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<li><a class="reference internal" href="#">Sorting Techniques</a><ul>
<li><a class="reference internal" href="#sorting-basics">Sorting Basics</a></li>
<li><a class="reference internal" href="#key-functions">Key Functions</a></li>
<li><a class="reference internal" href="#operator-module-functions-and-partial-function-evaluation">Operator Module Functions and Partial Function Evaluation</a></li>
<li><a class="reference internal" href="#ascending-and-descending">Ascending and Descending</a></li>
<li><a class="reference internal" href="#sort-stability-and-complex-sorts">Sort Stability and Complex Sorts</a></li>
<li><a class="reference internal" href="#decorate-sort-undecorate">Decorate-Sort-Undecorate</a></li>
<li><a class="reference internal" href="#comparison-functions">Comparison Functions</a></li>
<li><a class="reference internal" href="#odds-and-ends">Odds and Ends</a></li>
<li><a class="reference internal" href="#partial-sorts">Partial Sorts</a></li>
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 <section id="sorting-techniques">
<h1>Sorting Techniques<a class="headerlink" href="#sorting-techniques" title="Link to this heading">¶</a></h1>
<dl class="field-list simple">
<dt class="field-odd">Author:</dt>
<dd class="field-odd">Andrew Dalke and Raymond Hettinger
</dd>
</dl>
Python lists have a built-in <a class="reference internal" href="../library/stdtypes.html#list.sort" title="list.sort"><code class="xref py py-meth docutils literal notranslate">list.sort()</code></a> method that modifies the list
in-place. There is also a <a class="reference internal" href="../library/functions.html#sorted" title="sorted"><code class="xref py py-func docutils literal notranslate">sorted()</code></a> built-in function that builds a new
sorted list from an iterable.
In this document, we explore the various techniques for sorting data using Python.
<section id="sorting-basics">
<h2>Sorting Basics<a class="headerlink" href="#sorting-basics" title="Link to this heading">¶</a></h2>
A simple ascending sort is very easy: just call the <a class="reference internal" href="../library/functions.html#sorted" title="sorted"><code class="xref py py-func docutils literal notranslate">sorted()</code></a> function. It
returns a new sorted list:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; sorted([5, 2, 3, 1, 4])
[1, 2, 3, 4, 5]
</pre></div>
</div>
You can also use the <a class="reference internal" href="../library/stdtypes.html#list.sort" title="list.sort"><code class="xref py py-meth docutils literal notranslate">list.sort()</code></a> method. It modifies the list
in-place (and returns <code class="docutils literal notranslate">None</code> to avoid confusion). Usually it’s less convenient
than <a class="reference internal" href="../library/functions.html#sorted" title="sorted"><code class="xref py py-func docutils literal notranslate">sorted()</code></a> - but if you don’t need the original list, it’s slightly
more efficient.
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; a = [5, 2, 3, 1, 4]
&gt;&gt;&gt; a.sort()
&gt;&gt;&gt; a
[1, 2, 3, 4, 5]
</pre></div>
</div>
Another difference is that the <a class="reference internal" href="../library/stdtypes.html#list.sort" title="list.sort"><code class="xref py py-meth docutils literal notranslate">list.sort()</code></a> method is only defined for
lists. In contrast, the <a class="reference internal" href="../library/functions.html#sorted" title="sorted"><code class="xref py py-func docutils literal notranslate">sorted()</code></a> function accepts any iterable.
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; sorted({1: &#39;D&#39;, 2: &#39;B&#39;, 3: &#39;B&#39;, 4: &#39;E&#39;, 5: &#39;A&#39;})
[1, 2, 3, 4, 5]
</pre></div>
</div>
</section>
<section id="key-functions">
<h2>Key Functions<a class="headerlink" href="#key-functions" title="Link to this heading">¶</a></h2>
Both <a class="reference internal" href="../library/stdtypes.html#list.sort" title="list.sort"><code class="xref py py-meth docutils literal notranslate">list.sort()</code></a> and <a class="reference internal" href="../library/functions.html#sorted" title="sorted"><code class="xref py py-func docutils literal notranslate">sorted()</code></a> have a key parameter to specify a
function (or other callable) to be called on each list element prior to making
comparisons.
For example, here’s a case-insensitive string comparison:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; sorted(&quot;This is a test string from Andrew&quot;.split(), key=str.casefold)
[&#39;a&#39;, &#39;Andrew&#39;, &#39;from&#39;, &#39;is&#39;, &#39;string&#39;, &#39;test&#39;, &#39;This&#39;]
</pre></div>
</div>
The value of the key parameter should be a function (or other callable) that
takes a single argument and returns a key to use for sorting purposes. This
technique is fast because the key function is called exactly once for each
input record.
A common pattern is to sort complex objects using some of the object’s indices
as keys. For example:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; student_tuples = [
... (&#39;john&#39;, &#39;A&#39;, 15),
... (&#39;jane&#39;, &#39;B&#39;, 12),
... (&#39;dave&#39;, &#39;B&#39;, 10),
... ]
&gt;&gt;&gt; sorted(student_tuples, key=lambda student: student[2]) # sort by age
[(&#39;dave&#39;, &#39;B&#39;, 10), (&#39;jane&#39;, &#39;B&#39;, 12), (&#39;john&#39;, &#39;A&#39;, 15)]
</pre></div>
</div>
The same technique works for objects with named attributes. For example:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; class Student:
... def __init__(self, name, grade, age):
... self.name = name
... self.grade = grade
... self.age = age
... def __repr__(self):
... return repr((self.name, self.grade, self.age))

&gt;&gt;&gt; student_objects = [
... Student(&#39;john&#39;, &#39;A&#39;, 15),
... Student(&#39;jane&#39;, &#39;B&#39;, 12),
... Student(&#39;dave&#39;, &#39;B&#39;, 10),
... ]
&gt;&gt;&gt; sorted(student_objects, key=lambda student: student.age) # sort by age
[(&#39;dave&#39;, &#39;B&#39;, 10), (&#39;jane&#39;, &#39;B&#39;, 12), (&#39;john&#39;, &#39;A&#39;, 15)]
</pre></div>
</div>
Objects with named attributes can be made by a regular class as shown
above, or they can be instances of <a class="reference internal" href="../library/dataclasses.html#dataclasses.dataclass" title="dataclasses.dataclass"><code class="xref py py-class docutils literal notranslate">dataclass</code></a> or
a <a class="reference internal" href="../glossary.html#term-named-tuple">named tuple</a>.
</section>
<section id="operator-module-functions-and-partial-function-evaluation">
<h2>Operator Module Functions and Partial Function Evaluation<a class="headerlink" href="#operator-module-functions-and-partial-function-evaluation" title="Link to this heading">¶</a></h2>
The <a class="reference internal" href="../glossary.html#term-key-function">key function</a> patterns shown above are very common, so Python provides
convenience functions to make accessor functions easier and faster. The
<a class="reference internal" href="../library/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 has <a class="reference internal" href="../library/operator.html#operator.itemgetter" title="operator.itemgetter"><code class="xref py py-func docutils literal notranslate">itemgetter()</code></a>,
<a class="reference internal" href="../library/operator.html#operator.attrgetter" title="operator.attrgetter"><code class="xref py py-func docutils literal notranslate">attrgetter()</code></a>, and a <a class="reference internal" href="../library/operator.html#operator.methodcaller" title="operator.methodcaller"><code class="xref py py-func docutils literal notranslate">methodcaller()</code></a> function.
Using those functions, the above examples become simpler and faster:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; from operator import itemgetter, attrgetter

&gt;&gt;&gt; sorted(student_tuples, key=itemgetter(2))
[(&#39;dave&#39;, &#39;B&#39;, 10), (&#39;jane&#39;, &#39;B&#39;, 12), (&#39;john&#39;, &#39;A&#39;, 15)]

&gt;&gt;&gt; sorted(student_objects, key=attrgetter(&#39;age&#39;))
[(&#39;dave&#39;, &#39;B&#39;, 10), (&#39;jane&#39;, &#39;B&#39;, 12), (&#39;john&#39;, &#39;A&#39;, 15)]
</pre></div>
</div>
The operator module functions allow multiple levels of sorting. For example, to
sort by grade then by age:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; sorted(student_tuples, key=itemgetter(1,2))
[(&#39;john&#39;, &#39;A&#39;, 15), (&#39;dave&#39;, &#39;B&#39;, 10), (&#39;jane&#39;, &#39;B&#39;, 12)]

&gt;&gt;&gt; sorted(student_objects, key=attrgetter(&#39;grade&#39;, &#39;age&#39;))
[(&#39;john&#39;, &#39;A&#39;, 15), (&#39;dave&#39;, &#39;B&#39;, 10), (&#39;jane&#39;, &#39;B&#39;, 12)]
</pre></div>
</div>
The <a class="reference internal" href="../library/functools.html#module-functools" title="functools: Higher-order functions and operations on callable objects."><code class="xref py py-mod docutils literal notranslate">functools</code></a> module provides another helpful tool for making
key-functions. The <a class="reference internal" href="../library/functools.html#functools.partial" title="functools.partial"><code class="xref py py-func docutils literal notranslate">partial()</code></a> function can reduce the
<a class="reference external" href="https://en.wikipedia.org/wiki/Arity">arity</a> of a multi-argument
function making it suitable for use as a key-function.
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; from functools import partial
&gt;&gt;&gt; from unicodedata import normalize

&gt;&gt;&gt; names = &#39;Zoë Åbjørn Núñez Élana Zeke Abe Nubia Eloise&#39;.split()

&gt;&gt;&gt; sorted(names, key=partial(normalize, &#39;NFD&#39;))
[&#39;Abe&#39;, &#39;Åbjørn&#39;, &#39;Eloise&#39;, &#39;Élana&#39;, &#39;Nubia&#39;, &#39;Núñez&#39;, &#39;Zeke&#39;, &#39;Zoë&#39;]

&gt;&gt;&gt; sorted(names, key=partial(normalize, &#39;NFC&#39;))
[&#39;Abe&#39;, &#39;Eloise&#39;, &#39;Nubia&#39;, &#39;Núñez&#39;, &#39;Zeke&#39;, &#39;Zoë&#39;, &#39;Åbjørn&#39;, &#39;Élana&#39;]
</pre></div>
</div>
</section>
<section id="ascending-and-descending">
<h2>Ascending and Descending<a class="headerlink" href="#ascending-and-descending" title="Link to this heading">¶</a></h2>
Both <a class="reference internal" href="../library/stdtypes.html#list.sort" title="list.sort"><code class="xref py py-meth docutils literal notranslate">list.sort()</code></a> and <a class="reference internal" href="../library/functions.html#sorted" title="sorted"><code class="xref py py-func docutils literal notranslate">sorted()</code></a> accept a reverse parameter with a
boolean value. This is used to flag descending sorts. For example, to get the
student data in reverse age order:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; sorted(student_tuples, key=itemgetter(2), reverse=True)
[(&#39;john&#39;, &#39;A&#39;, 15), (&#39;jane&#39;, &#39;B&#39;, 12), (&#39;dave&#39;, &#39;B&#39;, 10)]

&gt;&gt;&gt; sorted(student_objects, key=attrgetter(&#39;age&#39;), reverse=True)
[(&#39;john&#39;, &#39;A&#39;, 15), (&#39;jane&#39;, &#39;B&#39;, 12), (&#39;dave&#39;, &#39;B&#39;, 10)]
</pre></div>
</div>
</section>
<section id="sort-stability-and-complex-sorts">
<h2>Sort Stability and Complex Sorts<a class="headerlink" href="#sort-stability-and-complex-sorts" title="Link to this heading">¶</a></h2>
Sorts are guaranteed to be <a class="reference external" href="https://en.wikipedia.org/wiki/Sorting_algorithm#Stability">stable</a>. That means that
when multiple records have the same key, their original order is preserved.
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; data = [(&#39;red&#39;, 1), (&#39;blue&#39;, 1), (&#39;red&#39;, 2), (&#39;blue&#39;, 2)]
&gt;&gt;&gt; sorted(data, key=itemgetter(0))
[(&#39;blue&#39;, 1), (&#39;blue&#39;, 2), (&#39;red&#39;, 1), (&#39;red&#39;, 2)]
</pre></div>
</div>
Notice how the two records for blue retain their original order so that
<code class="docutils literal notranslate">('blue', 1)</code> is guaranteed to precede <code class="docutils literal notranslate">('blue', 2)</code>.
This wonderful property lets you build complex sorts in a series of sorting
steps. For example, to sort the student data by descending grade and then
ascending age, do the age sort first and then sort again using grade:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; s = sorted(student_objects, key=attrgetter(&#39;age&#39;)) # sort on secondary key
&gt;&gt;&gt; sorted(s, key=attrgetter(&#39;grade&#39;), reverse=True) # now sort on primary key, descending
[(&#39;dave&#39;, &#39;B&#39;, 10), (&#39;jane&#39;, &#39;B&#39;, 12), (&#39;john&#39;, &#39;A&#39;, 15)]
</pre></div>
</div>
This can be abstracted out into a wrapper function that can take a list and
tuples of field and order to sort them on multiple passes.
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; def multisort(xs, specs):
... for key, reverse in reversed(specs):
... xs.sort(key=attrgetter(key), reverse=reverse)
... return xs

&gt;&gt;&gt; multisort(list(student_objects), ((&#39;grade&#39;, True), (&#39;age&#39;, False)))
[(&#39;dave&#39;, &#39;B&#39;, 10), (&#39;jane&#39;, &#39;B&#39;, 12), (&#39;john&#39;, &#39;A&#39;, 15)]
</pre></div>
</div>
The <a class="reference external" href="https://en.wikipedia.org/wiki/Timsort">Timsort</a> algorithm used in Python
does multiple sorts efficiently because it can take advantage of any ordering
already present in a dataset.
</section>
<section id="decorate-sort-undecorate">
<h2>Decorate-Sort-Undecorate<a class="headerlink" href="#decorate-sort-undecorate" title="Link to this heading">¶</a></h2>
This idiom is called Decorate-Sort-Undecorate after its three steps:
<ul class="simple">
<li>First, the initial list is decorated with new values that control the sort order.</li>
<li>Second, the decorated list is sorted.</li>
<li>Finally, the decorations are removed, creating a list that contains only the
initial values in the new order.</li>
</ul>
For example, to sort the student data by grade using the DSU approach:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; decorated = [(student.grade, i, student) for i, student in enumerate(student_objects)]
&gt;&gt;&gt; decorated.sort()
&gt;&gt;&gt; [student for grade, i, student in decorated] # undecorate
[(&#39;john&#39;, &#39;A&#39;, 15), (&#39;jane&#39;, &#39;B&#39;, 12), (&#39;dave&#39;, &#39;B&#39;, 10)]
</pre></div>
</div>
This idiom works because tuples are compared lexicographically; the first items
are compared; if they are the same then the second items are compared, and so
on.
It is not strictly necessary in all cases to include the index i in the
decorated list, but including it gives two benefits:
<ul class="simple">
<li>The sort is stable – if two items have the same key, their order will be
preserved in the sorted list.</li>
<li>The original items do not have to be comparable because the ordering of the
decorated tuples will be determined by at most the first two items. So for
example the original list could contain complex numbers which cannot be sorted
directly.</li>
</ul>
Another name for this idiom is
<a class="reference external" href="https://en.wikipedia.org/wiki/Schwartzian_transform">Schwartzian transform</a>,
after Randal L. Schwartz, who popularized it among Perl programmers.
Now that Python sorting provides key-functions, this technique is not often needed.
</section>
<section id="comparison-functions">
<h2>Comparison Functions<a class="headerlink" href="#comparison-functions" title="Link to this heading">¶</a></h2>
Unlike key functions that return an absolute value for sorting, a comparison
function computes the relative ordering for two inputs.
For example, a <a class="reference external" href="https://upload.wikimedia.org/wikipedia/commons/1/17/Balance_à_tabac_1850.JPG">balance scale</a>
compares two samples giving a relative ordering: lighter, equal, or heavier.
Likewise, a comparison function such as <code class="docutils literal notranslate">cmp(a, b)</code> will return a negative
value for less-than, zero if the inputs are equal, or a positive value for
greater-than.
It is common to encounter comparison functions when translating algorithms from
other languages. Also, some libraries provide comparison functions as part of
their API. For example, <a class="reference internal" href="../library/locale.html#locale.strcoll" title="locale.strcoll"><code class="xref py py-func docutils literal notranslate">locale.strcoll()</code></a> is a comparison function.
To accommodate those situations, Python provides
<a class="reference internal" href="../library/functools.html#functools.cmp_to_key" title="functools.cmp_to_key"><code class="xref py py-class docutils literal notranslate">functools.cmp_to_key</code></a> to wrap the comparison function
to make it usable as a key function:
<div class="highlight-python3 notranslate"><div class="highlight"><pre>sorted(words, key=cmp_to_key(strcoll)) # locale-aware sort order
</pre></div>
</div>
</section>
<section id="odds-and-ends">
<h2>Odds and Ends<a class="headerlink" href="#odds-and-ends" title="Link to this heading">¶</a></h2>
<ul>
<li>For locale aware sorting, use <a class="reference internal" href="../library/locale.html#locale.strxfrm" title="locale.strxfrm"><code class="xref py py-func docutils literal notranslate">locale.strxfrm()</code></a> for a key function or
<a class="reference internal" href="../library/locale.html#locale.strcoll" title="locale.strcoll"><code class="xref py py-func docutils literal notranslate">locale.strcoll()</code></a> for a comparison function. This is necessary
because “alphabetical” sort orderings can vary across cultures even
if the underlying alphabet is the same.</li>
<li>The reverse parameter still maintains sort stability (so that records with
equal keys retain the original order). Interestingly, that effect can be
simulated without the parameter by using the builtin <a class="reference internal" href="../library/functions.html#reversed" title="reversed"><code class="xref py py-func docutils literal notranslate">reversed()</code></a> function
twice:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; data = [(&#39;red&#39;, 1), (&#39;blue&#39;, 1), (&#39;red&#39;, 2), (&#39;blue&#39;, 2)]
&gt;&gt;&gt; standard_way = sorted(data, key=itemgetter(0), reverse=True)
&gt;&gt;&gt; double_reversed = list(reversed(sorted(reversed(data), key=itemgetter(0))))
&gt;&gt;&gt; assert standard_way == double_reversed
&gt;&gt;&gt; standard_way
[(&#39;red&#39;, 1), (&#39;red&#39;, 2), (&#39;blue&#39;, 1), (&#39;blue&#39;, 2)]
</pre></div>
</div>
</li>
<li>The sort routines use <code class="docutils literal notranslate">&lt;</code> when making comparisons
between two objects. So, it is easy to add a standard sort order to a class by
defining an <a class="reference internal" href="../reference/datamodel.html#object.__lt__" title="object.__lt__"><code class="xref py py-meth docutils literal notranslate">__lt__()</code></a> method:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; Student.__lt__ = lambda self, other: self.age &lt; other.age
&gt;&gt;&gt; sorted(student_objects)
[(&#39;dave&#39;, &#39;B&#39;, 10), (&#39;jane&#39;, &#39;B&#39;, 12), (&#39;john&#39;, &#39;A&#39;, 15)]
</pre></div>
</div>
However, note that <code class="docutils literal notranslate">&lt;</code> can fall back to using <a class="reference internal" href="../reference/datamodel.html#object.__gt__" title="object.__gt__"><code class="xref py py-meth docutils literal notranslate">__gt__()</code></a> if
<a class="reference internal" href="../reference/datamodel.html#object.__lt__" title="object.__lt__"><code class="xref py py-meth docutils literal notranslate">__lt__()</code></a> is not implemented (see <a class="reference internal" href="../reference/datamodel.html#object.__lt__" title="object.__lt__"><code class="xref py py-func docutils literal notranslate">object.__lt__()</code></a>
for details on the mechanics). To avoid surprises, <a class="pep reference external" href="https://peps.python.org/pep-0008/">PEP 8</a>
recommends that all six comparison methods be implemented.
The <a class="reference internal" href="../library/functools.html#functools.total_ordering" title="functools.total_ordering"><code class="xref py py-func docutils literal notranslate">total_ordering()</code></a> decorator is provided to make that
task easier.
</li>
<li>Key functions need not depend directly on the objects being sorted. A key
function can also access external resources. For instance, if the student grades
are stored in a dictionary, they can be used to sort a separate list of student
names:
<div class="highlight-pycon notranslate"><div class="highlight"><pre>&gt;&gt;&gt; students = [&#39;dave&#39;, &#39;john&#39;, &#39;jane&#39;]
&gt;&gt;&gt; newgrades = {&#39;john&#39;: &#39;F&#39;, &#39;jane&#39;:&#39;A&#39;, &#39;dave&#39;: &#39;C&#39;}
&gt;&gt;&gt; sorted(students, key=newgrades.__getitem__)
[&#39;jane&#39;, &#39;dave&#39;, &#39;john&#39;]
</pre></div>
</div>
</li>
</ul>
</section>
<section id="partial-sorts">
<h2>Partial Sorts<a class="headerlink" href="#partial-sorts" title="Link to this heading">¶</a></h2>
Some applications require only some of the data to be ordered. The standard
library provides several tools that do less work than a full sort:
<ul class="simple">
<li><a class="reference internal" href="../library/functions.html#min" title="min"><code class="xref py py-func docutils literal notranslate">min()</code></a> and <a class="reference internal" href="../library/functions.html#max" title="max"><code class="xref py py-func docutils literal notranslate">max()</code></a> return the smallest and largest values,
respectively. These functions make a single pass over the input data and
require almost no auxiliary memory.</li>
<li><a class="reference internal" href="../library/heapq.html#heapq.nsmallest" title="heapq.nsmallest"><code class="xref py py-func docutils literal notranslate">heapq.nsmallest()</code></a> and <a class="reference internal" href="../library/heapq.html#heapq.nlargest" title="heapq.nlargest"><code class="xref py py-func docutils literal notranslate">heapq.nlargest()</code></a> return
the n smallest and largest values, respectively. These functions
make a single pass over the data keeping only n elements in memory
at a time. For values of n that are small relative to the number of
inputs, these functions make far fewer comparisons than a full sort.</li>
<li><a class="reference internal" href="../library/heapq.html#heapq.heappush" title="heapq.heappush"><code class="xref py py-func docutils literal notranslate">heapq.heappush()</code></a> and <a class="reference internal" href="../library/heapq.html#heapq.heappop" title="heapq.heappop"><code class="xref py py-func docutils literal notranslate">heapq.heappop()</code></a> create and maintain a
partially sorted arrangement of data that keeps the smallest element
at position <code class="docutils literal notranslate">0</code>. These functions are suitable for implementing
priority queues which are commonly used for task scheduling.</li>
</ul>
</section>
</section>

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 <h3><a href="../contents.html">Table of Contents</a></h3>
 <ul>
<li><a class="reference internal" href="#">Sorting Techniques</a><ul>
<li><a class="reference internal" href="#sorting-basics">Sorting Basics</a></li>
<li><a class="reference internal" href="#key-functions">Key Functions</a></li>
<li><a class="reference internal" href="#operator-module-functions-and-partial-function-evaluation">Operator Module Functions and Partial Function Evaluation</a></li>
<li><a class="reference internal" href="#ascending-and-descending">Ascending and Descending</a></li>
<li><a class="reference internal" href="#sort-stability-and-complex-sorts">Sort Stability and Complex Sorts</a></li>
<li><a class="reference internal" href="#decorate-sort-undecorate">Decorate-Sort-Undecorate</a></li>
<li><a class="reference internal" href="#comparison-functions">Comparison Functions</a></li>
<li><a class="reference internal" href="#odds-and-ends">Odds and Ends</a></li>
<li><a class="reference internal" href="#partial-sorts">Partial Sorts</a></li>
</ul>
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