre></div>

</div>

Here’s what the output looks like:

>>> print_hist(h)

a 1

p 1

r 2

t 1

o 1

</pre></div>

</div>

Again, the keys are in no particular order. To traverse the keys in

sorted order, you can use the built-in function sorted:

<div class="highlight-python"><div class="highlight"><pre>>>> for key in sorted(h):

... print(key, h[key])

a 1

o 1

p 1

r 2

t 1

</pre></div>

</div>

<h2>Reverse lookup<a class="headerlink" href="#reverse-lookup" title="Permalink to this headline">¶</a></h2>

Given a dictionary d and a key k, it is easy to find the corresponding

value v = d[k]. This operation is called a lookup.

But what if you have v and you want to find k? You have two problems:

first, there might be more than one key that maps to the value v.

Depending on the application, you might be able to pick one, or you

might have to make a list that contains all of them. Second, there is no

simple syntax to do a reverse lookup; you have to search.

Here is a function that takes a value and returns the first key that

maps to that value:

<div class="highlight-python"><div class="highlight"><pre>def reverse_lookup(d, v):

for k in d:

if d[k] == v:

return k

raise LookupError()

</pre></div>

</div>

This function is yet another example of the search pattern, but it uses

a feature we haven’t seen before, raise. The raise statement causes

an exception; in this case it causes a LookupError, which is a built-in

exception used to indicate that a lookup operation failed.

If we get to the end of the loop, that means v doesn’t appear in the

dictionary as a value, so we raise an exception.

Here is an example of a successful reverse lookup:

>>> key = reverse_lookup(h, 2)

>>> key

'r'

</pre></div>

</div>

And an unsuccessful one:

<div class="highlight-python"><div class="highlight"><pre>>>> key = reverse_lookup(h, 3)

Traceback (most recent call last):

File "<stdin>", line 1, in <module>

File "<stdin>", line 5, in reverse_lookup

LookupError

</pre></div>

</div>

The effect when you raise an exception is the same as when Python raises

one: it prints a traceback and an error message.

The raise statement can take a detailed error message as an optional

argument. For example:

<div class="highlight-python"><div class="highlight"><pre>>>> raise LookupError('value does not appear in the dictionary')

Traceback (most recent call last):

File "<stdin>", line 1, in ?

LookupError: value does not appear in the dictionary

</pre></div>

</div>

A reverse lookup is much slower than a forward lookup; if you have to do

it often, or if the dictionary gets big, the performance of your program

will suffer.

</div>

<h2>Dictionaries and lists<a class="headerlink" href="#dictionaries-and-lists" title="Permalink to this headline">¶</a></h2>

Lists can appear as values in a dictionary. For example, if you are

given a dictionary that maps from letters to frequencies, you might want

to invert it; that is, create a dictionary that maps from frequencies to

letters. Since there might be several letters with the same frequency,

each value in the inverted dictionary should be a list of letters.

Here is a function that inverts a dictionary:

<div class="highlight-python"><div class="highlight"><pre>def invert_dict(d):

inverse = dict()

for key in d:

val = d[key]

if val not in inverse:

inverse[val] = [key]

else:

inverse[val].append(key)

return inverse

</pre></div>

</div>

Each time through the loop, key gets a key from d and val gets the

corresponding value. If val is not in inverse, that means we haven’t

seen it before, so we create a new item and initialize it with a

singleton (a list that contains a single element). Otherwise we have

seen this value before, so we append the corresponding key to the list.

Here is an example:

<div class="highlight-python"><div class="highlight"><pre>>>> hist = histogram('parrot')

>>> hist

{'a': 1, 'p': 1, 'r': 2, 't': 1, 'o': 1}

>>> inverse = invert_dict(hist)

>>> inverse

{1: ['a', 'p', 't', 'o'], 2: ['r']}

</pre></div>

</div>

State diagram.

</div>

Figure [fig.dict1] is a state diagram showing hist and inverse. A

dictionary is represented as a box with the type dict above it and the

key-value pairs inside. If the values are integers, floats or strings, I

draw them inside the box, but I usually draw lists outside the box, just

to keep the diagram simple.

Lists can be values in a dictionary, as this example shows, but they

cannot be keys. Here’s what happens if you try:

<div class="highlight-python"><div class="highlight"><pre>>>> t = [1, 2, 3]

>>> d = dict()

>>> d[t] = 'oops'

Traceback (most recent call last):

File "<stdin>", line 1, in ?

TypeError: list objects are unhashable

</pre></div>

</div>

I mentioned earlier that a dictionary is implemented using a hashtable

and that means that the keys have to be hashable.

A hash is a function that takes a value (of any kind) and returns an

integer. Dictionaries use these integers, called hash values, to store

and look up key-value pairs.

This system works fine if the keys are immutable. But if the keys are

mutable, like lists, bad things happen. For example, when you create a

key-value pair, Python hashes the key and stores it in the corresponding

location. If you modify the key and then hash it again, it would go to a

different location. In that case you might have two entries for the same

key, or you might not be able to find a key. Either way, the dictionary

wouldn’t work correctly.

That’s why keys have to be hashable, and why mutable types like lists

aren’t. The simplest way to get around this limitation is to use tuples,

which we will see in the next chapter.

Since dictionaries are mutable, they can’t be used as keys, but they

can be used as values.

</div>

<h2>Memos<a class="headerlink" href="#memos" title="Permalink to this headline">¶</a></h2>

If you played with the fibonacci function from

Section [one.more.example], you might have noticed that the bigger the

argument you provide, the longer the function takes to run. Furthermore,

the run time increases quickly.

To understand why, consider Figure [fig.fibonacci], which shows the

call graph for fibonacci with n=4:

Call graph.

</div>

A call graph shows a set of function frames, with lines connecting each

frame to the frames of the functions it calls. At the top of the graph,

fibonacci with n=4 calls fibonacci with n=3 and n=2. In turn, fibonacci

with n=3 calls fibonacci with n=2 and n=1. And so on.

Count how many times fibonacci(0) and fibonacci(1) are called. This is

an inefficient solution to the problem, and it gets worse as the

argument gets bigger.

One solution is to keep track of values that have already been computed

by storing them in a dictionary. A previously computed value that is

stored for later use is called a memo. Here is a “memoized” version

of fibonacci:

<div class="highlight-python"><div class="highlight"><pre>known = {0:0, 1:1}

def fibonacci(n):

if n in known:

return known[n]

res = fibonacci(n-1) + fibonacci(n-2)

known[n] = res

return res

</pre></div>

</div>

known is a dictionary that keeps track of the Fibonacci numbers we

already know. It starts with two items: 0 maps to 0 and 1 maps to 1.

Whenever fibonacci is called, it checks known. If the result is already

there, it can return immediately. Otherwise it has to compute the new

value, add it to the dictionary, and return it.

If you run this version of fibonacci and compare it with the original,

you will find that it is much faster.

</div>

<h2>Global variables<a class="headerlink" href="#global-variables" title="Permalink to this headline">¶</a></h2>

In the previous example, known is created outside the function, so it

belongs to the special frame called <code class="docutils literal">__main__</code>. Variables in

<code class="docutils literal">__main__</code> are sometimes called global because they can be

accessed from any function. Unlike local variables, which disappear when

their function ends, global variables persist from one function call to

the next.

It is common to use global variables for flags; that is, boolean

variables that indicate (“flag”) whether a condition is true. For

example, some programs use a flag named verbose to control the level of

detail in the output:

<div class="highlight-python"><div class="highlight"><pre>verbose = True

def example1():

if verbose:

print('Running example1')

</pre></div>

</div>

If you try to reassign a global variable, you might be surprised. The

following example is supposed to keep track of whether the function has

been called:

<div class="highlight-python"><div class="highlight"><pre>been_called = False

def example2():

been_called = True # WRONG

</pre></div>

</div>

But if you run it you will see that the value of <code class="docutils literal">been_called</code> doesn’t

change. The problem is that example2 creates a new local variable named

<code class="docutils literal">been_called</code>. The local variable goes away when the function ends,

and has no effect on the global variable.

To reassign a global variable inside a function you have to declare

the global variable before you use it:

<div class="highlight-python"><div class="highlight"><pre>been_called = False

def example2():

global been_called

been_called = True

</pre></div>

</div>

The global statement tells the interpreter something like, “In this

function, when I say <code class="docutils literal">been_called</code>, I mean the global variable; don’t

create a local one.”

Here’s an example that tries to update a global variable:

<div class="highlight-python"><div class="highlight"><pre>count = 0

def example3():

count = count + 1 # WRONG

</pre></div>

</div>

If you run it you get:

<div class="highlight-python"><div class="highlight"><pre>UnboundLocalError: local variable 'count' referenced before assignment

</pre></div>

</div>

Python assumes that count is local, and under that assumption you are

reading it before writing it. The solution, again, is to declare count

global.

<div class="highlight-python"><div class="highlight"><pre>def example3():

global count

count += 1

</pre></div>

</div>

If a global variable refers to a mutable value, you can modify the value

without declaring the variable:

def example4():

known[2] = 1

</pre></div>

</div>

So you can add, remove and replace elements of a global list or

dictionary, but if you want to reassign the variable, you have to

declare it:

<div class="highlight-python"><div class="highlight"><pre>def example5():

global known

known = dict()

</pre></div>

</div>

Global variables can be useful, but if you have a lot of them, and you

modify them frequently, they can make programs hard to debug.

</div>

<h2>Debugging<a class="headerlink" href="#debugging" title="Permalink to this headline">¶</a></h2>

As you work with bigger datasets it can become unwieldy to debug by

printing and checking the output by hand. Here are some suggestions for

debugging large datasets:

<dt>Scale down the input:</dt>

<dd>If possible, reduce the size of the dataset. For example if the

program reads a text file, start with just the first 10 lines, or

with the smallest example you can find. You can either edit the

files themselves, or (better) modify the program so it reads only

the first n lines.

If there is an error, you can reduce n to the smallest value that

manifests the error, and then increase it gradually as you find and

correct errors.

</dd>

<dt>Check summaries and types:</dt>

<dd>Instead of printing and checking the entire dataset, consider

printing summaries of the data: for example, the number of items in

a dictionary or the total of a list of numbers.

A common cause of runtime errors is a value that is not the right

type. For debugging this kind of error, it is often enough to print

the type of a value.

</dd>

<dt>Write self-checks:</dt>

<dd>Sometimes you can write code to check for errors automatically. For

example, if you are computing the average of a list of numbers, you

could check that the result is not greater than the largest element

in the list or less than the smallest. This is called a “sanity

check” because it detects results that are “insane”.

Another kind of check compares the results of two different

computations to see if they are consistent. This is called a

“consistency check”.

</dd>

<dt>Format the output:</dt>

<dd>Formatting debugging output can make it easier to spot an error. We

saw an example in Section [factdebug]. The pprint module provides a

pprint function that displays built-in types in a more

human-readable format (pprint stands for “pretty print”).</dd>

</dl>

Again, time you spend building scaffolding can reduce the time you spend

debugging.

</div>

<h2>Glossary<a class="headerlink" href="#glossary" title="Permalink to this headline">¶</a></h2>

<dt>mapeamento (mapping)</dt>

<dd>A relationship in which each element of one set corresponds to an element of another set.</dd>

<dt>dicionário (dictionary)</dt>

<dd>A mapping from keys to their corresponding values.</dd>

<dt>par chave-valor (key-value pair)</dt>

<dd>The representation of the mapping from a key to a value.</dd>

<dd>In a dictionary, another name for a key-value pair.</dd>

<dt>chave (key)</dt>

<dd>An object that appears in a dictionary as the first part of a key-value pair.</dd>

<dt>valor (value)</dt>

<dd>An object that appears in a dictionary as the second part of a key-value pair. This is more specific than our previous use of the word “value”.</dd>

<dt>implementação (implementation)</dt>

<dd>A way of performing a computation.</dd>

<dt>tabela de hash (hashtable)</dt>

<dd>The algorithm used to implement Python dictionaries.</dd>

<dt>função de hash (hash function)</dt>

<dd>A function used by a hashtable to compute the location for a key.</dd>

<dt>hashable ()</dt>

<dd>A type that has a hash function. Immutable types like integers, floats and strings are hashable; mutable types like lists and dictionaries are not.</dd>

<dt>busca (lookup)</dt>

<dd>A dictionary operation that takes a key and finds the corresponding value.</dd>

<dt>busca invertida (reverse lookup)</dt>

<dd>A dictionary operation that takes a value and finds one or more keys that map to it.</dd>

<dt><code class="docutils literal">raise</code>, instrução (<code class="docutils literal">raise</code> statement)</dt>

<dd>A statement that (deliberately) raises an exception.</dd>

<dt>singleton</dt>

<dd>A list (or other sequence) with a single element.</dd>

<dt>diagrama de chamadas (call graph)</dt>

<dd>A diagram that shows every frame created during the execution of a program, with an arrow from each caller to each callee.</dd>

<dt>memo (“lembrete”)</dt>

<dd>A computed value stored to avoid unnecessary future computation.</dd>

<dt>variável global (global variable)</dt>

<dd>A variable defined outside a function. Global variables can be accessed from any function.</dd>

<dt><code class="docutils literal">global</code>, declaração (<code class="docutils literal">global</code> statement)</dt>

<dd>A statement that declares a variable name global.</dd>

<dt>flag (“indicador”)</dt>

<dd>A boolean variable used to indicate whether a condition is true.</dd>

<dt>declaração (declaration)</dt>

<dd>A statement like global that tells the interpreter something about a variable.</dd>

</dl>

</div>

<h2>Exercises<a class="headerlink" href="#exercises" title="Permalink to this headline">¶</a></h2>

[wordlist2]

Write a function that reads the words in words.txt and stores them as

keys in a dictionary. It doesn’t matter what the values are. Then you

can use the in operator as a fast way to check whether a string is in

the dictionary.

If you did Exercise [wordlist1], you can compare the speed of this

implementation with the list in operator and the bisection search.

[setdefault]

Read the documentation of the dictionary method setdefault and use it to

write a more concise version of <code class="docutils literal">invert_dict</code>. Solution:

<a class="reference external" href="http://thinkpython2.com/code/invert_dict.py">http://thinkpython2.com/code/invert_dict.py</a>.

Memoize the Ackermann function from Exercise [ackermann] and see if

memoization makes it possible to evaluate the function with bigger

arguments. Hint: no. Solution:

<a class="reference external" href="http://thinkpython2.com/code/ackermann_memo.py">http://thinkpython2.com/code/ackermann_memo.py</a>.

If you did Exercise [duplicate], you already have a function named

<code class="docutils literal">has_duplicates</code> that takes a list as a parameter and returns True if

there is any object that appears more than once in the list.

Use a dictionary to write a faster, simpler version of

<code class="docutils literal">has_duplicates</code>. Solution:

<a class="reference external" href="http://thinkpython2.com/code/has_duplicates.py">http://thinkpython2.com/code/has_duplicates.py</a>.

[exrotatepairs]

Two words are “rotate pairs” if you can rotate one of them and get the

other (see <code class="docutils literal">rotate_word</code> in Exercise [exrotate]).

Write a program that reads a wordlist and finds all the rotate pairs.

Solution: <a class="reference external" href="http://thinkpython2.com/code/rotate_pairs.py">http://thinkpython2.com/code/rotate_pairs.py</a>.

Here’s another Puzzler from Car Talk

(<a class="reference external" href="http://www.cartalk.com/content/puzzlers">http://www.cartalk.com/content/puzzlers</a>):

<div>This was sent in by a fellow named Dan O’Leary. He came upon a

common one-syllable, five-letter word recently that has the

following unique property. When you remove the first letter, the

remaining letters form a homophone of the original word, that is a

word that sounds exactly the same. Replace the first letter, that

is, put it back and remove the second letter and the result is yet

another homophone of the original word. And the question is, what’s

the word?

Now I’m going to give you an example that doesn’t work. Let’s look

at the five-letter word, ‘wrack.’ W-R-A-C-K, you know like to ‘wrack

with pain.’ If I remove the first letter, I am left with a

four-letter word, ’R-A-C-K.’ As in, ‘Holy cow, did you see the rack

on that buck! It must have been a nine-pointer!’ It’s a perfect

homophone. If you put the ‘w’ back, and remove the ‘r,’ instead,

you’re left with the word, ‘wack,’ which is a real word, it’s just

not a homophone of the other two words.

But there is, however, at least one word that Dan and we know of,

which will yield two homophones if you remove either of the first

two letters to make two, new four-letter words. The question is,

what’s the word?

</div></blockquote>

You can use the dictionary from Exercise [wordlist2] to check whether a

string is in the word list.

To check whether two words are homophones, you can use the CMU

Pronouncing Dictionary. You can download it from

<a class="reference external" href="http://www.speech.cs.cmu.edu/cgi-bin/cmudict">http://www.speech.cs.cmu.edu/cgi-bin/cmudict</a> or from

<a class="reference external" href="http://thinkpython2.com/code/c06d">http://thinkpython2.com/code/c06d</a> and you can also download

<a class="reference external" href="http://thinkpython2.com/code/pronounce.py">http://thinkpython2.com/code/pronounce.py</a>, which provides a function

named <code class="docutils literal">read_dictionary</code> that reads the pronouncing dictionary and

returns a Python dictionary that maps from each word to a string that

describes its primary pronunciation.

Write a program that lists all the words that solve the Puzzler.

Solution: <a class="reference external" href="http://thinkpython2.com/code/homophone.py">http://thinkpython2.com/code/homophone.py</a>.

</div>

<h3><a href="index.html">Table Of Contents</a></h3>

<ul>

<li><a class="reference internal" href="#">Dictionaries</a><ul>

<li><a class="reference internal" href="#a-dictionary-is-a-mapping">A dictionary is a mapping</a></li>

<li><a class="reference internal" href="#dictionary-as-a-collection-of-counters">Dictionary as a collection of counters</a></li>

<li><a class="reference internal" href="#looping-and-dictionaries">Looping and dictionaries</a></li>

<li><a class="reference internal" href="#reverse-lookup">Reverse lookup</a></li>

<li><a class="reference internal" href="#dictionaries-and-lists">Dictionaries and lists</a></li>

<li><a class="reference internal" href="#memos">Memos</a></li>

<li><a class="reference internal" href="#global-variables">Global variables</a></li>

<li><a class="reference internal" href="#debugging">Debugging</a></li>

<li><a class="reference internal" href="#glossary">Glossary</a></li>

<li><a class="reference internal" href="#exercises">Exercises</a></li>

</ul>

</li>

</ul>

<h3>Related Topics</h3>

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<li>Next: <a href="12-tuple.html" title="next chapter">Tuples</a></li>

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