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 <div class="section" id="module-itertools">
<h1>10.1. <a class="reference internal" href="#module-itertools" title="itertools: Functions creating iterators for efficient looping."><code class="xref py py-mod docutils literal notranslate">itertools</code></a> — Functions creating iterators for efficient looping<a class="headerlink" href="#module-itertools" title="Permalink to this headline">¶</a></h1>
<hr class="docutils" />
This module implements a number of <a class="reference internal" href="../glossary.html#term-iterator">iterator</a> building blocks inspired
by constructs from APL, Haskell, and SML. Each has been recast in a form
suitable for Python.
The module standardizes a core set of fast, memory efficient tools that are
useful by themselves or in combination. Together, they form an “iterator
algebra” making it possible to construct specialized tools succinctly and
efficiently in pure Python.
For instance, SML provides a tabulation tool: <code class="docutils literal notranslate">tabulate(f)</code> which produces a
sequence <code class="docutils literal notranslate">f(0), f(1), ...</code>. The same effect can be achieved in Python
by combining <a class="reference internal" href="functions.html#map" title="map"><code class="xref py py-func docutils literal notranslate">map()</code></a> and <a class="reference internal" href="#itertools.count" title="itertools.count"><code class="xref py py-func docutils literal notranslate">count()</code></a> to form <code class="docutils literal notranslate">map(f, count())</code>.
These tools and their built-in counterparts also work well with the high-speed
functions in the <a class="reference internal" href="operator.html#module-operator" title="operator: Functions corresponding to the standard operators."><code class="xref py py-mod docutils literal notranslate">operator</code></a> module. For example, the multiplication
operator can be mapped across two vectors to form an efficient dot-product:
<code class="docutils literal notranslate">sum(map(operator.mul, vector1, vector2))</code>.
Infinite iterators:
<table border="1" class="docutils">
<colgroup>
<col width="14%" />
<col width="14%" />
<col width="39%" />
<col width="33%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">Iterator</th>
<th class="head">Arguments</th>
<th class="head">Results</th>
<th class="head">Example</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><a class="reference internal" href="#itertools.count" title="itertools.count"><code class="xref py py-func docutils literal notranslate">count()</code></a></td>
<td>start, [step]</td>
<td>start, start+step, start+2*step, …</td>
<td><code class="docutils literal notranslate">count(10) --&gt; 10 11 12 13 14 ...</code></td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#itertools.cycle" title="itertools.cycle"><code class="xref py py-func docutils literal notranslate">cycle()</code></a></td>
<td>p</td>
<td>p0, p1, … plast, p0, p1, …</td>
<td><code class="docutils literal notranslate">cycle('ABCD') --&gt; A B C D A B C D ...</code></td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#itertools.repeat" title="itertools.repeat"><code class="xref py py-func docutils literal notranslate">repeat()</code></a></td>
<td>elem [,n]</td>
<td>elem, elem, elem, … endlessly or up to n times</td>
<td><code class="docutils literal notranslate">repeat(10, 3) --&gt; 10 10 10</code></td>
</tr>
</tbody>
</table>
Iterators terminating on the shortest input sequence:
<table border="1" class="docutils">
<colgroup>
<col width="17%" />
<col width="17%" />
<col width="30%" />
<col width="37%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">Iterator</th>
<th class="head">Arguments</th>
<th class="head">Results</th>
<th class="head">Example</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><a class="reference internal" href="#itertools.accumulate" title="itertools.accumulate"><code class="xref py py-func docutils literal notranslate">accumulate()</code></a></td>
<td>p [,func]</td>
<td>p0, p0+p1, p0+p1+p2, …</td>
<td><code class="docutils literal notranslate">accumulate([1,2,3,4,5]) --&gt; 1 3 6 10 15</code></td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#itertools.chain" title="itertools.chain"><code class="xref py py-func docutils literal notranslate">chain()</code></a></td>
<td>p, q, …</td>
<td>p0, p1, … plast, q0, q1, …</td>
<td><code class="docutils literal notranslate">chain('ABC', 'DEF') --&gt; A B C D E F</code></td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#itertools.chain.from_iterable" title="itertools.chain.from_iterable"><code class="xref py py-func docutils literal notranslate">chain.from_iterable()</code></a></td>
<td>iterable</td>
<td>p0, p1, … plast, q0, q1, …</td>
<td><code class="docutils literal notranslate">chain.from_iterable(['ABC', 'DEF']) --&gt; A B C D E F</code></td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#itertools.compress" title="itertools.compress"><code class="xref py py-func docutils literal notranslate">compress()</code></a></td>
<td>data, selectors</td>
<td>(d[0] if s[0]), (d[1] if s[1]), …</td>
<td><code class="docutils literal notranslate">compress('ABCDEF', [1,0,1,0,1,1]) --&gt; A C E F</code></td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#itertools.dropwhile" title="itertools.dropwhile"><code class="xref py py-func docutils literal notranslate">dropwhile()</code></a></td>
<td>pred, seq</td>
<td>seq[n], seq[n+1], starting when pred fails</td>
<td><code class="docutils literal notranslate">dropwhile(lambda x: x&lt;5, [1,4,6,4,1]) --&gt; 6 4 1</code></td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#itertools.filterfalse" title="itertools.filterfalse"><code class="xref py py-func docutils literal notranslate">filterfalse()</code></a></td>
<td>pred, seq</td>
<td>elements of seq where pred(elem) is false</td>
<td><code class="docutils literal notranslate">filterfalse(lambda x: x%2, range(10)) --&gt; 0 2 4 6 8</code></td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#itertools.groupby" title="itertools.groupby"><code class="xref py py-func docutils literal notranslate">groupby()</code></a></td>
<td>iterable[, key]</td>
<td>sub-iterators grouped by value of key(v)</td>
<td>&#160;</td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#itertools.islice" title="itertools.islice"><code class="xref py py-func docutils literal notranslate">islice()</code></a></td>
<td>seq, [start,] stop [, step]</td>
<td>elements from seq[start:stop:step]</td>
<td><code class="docutils literal notranslate">islice('ABCDEFG', 2, None) --&gt; C D E F G</code></td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#itertools.starmap" title="itertools.starmap"><code class="xref py py-func docutils literal notranslate">starmap()</code></a></td>
<td>func, seq</td>
<td>func(*seq[0]), func(*seq[1]), …</td>
<td><code class="docutils literal notranslate">starmap(pow, [(2,5), (3,2), (10,3)]) --&gt; 32 9 1000</code></td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#itertools.takewhile" title="itertools.takewhile"><code class="xref py py-func docutils literal notranslate">takewhile()</code></a></td>
<td>pred, seq</td>
<td>seq[0], seq[1], until pred fails</td>
<td><code class="docutils literal notranslate">takewhile(lambda x: x&lt;5, [1,4,6,4,1]) --&gt; 1 4</code></td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#itertools.tee" title="itertools.tee"><code class="xref py py-func docutils literal notranslate">tee()</code></a></td>
<td>it, n</td>
<td>it1, it2, … itn splits one iterator into n</td>
<td>&#160;</td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#itertools.zip_longest" title="itertools.zip_longest"><code class="xref py py-func docutils literal notranslate">zip_longest()</code></a></td>
<td>p, q, …</td>
<td>(p[0], q[0]), (p[1], q[1]), …</td>
<td><code class="docutils literal notranslate">zip_longest('ABCD', 'xy', fillvalue='-') --&gt; Ax By C- D-</code></td>
</tr>
</tbody>
</table>
Combinatoric iterators:
<table border="1" class="docutils">
<colgroup>
<col width="36%" />
<col width="16%" />
<col width="48%" />
</colgroup>
<thead valign="bottom">
<tr class="row-odd"><th class="head">Iterator</th>
<th class="head">Arguments</th>
<th class="head">Results</th>
</tr>
</thead>
<tbody valign="top">
<tr class="row-even"><td><a class="reference internal" href="#itertools.product" title="itertools.product"><code class="xref py py-func docutils literal notranslate">product()</code></a></td>
<td>p, q, … [repeat=1]</td>
<td>cartesian product, equivalent to a nested for-loop</td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#itertools.permutations" title="itertools.permutations"><code class="xref py py-func docutils literal notranslate">permutations()</code></a></td>
<td>p[, r]</td>
<td>r-length tuples, all possible orderings, no repeated elements</td>
</tr>
<tr class="row-even"><td><a class="reference internal" href="#itertools.combinations" title="itertools.combinations"><code class="xref py py-func docutils literal notranslate">combinations()</code></a></td>
<td>p, r</td>
<td>r-length tuples, in sorted order, no repeated elements</td>
</tr>
<tr class="row-odd"><td><a class="reference internal" href="#itertools.combinations_with_replacement" title="itertools.combinations_with_replacement"><code class="xref py py-func docutils literal notranslate">combinations_with_replacement()</code></a></td>
<td>p, r</td>
<td>r-length tuples, in sorted order, with repeated elements</td>
</tr>
<tr class="row-even"><td><code class="docutils literal notranslate">product('ABCD', repeat=2)</code></td>
<td>&#160;</td>
<td><code class="docutils literal notranslate">AA AB AC AD BA BB BC BD CA CB CC CD DA DB DC DD</code></td>
</tr>
<tr class="row-odd"><td><code class="docutils literal notranslate">permutations('ABCD', 2)</code></td>
<td>&#160;</td>
<td><code class="docutils literal notranslate">AB AC AD BA BC BD CA CB CD DA DB DC</code></td>
</tr>
<tr class="row-even"><td><code class="docutils literal notranslate">combinations('ABCD', 2)</code></td>
<td>&#160;</td>
<td><code class="docutils literal notranslate">AB AC AD BC BD CD</code></td>
</tr>
<tr class="row-odd"><td><code class="docutils literal notranslate">combinations_with_replacement('ABCD', 2)</code></td>
<td>&#160;</td>
<td><code class="docutils literal notranslate">AA AB AC AD BB BC BD CC CD DD</code></td>
</tr>
</tbody>
</table>
<div class="section" id="itertool-functions">
<h2>10.1.1. Itertool functions<a class="headerlink" href="#itertool-functions" title="Permalink to this headline">¶</a></h2>
The following module functions all construct and return iterators. Some provide
streams of infinite length, so they should only be accessed by functions or
loops that truncate the stream.
<dl class="function">
<dt id="itertools.accumulate">
<code class="descclassname">itertools.</code><code class="descname">accumulate</code>(iterable[, func])<a class="headerlink" href="#itertools.accumulate" title="Permalink to this definition">¶</a></dt>
<dd>Make an iterator that returns accumulated sums, or accumulated
results of other binary functions (specified via the optional
func argument). If func is supplied, it should be a function
of two arguments. Elements of the input iterable may be any type
that can be accepted as arguments to func. (For example, with
the default operation of addition, elements may be any addable
type including <a class="reference internal" href="decimal.html#decimal.Decimal" title="decimal.Decimal"><code class="xref py py-class docutils literal notranslate">Decimal</code></a> or
<a class="reference internal" href="fractions.html#fractions.Fraction" title="fractions.Fraction"><code class="xref py py-class docutils literal notranslate">Fraction</code></a>.) If the input iterable is empty, the
output iterable will also be empty.
Roughly equivalent to:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def accumulate(iterable, func=operator.add):
 &#39;Return running totals&#39;
 # accumulate([1,2,3,4,5]) --&gt; 1 3 6 10 15
 # accumulate([1,2,3,4,5], operator.mul) --&gt; 1 2 6 24 120
 it = iter(iterable)
 try:
 total = next(it)
 except StopIteration:
 return
 yield total
 for element in it:
 total = func(total, element)
 yield total
</pre></div>
</div>
There are a number of uses for the func argument. It can be set to
<a class="reference internal" href="functions.html#min" title="min"><code class="xref py py-func docutils literal notranslate">min()</code></a> for a running minimum, <a class="reference internal" href="functions.html#max" title="max"><code class="xref py py-func docutils literal notranslate">max()</code></a> for a running maximum, or
<a class="reference internal" href="operator.html#operator.mul" title="operator.mul"><code class="xref py py-func docutils literal notranslate">operator.mul()</code></a> for a running product. Amortization tables can be
built by accumulating interest and applying payments. First-order
<a class="reference external" href="https://en.wikipedia.org/wiki/Recurrence_relation">recurrence relations</a>
can be modeled by supplying the initial value in the iterable and using only
the accumulated total in func argument:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; data = [3, 4, 6, 2, 1, 9, 0, 7, 5, 8]
&gt;&gt;&gt; list(accumulate(data, operator.mul)) # running product
[3, 12, 72, 144, 144, 1296, 0, 0, 0, 0]
&gt;&gt;&gt; list(accumulate(data, max)) # running maximum
[3, 4, 6, 6, 6, 9, 9, 9, 9, 9]

# Amortize a 5% loan of 1000 with 4 annual payments of 90
&gt;&gt;&gt; cashflows = [1000, -90, -90, -90, -90]
&gt;&gt;&gt; list(accumulate(cashflows, lambda bal, pmt: bal*1.05 + pmt))
[1000, 960.0, 918.0, 873.9000000000001, 827.5950000000001]

# Chaotic recurrence relation https://en.wikipedia.org/wiki/Logistic_map
&gt;&gt;&gt; logistic_map = lambda x, _: r * x * (1 - x)
&gt;&gt;&gt; r = 3.8
&gt;&gt;&gt; x0 = 0.4
&gt;&gt;&gt; inputs = repeat(x0, 36) # only the initial value is used
&gt;&gt;&gt; [format(x, &#39;.2f&#39;) for x in accumulate(inputs, logistic_map)]
[&#39;0.40&#39;, &#39;0.91&#39;, &#39;0.30&#39;, &#39;0.81&#39;, &#39;0.60&#39;, &#39;0.92&#39;, &#39;0.29&#39;, &#39;0.79&#39;, &#39;0.63&#39;,
 &#39;0.88&#39;, &#39;0.39&#39;, &#39;0.90&#39;, &#39;0.33&#39;, &#39;0.84&#39;, &#39;0.52&#39;, &#39;0.95&#39;, &#39;0.18&#39;, &#39;0.57&#39;,
 &#39;0.93&#39;, &#39;0.25&#39;, &#39;0.71&#39;, &#39;0.79&#39;, &#39;0.63&#39;, &#39;0.88&#39;, &#39;0.39&#39;, &#39;0.91&#39;, &#39;0.32&#39;,
 &#39;0.83&#39;, &#39;0.54&#39;, &#39;0.95&#39;, &#39;0.20&#39;, &#39;0.60&#39;, &#39;0.91&#39;, &#39;0.30&#39;, &#39;0.80&#39;, &#39;0.60&#39;]
</pre></div>
</div>
See <a class="reference internal" href="functools.html#functools.reduce" title="functools.reduce"><code class="xref py py-func docutils literal notranslate">functools.reduce()</code></a> for a similar function that returns only the
final accumulated value.
<div class="versionadded">
New in version 3.2.
</div>
<div class="versionchanged">
Changed in version 3.3: Added the optional func parameter.
</div>
</dd></dl>

<dl class="function">
<dt id="itertools.chain">
<code class="descclassname">itertools.</code><code class="descname">chain</code>(*iterables)<a class="headerlink" href="#itertools.chain" title="Permalink to this definition">¶</a></dt>
<dd>Make an iterator that returns elements from the first iterable until it is
exhausted, then proceeds to the next iterable, until all of the iterables are
exhausted. Used for treating consecutive sequences as a single sequence.
Roughly equivalent to:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def chain(*iterables):
 # chain(&#39;ABC&#39;, &#39;DEF&#39;) --&gt; A B C D E F
 for it in iterables:
 for element in it:
 yield element
</pre></div>
</div>
</dd></dl>

<dl class="classmethod">
<dt id="itertools.chain.from_iterable">
classmethod <code class="descclassname">chain.</code><code class="descname">from_iterable</code>(iterable)<a class="headerlink" href="#itertools.chain.from_iterable" title="Permalink to this definition">¶</a></dt>
<dd>Alternate constructor for <a class="reference internal" href="#itertools.chain" title="itertools.chain"><code class="xref py py-func docutils literal notranslate">chain()</code></a>. Gets chained inputs from a
single iterable argument that is evaluated lazily. Roughly equivalent to:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def from_iterable(iterables):
 # chain.from_iterable([&#39;ABC&#39;, &#39;DEF&#39;]) --&gt; A B C D E F
 for it in iterables:
 for element in it:
 yield element
</pre></div>
</div>
</dd></dl>

<dl class="function">
<dt id="itertools.combinations">
<code class="descclassname">itertools.</code><code class="descname">combinations</code>(iterable, r)<a class="headerlink" href="#itertools.combinations" title="Permalink to this definition">¶</a></dt>
<dd>Return r length subsequences of elements from the input iterable.
Combinations are emitted in lexicographic sort order. So, if the
input iterable is sorted, the combination tuples will be produced
in sorted order.
Elements are treated as unique based on their position, not on their
value. So if the input elements are unique, there will be no repeat
values in each combination.
Roughly equivalent to:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def combinations(iterable, r):
 # combinations(&#39;ABCD&#39;, 2) --&gt; AB AC AD BC BD CD
 # combinations(range(4), 3) --&gt; 012 013 023 123
 pool = tuple(iterable)
 n = len(pool)
 if r &gt; n:
 return
 indices = list(range(r))
 yield tuple(pool[i] for i in indices)
 while True:
 for i in reversed(range(r)):
 if indices[i] != i + n - r:
 break
 else:
 return
 indices[i] += 1
 for j in range(i+1, r):
 indices[j] = indices[j-1] + 1
 yield tuple(pool[i] for i in indices)
</pre></div>
</div>
The code for <a class="reference internal" href="#itertools.combinations" title="itertools.combinations"><code class="xref py py-func docutils literal notranslate">combinations()</code></a> can be also expressed as a subsequence
of <a class="reference internal" href="#itertools.permutations" title="itertools.permutations"><code class="xref py py-func docutils literal notranslate">permutations()</code></a> after filtering entries where the elements are not
in sorted order (according to their position in the input pool):
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def combinations(iterable, r):
 pool = tuple(iterable)
 n = len(pool)
 for indices in permutations(range(n), r):
 if sorted(indices) == list(indices):
 yield tuple(pool[i] for i in indices)
</pre></div>
</div>
The number of items returned is <code class="docutils literal notranslate">n! / r! / (n-r)!</code> when <code class="docutils literal notranslate">0 &lt;= r &lt;= n</code>
or zero when <code class="docutils literal notranslate">r &gt; n</code>.
</dd></dl>

<dl class="function">
<dt id="itertools.combinations_with_replacement">
<code class="descclassname">itertools.</code><code class="descname">combinations_with_replacement</code>(iterable, r)<a class="headerlink" href="#itertools.combinations_with_replacement" title="Permalink to this definition">¶</a></dt>
<dd>Return r length subsequences of elements from the input iterable
allowing individual elements to be repeated more than once.
Combinations are emitted in lexicographic sort order. So, if the
input iterable is sorted, the combination tuples will be produced
in sorted order.
Elements are treated as unique based on their position, not on their
value. So if the input elements are unique, the generated combinations
will also be unique.
Roughly equivalent to:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def combinations_with_replacement(iterable, r):
 # combinations_with_replacement(&#39;ABC&#39;, 2) --&gt; AA AB AC BB BC CC
 pool = tuple(iterable)
 n = len(pool)
 if not n and r:
 return
 indices = [0] * r
 yield tuple(pool[i] for i in indices)
 while True:
 for i in reversed(range(r)):
 if indices[i] != n - 1:
 break
 else:
 return
 indices[i:] = [indices[i] + 1] * (r - i)
 yield tuple(pool[i] for i in indices)
</pre></div>
</div>
The code for <a class="reference internal" href="#itertools.combinations_with_replacement" title="itertools.combinations_with_replacement"><code class="xref py py-func docutils literal notranslate">combinations_with_replacement()</code></a> can be also expressed as
a subsequence of <a class="reference internal" href="#itertools.product" title="itertools.product"><code class="xref py py-func docutils literal notranslate">product()</code></a> after filtering entries where the elements
are not in sorted order (according to their position in the input pool):
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def combinations_with_replacement(iterable, r):
 pool = tuple(iterable)
 n = len(pool)
 for indices in product(range(n), repeat=r):
 if sorted(indices) == list(indices):
 yield tuple(pool[i] for i in indices)
</pre></div>
</div>
The number of items returned is <code class="docutils literal notranslate">(n+r-1)! / r! / (n-1)!</code> when <code class="docutils literal notranslate">n &gt; 0</code>.
<div class="versionadded">
New in version 3.1.
</div>
</dd></dl>

<dl class="function">
<dt id="itertools.compress">
<code class="descclassname">itertools.</code><code class="descname">compress</code>(data, selectors)<a class="headerlink" href="#itertools.compress" title="Permalink to this definition">¶</a></dt>
<dd>Make an iterator that filters elements from data returning only those that
have a corresponding element in selectors that evaluates to <code class="docutils literal notranslate">True</code>.
Stops when either the data or selectors iterables has been exhausted.
Roughly equivalent to:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def compress(data, selectors):
 # compress(&#39;ABCDEF&#39;, [1,0,1,0,1,1]) --&gt; A C E F
 return (d for d, s in zip(data, selectors) if s)
</pre></div>
</div>
<div class="versionadded">
New in version 3.1.
</div>
</dd></dl>

<dl class="function">
<dt id="itertools.count">
<code class="descclassname">itertools.</code><code class="descname">count</code>(start=0, step=1)<a class="headerlink" href="#itertools.count" title="Permalink to this definition">¶</a></dt>
<dd>Make an iterator that returns evenly spaced values starting with number start. Often
used as an argument to <a class="reference internal" href="functions.html#map" title="map"><code class="xref py py-func docutils literal notranslate">map()</code></a> to generate consecutive data points.
Also, used with <a class="reference internal" href="functions.html#zip" title="zip"><code class="xref py py-func docutils literal notranslate">zip()</code></a> to add sequence numbers. Roughly equivalent to:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def count(start=0, step=1):
 # count(10) --&gt; 10 11 12 13 14 ...
 # count(2.5, 0.5) -&gt; 2.5 3.0 3.5 ...
 n = start
 while True:
 yield n
 n += step
</pre></div>
</div>
When counting with floating point numbers, better accuracy can sometimes be
achieved by substituting multiplicative code such as: <code class="docutils literal notranslate">(start + step * i
for i in count())</code>.
<div class="versionchanged">
Changed in version 3.1: Added step argument and allowed non-integer arguments.
</div>
</dd></dl>

<dl class="function">
<dt id="itertools.cycle">
<code class="descclassname">itertools.</code><code class="descname">cycle</code>(iterable)<a class="headerlink" href="#itertools.cycle" title="Permalink to this definition">¶</a></dt>
<dd>Make an iterator returning elements from the iterable and saving a copy of each.
When the iterable is exhausted, return elements from the saved copy. Repeats
indefinitely. Roughly equivalent to:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def cycle(iterable):
 # cycle(&#39;ABCD&#39;) --&gt; A B C D A B C D A B C D ...
 saved = []
 for element in iterable:
 yield element
 saved.append(element)
 while saved:
 for element in saved:
 yield element
</pre></div>
</div>
Note, this member of the toolkit may require significant auxiliary storage
(depending on the length of the iterable).
</dd></dl>

<dl class="function">
<dt id="itertools.dropwhile">
<code class="descclassname">itertools.</code><code class="descname">dropwhile</code>(predicate, iterable)<a class="headerlink" href="#itertools.dropwhile" title="Permalink to this definition">¶</a></dt>
<dd>Make an iterator that drops elements from the iterable as long as the predicate
is true; afterwards, returns every element. Note, the iterator does not produce
any output until the predicate first becomes false, so it may have a lengthy
start-up time. Roughly equivalent to:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def dropwhile(predicate, iterable):
 # dropwhile(lambda x: x&lt;5, [1,4,6,4,1]) --&gt; 6 4 1
 iterable = iter(iterable)
 for x in iterable:
 if not predicate(x):
 yield x
 break
 for x in iterable:
 yield x
</pre></div>
</div>
</dd></dl>

<dl class="function">
<dt id="itertools.filterfalse">
<code class="descclassname">itertools.</code><code class="descname">filterfalse</code>(predicate, iterable)<a class="headerlink" href="#itertools.filterfalse" title="Permalink to this definition">¶</a></dt>
<dd>Make an iterator that filters elements from iterable returning only those for
which the predicate is <code class="docutils literal notranslate">False</code>. If predicate is <code class="docutils literal notranslate">None</code>, return the items
that are false. Roughly equivalent to:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def filterfalse(predicate, iterable):
 # filterfalse(lambda x: x%2, range(10)) --&gt; 0 2 4 6 8
 if predicate is None:
 predicate = bool
 for x in iterable:
 if not predicate(x):
 yield x
</pre></div>
</div>
</dd></dl>

<dl class="function">
<dt id="itertools.groupby">
<code class="descclassname">itertools.</code><code class="descname">groupby</code>(iterable, key=None)<a class="headerlink" href="#itertools.groupby" title="Permalink to this definition">¶</a></dt>
<dd>Make an iterator that returns consecutive keys and groups from the iterable.
The key is a function computing a key value for each element. If not
specified or is <code class="docutils literal notranslate">None</code>, key defaults to an identity function and returns
the element unchanged. Generally, the iterable needs to already be sorted on
the same key function.
The operation of <a class="reference internal" href="#itertools.groupby" title="itertools.groupby"><code class="xref py py-func docutils literal notranslate">groupby()</code></a> is similar to the <code class="docutils literal notranslate">uniq</code> filter in Unix. It
generates a break or new group every time the value of the key function changes
(which is why it is usually necessary to have sorted the data using the same key
function). That behavior differs from SQL’s GROUP BY which aggregates common
elements regardless of their input order.
The returned group is itself an iterator that shares the underlying iterable
with <a class="reference internal" href="#itertools.groupby" title="itertools.groupby"><code class="xref py py-func docutils literal notranslate">groupby()</code></a>. Because the source is shared, when the <a class="reference internal" href="#itertools.groupby" title="itertools.groupby"><code class="xref py py-func docutils literal notranslate">groupby()</code></a>
object is advanced, the previous group is no longer visible. So, if that data
is needed later, it should be stored as a list:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>groups = []
uniquekeys = []
data = sorted(data, key=keyfunc)
for k, g in groupby(data, keyfunc):
 groups.append(list(g)) # Store group iterator as a list
 uniquekeys.append(k)
</pre></div>
</div>
<a class="reference internal" href="#itertools.groupby" title="itertools.groupby"><code class="xref py py-func docutils literal notranslate">groupby()</code></a> is roughly equivalent to:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>class groupby:
 # [k for k, g in groupby(&#39;AAAABBBCCDAABBB&#39;)] --&gt; A B C D A B
 # [list(g) for k, g in groupby(&#39;AAAABBBCCD&#39;)] --&gt; AAAA BBB CC D
 def __init__(self, iterable, key=None):
 if key is None:
 key = lambda x: x
 self.keyfunc = key
 self.it = iter(iterable)
 self.tgtkey = self.currkey = self.currvalue = object()
 def __iter__(self):
 return self
 def __next__(self):
 while self.currkey == self.tgtkey:
 self.currvalue = next(self.it) # Exit on StopIteration
 self.currkey = self.keyfunc(self.currvalue)
 self.tgtkey = self.currkey
 return (self.currkey, self._grouper(self.tgtkey))
 def _grouper(self, tgtkey):
 while self.currkey == tgtkey:
 yield self.currvalue
 try:
 self.currvalue = next(self.it)
 except StopIteration:
 return
 self.currkey = self.keyfunc(self.currvalue)
</pre></div>
</div>
</dd></dl>

<dl class="function">
<dt id="itertools.islice">
<code class="descclassname">itertools.</code><code class="descname">islice</code>(iterable, stop)<a class="headerlink" href="#itertools.islice" title="Permalink to this definition">¶</a></dt>
<dt>
<code class="descclassname">itertools.</code><code class="descname">islice</code>(iterable, start, stop[, step])</dt>
<dd>Make an iterator that returns selected elements from the iterable. If start is
non-zero, then elements from the iterable are skipped until start is reached.
Afterward, elements are returned consecutively unless step is set higher than
one which results in items being skipped. If stop is <code class="docutils literal notranslate">None</code>, then iteration
continues until the iterator is exhausted, if at all; otherwise, it stops at the
specified position. Unlike regular slicing, <a class="reference internal" href="#itertools.islice" title="itertools.islice"><code class="xref py py-func docutils literal notranslate">islice()</code></a> does not support
negative values for start, stop, or step. Can be used to extract related
fields from data where the internal structure has been flattened (for example, a
multi-line report may list a name field on every third line). Roughly equivalent to:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def islice(iterable, *args):
 # islice(&#39;ABCDEFG&#39;, 2) --&gt; A B
 # islice(&#39;ABCDEFG&#39;, 2, 4) --&gt; C D
 # islice(&#39;ABCDEFG&#39;, 2, None) --&gt; C D E F G
 # islice(&#39;ABCDEFG&#39;, 0, None, 2) --&gt; A C E G
 s = slice(*args)
 start, stop, step = s.start or 0, s.stop or sys.maxsize, s.step or 1
 it = iter(range(start, stop, step))
 try:
 nexti = next(it)
 except StopIteration:
 # Consume *iterable* up to the *start* position.
 for i, element in zip(range(start), iterable):
 pass
 return
 try:
 for i, element in enumerate(iterable):
 if i == nexti:
 yield element
 nexti = next(it)
 except StopIteration:
 # Consume to *stop*.
 for i, element in zip(range(i + 1, stop), iterable):
 pass
</pre></div>
</div>
If start is <code class="docutils literal notranslate">None</code>, then iteration starts at zero. If step is <code class="docutils literal notranslate">None</code>,
then the step defaults to one.
</dd></dl>

<dl class="function">
<dt id="itertools.permutations">
<code class="descclassname">itertools.</code><code class="descname">permutations</code>(iterable, r=None)<a class="headerlink" href="#itertools.permutations" title="Permalink to this definition">¶</a></dt>
<dd>Return successive r length permutations of elements in the iterable.
If r is not specified or is <code class="docutils literal notranslate">None</code>, then r defaults to the length
of the iterable and all possible full-length permutations
are generated.
Permutations are emitted in lexicographic sort order. So, if the
input iterable is sorted, the permutation tuples will be produced
in sorted order.
Elements are treated as unique based on their position, not on their
value. So if the input elements are unique, there will be no repeat
values in each permutation.
Roughly equivalent to:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def permutations(iterable, r=None):
 # permutations(&#39;ABCD&#39;, 2) --&gt; AB AC AD BA BC BD CA CB CD DA DB DC
 # permutations(range(3)) --&gt; 012 021 102 120 201 210
 pool = tuple(iterable)
 n = len(pool)
 r = n if r is None else r
 if r &gt; n:
 return
 indices = list(range(n))
 cycles = list(range(n, n-r, -1))
 yield tuple(pool[i] for i in indices[:r])
 while n:
 for i in reversed(range(r)):
 cycles[i] -= 1
 if cycles[i] == 0:
 indices[i:] = indices[i+1:] + indices[i:i+1]
 cycles[i] = n - i
 else:
 j = cycles[i]
 indices[i], indices[-j] = indices[-j], indices[i]
 yield tuple(pool[i] for i in indices[:r])
 break
 else:
 return
</pre></div>
</div>
The code for <a class="reference internal" href="#itertools.permutations" title="itertools.permutations"><code class="xref py py-func docutils literal notranslate">permutations()</code></a> can be also expressed as a subsequence of
<a class="reference internal" href="#itertools.product" title="itertools.product"><code class="xref py py-func docutils literal notranslate">product()</code></a>, filtered to exclude entries with repeated elements (those
from the same position in the input pool):
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def permutations(iterable, r=None):
 pool = tuple(iterable)
 n = len(pool)
 r = n if r is None else r
 for indices in product(range(n), repeat=r):
 if len(set(indices)) == r:
 yield tuple(pool[i] for i in indices)
</pre></div>
</div>
The number of items returned is <code class="docutils literal notranslate">n! / (n-r)!</code> when <code class="docutils literal notranslate">0 &lt;= r &lt;= n</code>
or zero when <code class="docutils literal notranslate">r &gt; n</code>.
</dd></dl>

<dl class="function">
<dt id="itertools.product">
<code class="descclassname">itertools.</code><code class="descname">product</code>(*iterables, repeat=1)<a class="headerlink" href="#itertools.product" title="Permalink to this definition">¶</a></dt>
<dd>Cartesian product of input iterables.
Roughly equivalent to nested for-loops in a generator expression. For example,
<code class="docutils literal notranslate">product(A, B)</code> returns the same as <code class="docutils literal notranslate">((x,y) for x in A for y in B)</code>.
The nested loops cycle like an odometer with the rightmost element advancing
on every iteration. This pattern creates a lexicographic ordering so that if
the input’s iterables are sorted, the product tuples are emitted in sorted
order.
To compute the product of an iterable with itself, specify the number of
repetitions with the optional repeat keyword argument. For example,
<code class="docutils literal notranslate">product(A, repeat=4)</code> means the same as <code class="docutils literal notranslate">product(A, A, A, A)</code>.
This function is roughly equivalent to the following code, except that the
actual implementation does not build up intermediate results in memory:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def product(*args, repeat=1):
 # product(&#39;ABCD&#39;, &#39;xy&#39;) --&gt; Ax Ay Bx By Cx Cy Dx Dy
 # product(range(2), repeat=3) --&gt; 000 001 010 011 100 101 110 111
 pools = [tuple(pool) for pool in args] * repeat
 result = [[]]
 for pool in pools:
 result = [x+[y] for x in result for y in pool]
 for prod in result:
 yield tuple(prod)
</pre></div>
</div>
</dd></dl>

<dl class="function">
<dt id="itertools.repeat">
<code class="descclassname">itertools.</code><code class="descname">repeat</code>(object[, times])<a class="headerlink" href="#itertools.repeat" title="Permalink to this definition">¶</a></dt>
<dd>Make an iterator that returns object over and over again. Runs indefinitely
unless the times argument is specified. Used as argument to <a class="reference internal" href="functions.html#map" title="map"><code class="xref py py-func docutils literal notranslate">map()</code></a> for
invariant parameters to the called function. Also used with <a class="reference internal" href="functions.html#zip" title="zip"><code class="xref py py-func docutils literal notranslate">zip()</code></a> to
create an invariant part of a tuple record.
Roughly equivalent to:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def repeat(object, times=None):
 # repeat(10, 3) --&gt; 10 10 10
 if times is None:
 while True:
 yield object
 else:
 for i in range(times):
 yield object
</pre></div>
</div>
A common use for repeat is to supply a stream of constant values to map
or zip:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>&gt;&gt;&gt; list(map(pow, range(10), repeat(2)))
[0, 1, 4, 9, 16, 25, 36, 49, 64, 81]
</pre></div>
</div>
</dd></dl>

<dl class="function">
<dt id="itertools.starmap">
<code class="descclassname">itertools.</code><code class="descname">starmap</code>(function, iterable)<a class="headerlink" href="#itertools.starmap" title="Permalink to this definition">¶</a></dt>
<dd>Make an iterator that computes the function using arguments obtained from
the iterable. Used instead of <a class="reference internal" href="functions.html#map" title="map"><code class="xref py py-func docutils literal notranslate">map()</code></a> when argument parameters are already
grouped in tuples from a single iterable (the data has been “pre-zipped”). The
difference between <a class="reference internal" href="functions.html#map" title="map"><code class="xref py py-func docutils literal notranslate">map()</code></a> and <a class="reference internal" href="#itertools.starmap" title="itertools.starmap"><code class="xref py py-func docutils literal notranslate">starmap()</code></a> parallels the distinction
between <code class="docutils literal notranslate">function(a,b)</code> and <code class="docutils literal notranslate">function(*c)</code>. Roughly equivalent to:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def starmap(function, iterable):
 # starmap(pow, [(2,5), (3,2), (10,3)]) --&gt; 32 9 1000
 for args in iterable:
 yield function(*args)
</pre></div>
</div>
</dd></dl>

<dl class="function">
<dt id="itertools.takewhile">
<code class="descclassname">itertools.</code><code class="descname">takewhile</code>(predicate, iterable)<a class="headerlink" href="#itertools.takewhile" title="Permalink to this definition">¶</a></dt>
<dd>Make an iterator that returns elements from the iterable as long as the
predicate is true. Roughly equivalent to:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def takewhile(predicate, iterable):
 # takewhile(lambda x: x&lt;5, [1,4,6,4,1]) --&gt; 1 4
 for x in iterable:
 if predicate(x):
 yield x
 else:
 break
</pre></div>
</div>
</dd></dl>

<dl class="function">
<dt id="itertools.tee">
<code class="descclassname">itertools.</code><code class="descname">tee</code>(iterable, n=2)<a class="headerlink" href="#itertools.tee" title="Permalink to this definition">¶</a></dt>
<dd>Return n independent iterators from a single iterable.
The following Python code helps explain what tee does (although the actual
implementation is more complex and uses only a single underlying
<abbr title="first-in, first-out">FIFO</abbr> queue).
Roughly equivalent to:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def tee(iterable, n=2):
 it = iter(iterable)
 deques = [collections.deque() for i in range(n)]
 def gen(mydeque):
 while True:
 if not mydeque: # when the local deque is empty
 try:
 newval = next(it) # fetch a new value and
 except StopIteration:
 return
 for d in deques: # load it to all the deques
 d.append(newval)
 yield mydeque.popleft()
 return tuple(gen(d) for d in deques)
</pre></div>
</div>
Once <a class="reference internal" href="#itertools.tee" title="itertools.tee"><code class="xref py py-func docutils literal notranslate">tee()</code></a> has made a split, the original iterable should not be
used anywhere else; otherwise, the iterable could get advanced without
the tee objects being informed.
This itertool may require significant auxiliary storage (depending on how
much temporary data needs to be stored). In general, if one iterator uses
most or all of the data before another iterator starts, it is faster to use
<a class="reference internal" href="stdtypes.html#list" title="list"><code class="xref py py-func docutils literal notranslate">list()</code></a> instead of <a class="reference internal" href="#itertools.tee" title="itertools.tee"><code class="xref py py-func docutils literal notranslate">tee()</code></a>.
</dd></dl>

<dl class="function">
<dt id="itertools.zip_longest">
<code class="descclassname">itertools.</code><code class="descname">zip_longest</code>(*iterables, fillvalue=None)<a class="headerlink" href="#itertools.zip_longest" title="Permalink to this definition">¶</a></dt>
<dd>Make an iterator that aggregates elements from each of the iterables. If the
iterables are of uneven length, missing values are filled-in with fillvalue.
Iteration continues until the longest iterable is exhausted. Roughly equivalent to:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>class ZipExhausted(Exception):
 pass

def zip_longest(*args, **kwds):
 # zip_longest(&#39;ABCD&#39;, &#39;xy&#39;, fillvalue=&#39;-&#39;) --&gt; Ax By C- D-
 fillvalue = kwds.get(&#39;fillvalue&#39;)
 counter = len(args) - 1
 def sentinel():
 nonlocal counter
 if not counter:
 raise ZipExhausted
 counter -= 1
 yield fillvalue
 fillers = repeat(fillvalue)
 iterators = [chain(it, sentinel(), fillers) for it in args]
 try:
 while iterators:
 yield tuple(map(next, iterators))
 except ZipExhausted:
 pass
</pre></div>
</div>
If one of the iterables is potentially infinite, then the <a class="reference internal" href="#itertools.zip_longest" title="itertools.zip_longest"><code class="xref py py-func docutils literal notranslate">zip_longest()</code></a>
function should be wrapped with something that limits the number of calls
(for example <a class="reference internal" href="#itertools.islice" title="itertools.islice"><code class="xref py py-func docutils literal notranslate">islice()</code></a> or <a class="reference internal" href="#itertools.takewhile" title="itertools.takewhile"><code class="xref py py-func docutils literal notranslate">takewhile()</code></a>). If not specified,
fillvalue defaults to <code class="docutils literal notranslate">None</code>.
</dd></dl>

</div>
<div class="section" id="itertools-recipes">
<h2>10.1.2. Itertools Recipes<a class="headerlink" href="#itertools-recipes" title="Permalink to this headline">¶</a></h2>
This section shows recipes for creating an extended toolset using the existing
itertools as building blocks.
The extended tools offer the same high performance as the underlying toolset.
The superior memory performance is kept by processing elements one at a time
rather than bringing the whole iterable into memory all at once. Code volume is
kept small by linking the tools together in a functional style which helps
eliminate temporary variables. High speed is retained by preferring
“vectorized” building blocks over the use of for-loops and <a class="reference internal" href="../glossary.html#term-generator">generator</a>s
which incur interpreter overhead.
<div class="highlight-python notranslate"><div class="highlight"><pre>def take(n, iterable):
 &quot;Return first n items of the iterable as a list&quot;
 return list(islice(iterable, n))

def prepend(value, iterator):
 &quot;Prepend a single value in front of an iterator&quot;
 # prepend(1, [2, 3, 4]) -&gt; 1 2 3 4
 return chain([value], iterator)

def tabulate(function, start=0):
 &quot;Return function(0), function(1), ...&quot;
 return map(function, count(start))

def tail(n, iterable):
 &quot;Return an iterator over the last n items&quot;
 # tail(3, &#39;ABCDEFG&#39;) --&gt; E F G
 return iter(collections.deque(iterable, maxlen=n))

def consume(iterator, n=None):
 &quot;Advance the iterator n-steps ahead. If n is None, consume entirely.&quot;
 # Use functions that consume iterators at C speed.
 if n is None:
 # feed the entire iterator into a zero-length deque
 collections.deque(iterator, maxlen=0)
 else:
 # advance to the empty slice starting at position n
 next(islice(iterator, n, n), None)

def nth(iterable, n, default=None):
 &quot;Returns the nth item or a default value&quot;
 return next(islice(iterable, n, None), default)

def all_equal(iterable):
 &quot;Returns True if all the elements are equal to each other&quot;
 g = groupby(iterable)
 return next(g, True) and not next(g, False)

def quantify(iterable, pred=bool):
 &quot;Count how many times the predicate is true&quot;
 return sum(map(pred, iterable))

def padnone(iterable):
 &quot;&quot;&quot;Returns the sequence elements and then returns None indefinitely.

Useful for emulating the behavior of the built-in map() function.
 &quot;&quot;&quot;
 return chain(iterable, repeat(None))

def ncycles(iterable, n):
 &quot;Returns the sequence elements n times&quot;
 return chain.from_iterable(repeat(tuple(iterable), n))

def dotproduct(vec1, vec2):
 return sum(map(operator.mul, vec1, vec2))

def flatten(listOfLists):
 &quot;Flatten one level of nesting&quot;
 return chain.from_iterable(listOfLists)

def repeatfunc(func, times=None, *args):
 &quot;&quot;&quot;Repeat calls to func with specified arguments.

Example: repeatfunc(random.random)
 &quot;&quot;&quot;
 if times is None:
 return starmap(func, repeat(args))
 return starmap(func, repeat(args, times))

def pairwise(iterable):
 &quot;s -&gt; (s0,s1), (s1,s2), (s2, s3), ...&quot;
 a, b = tee(iterable)
 next(b, None)
 return zip(a, b)

def grouper(iterable, n, fillvalue=None):
 &quot;Collect data into fixed-length chunks or blocks&quot;
 # grouper(&#39;ABCDEFG&#39;, 3, &#39;x&#39;) --&gt; ABC DEF Gxx&quot;
 args = [iter(iterable)] * n
 return zip_longest(*args, fillvalue=fillvalue)

def roundrobin(*iterables):
 &quot;roundrobin(&#39;ABC&#39;, &#39;D&#39;, &#39;EF&#39;) --&gt; A D E B F C&quot;
 # Recipe credited to George Sakkis
 num_active = len(iterables)
 nexts = cycle(iter(it).__next__ for it in iterables)
 while num_active:
 try:
 for next in nexts:
 yield next()
 except StopIteration:
 # Remove the iterator we just exhausted from the cycle.
 num_active -= 1
 nexts = cycle(islice(nexts, num_active))

def partition(pred, iterable):
 &#39;Use a predicate to partition entries into false entries and true entries&#39;
 # partition(is_odd, range(10)) --&gt; 0 2 4 6 8 and 1 3 5 7 9
 t1, t2 = tee(iterable)
 return filterfalse(pred, t1), filter(pred, t2)

def powerset(iterable):
 &quot;powerset([1,2,3]) --&gt; () (1,) (2,) (3,) (1,2) (1,3) (2,3) (1,2,3)&quot;
 s = list(iterable)
 return chain.from_iterable(combinations(s, r) for r in range(len(s)+1))

def unique_everseen(iterable, key=None):
 &quot;List unique elements, preserving order. Remember all elements ever seen.&quot;
 # unique_everseen(&#39;AAAABBBCCDAABBB&#39;) --&gt; A B C D
 # unique_everseen(&#39;ABBCcAD&#39;, str.lower) --&gt; A B C D
 seen = set()
 seen_add = seen.add
 if key is None:
 for element in filterfalse(seen.__contains__, iterable):
 seen_add(element)
 yield element
 else:
 for element in iterable:
 k = key(element)
 if k not in seen:
 seen_add(k)
 yield element

def unique_justseen(iterable, key=None):
 &quot;List unique elements, preserving order. Remember only the element just seen.&quot;
 # unique_justseen(&#39;AAAABBBCCDAABBB&#39;) --&gt; A B C D A B
 # unique_justseen(&#39;ABBCcAD&#39;, str.lower) --&gt; A B C A D
 return map(next, map(itemgetter(1), groupby(iterable, key)))

def iter_except(func, exception, first=None):
 &quot;&quot;&quot; Call a function repeatedly until an exception is raised.

Converts a call-until-exception interface to an iterator interface.
 Like builtins.iter(func, sentinel) but uses an exception instead
 of a sentinel to end the loop.

Examples:
 iter_except(functools.partial(heappop, h), IndexError) # priority queue iterator
 iter_except(d.popitem, KeyError) # non-blocking dict iterator
 iter_except(d.popleft, IndexError) # non-blocking deque iterator
 iter_except(q.get_nowait, Queue.Empty) # loop over a producer Queue
 iter_except(s.pop, KeyError) # non-blocking set iterator

&quot;&quot;&quot;
 try:
 if first is not None:
 yield first() # For database APIs needing an initial cast to db.first()
 while True:
 yield func()
 except exception:
 pass

def first_true(iterable, default=False, pred=None):
 &quot;&quot;&quot;Returns the first true value in the iterable.

If no true value is found, returns *default*

If *pred* is not None, returns the first item
 for which pred(item) is true.

&quot;&quot;&quot;
 # first_true([a,b,c], x) --&gt; a or b or c or x
 # first_true([a,b], x, f) --&gt; a if f(a) else b if f(b) else x
 return next(filter(pred, iterable), default)

def random_product(*args, repeat=1):
 &quot;Random selection from itertools.product(*args, **kwds)&quot;
 pools = [tuple(pool) for pool in args] * repeat
 return tuple(random.choice(pool) for pool in pools)

def random_permutation(iterable, r=None):
 &quot;Random selection from itertools.permutations(iterable, r)&quot;
 pool = tuple(iterable)
 r = len(pool) if r is None else r
 return tuple(random.sample(pool, r))

def random_combination(iterable, r):
 &quot;Random selection from itertools.combinations(iterable, r)&quot;
 pool = tuple(iterable)
 n = len(pool)
 indices = sorted(random.sample(range(n), r))
 return tuple(pool[i] for i in indices)

def random_combination_with_replacement(iterable, r):
 &quot;Random selection from itertools.combinations_with_replacement(iterable, r)&quot;
 pool = tuple(iterable)
 n = len(pool)
 indices = sorted(random.randrange(n) for i in range(r))
 return tuple(pool[i] for i in indices)

def nth_combination(iterable, r, index):
 &#39;Equivalent to list(combinations(iterable, r))[index]&#39;
 pool = tuple(iterable)
 n = len(pool)
 if r &lt; 0 or r &gt; n:
 raise ValueError
 c = 1
 k = min(r, n-r)
 for i in range(1, k+1):
 c = c * (n - k + i) // i
 if index &lt; 0:
 index += c
 if index &lt; 0 or index &gt;= c:
 raise IndexError
 result = []
 while r:
 c, n, r = c*r//n, n-1, r-1
 while index &gt;= c:
 index -= c
 c, n = c*(n-r)//n, n-1
 result.append(pool[-1-n])
 return tuple(result)
</pre></div>
</div>
Note, many of the above recipes can be optimized by replacing global lookups
with local variables defined as default values. For example, the
dotproduct recipe can be written as:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>def dotproduct(vec1, vec2, sum=sum, map=map, mul=operator.mul):
 return sum(map(mul, vec1, vec2))
</pre></div>
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