Sorting Techniques
******************

Author:
   Andrew Dalke and Raymond Hettinger

Python lists have a built-in "list.sort()" method that modifies the
list in-place.  There is also a "sorted()" 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.


Sorting Basics
==============

A simple ascending sort is very easy: just call the "sorted()"
function. It returns a new sorted list:

   >>> sorted([5, 2, 3, 1, 4])
   [1, 2, 3, 4, 5]

You can also use the "list.sort()" method. It modifies the list in-
place (and returns "None" to avoid confusion). Usually it's less
convenient than "sorted()" - but if you don't need the original list,
it's slightly more efficient.

   >>> a = [5, 2, 3, 1, 4]
   >>> a.sort()
   >>> a
   [1, 2, 3, 4, 5]

Another difference is that the "list.sort()" method is only defined
for lists. In contrast, the "sorted()" function accepts any iterable.

   >>> sorted({1: 'D', 2: 'B', 3: 'B', 4: 'E', 5: 'A'})
   [1, 2, 3, 4, 5]


Key Functions
=============

The "list.sort()" method and the functions "sorted()", "min()",
"max()", "heapq.nsmallest()", and "heapq.nlargest()" 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 using
"str.casefold()":

   >>> sorted("This is a test string from Andrew".split(), key=str.casefold)
   ['a', 'Andrew', 'from', 'is', 'string', 'test', 'This']

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:

   >>> student_tuples = [
   ...     ('john', 'A', 15),
   ...     ('jane', 'B', 12),
   ...     ('dave', 'B', 10),
   ... ]
   >>> sorted(student_tuples, key=lambda student: student[2])   # sort by age
   [('dave', 'B', 10), ('jane', 'B', 12), ('john', 'A', 15)]

The same technique works for objects with named attributes. For
example:

   >>> 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))

   >>> student_objects = [
   ...     Student('john', 'A', 15),
   ...     Student('jane', 'B', 12),
   ...     Student('dave', 'B', 10),
   ... ]
   >>> sorted(student_objects, key=lambda student: student.age)   # sort by age
   [('dave', 'B', 10), ('jane', 'B', 12), ('john', 'A', 15)]

Objects with named attributes can be made by a regular class as shown
above, or they can be instances of "dataclass" or a *named tuple*.


Operator Module Functions and Partial Function Evaluation
=========================================================

The *key function* patterns shown above are very common, so Python
provides convenience functions to make accessor functions easier and
faster. The "operator" module has "itemgetter()", "attrgetter()", and
a "methodcaller()" function.

Using those functions, the above examples become simpler and faster:

   >>> from operator import itemgetter, attrgetter

   >>> sorted(student_tuples, key=itemgetter(2))
   [('dave', 'B', 10), ('jane', 'B', 12), ('john', 'A', 15)]

   >>> sorted(student_objects, key=attrgetter('age'))
   [('dave', 'B', 10), ('jane', 'B', 12), ('john', 'A', 15)]

The operator module functions allow multiple levels of sorting. For
example, to sort by *grade* then by *age*:

   >>> sorted(student_tuples, key=itemgetter(1,2))
   [('john', 'A', 15), ('dave', 'B', 10), ('jane', 'B', 12)]

   >>> sorted(student_objects, key=attrgetter('grade', 'age'))
   [('john', 'A', 15), ('dave', 'B', 10), ('jane', 'B', 12)]

The "functools" module provides another helpful tool for making key-
functions.  The "partial()" function can reduce the arity of a multi-
argument function making it suitable for use as a key-function.

   >>> from functools import partial
   >>> from unicodedata import normalize

   >>> names = 'Zoë Åbjørn Núñez Élana Zeke Abe Nubia Eloise'.split()

   >>> sorted(names, key=partial(normalize, 'NFD'))
   ['Abe', 'Åbjørn', 'Eloise', 'Élana', 'Nubia', 'Núñez', 'Zeke', 'Zoë']

   >>> sorted(names, key=partial(normalize, 'NFC'))
   ['Abe', 'Eloise', 'Nubia', 'Núñez', 'Zeke', 'Zoë', 'Åbjørn', 'Élana']


Ascending and Descending
========================

Both "list.sort()" and "sorted()" 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:

   >>> sorted(student_tuples, key=itemgetter(2), reverse=True)
   [('john', 'A', 15), ('jane', 'B', 12), ('dave', 'B', 10)]

   >>> sorted(student_objects, key=attrgetter('age'), reverse=True)
   [('john', 'A', 15), ('jane', 'B', 12), ('dave', 'B', 10)]


Sort Stability and Complex Sorts
================================

Sorts are guaranteed to be stable. That means that when multiple
records have the same key, their original order is preserved.

   >>> data = [('red', 1), ('blue', 1), ('red', 2), ('blue', 2)]
   >>> sorted(data, key=itemgetter(0))
   [('blue', 1), ('blue', 2), ('red', 1), ('red', 2)]

Notice how the two records for *blue* retain their original order so
that "('blue', 1)" is guaranteed to precede "('blue', 2)".

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*:

   >>> s = sorted(student_objects, key=attrgetter('age'))     # sort on secondary key
   >>> sorted(s, key=attrgetter('grade'), reverse=True)       # now sort on primary key, descending
   [('dave', 'B', 10), ('jane', 'B', 12), ('john', 'A', 15)]

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.

   >>> def multisort(xs, specs):
   ...     for key, reverse in reversed(specs):
   ...         xs.sort(key=attrgetter(key), reverse=reverse)
   ...     return xs

   >>> multisort(list(student_objects), (('grade', True), ('age', False)))
   [('dave', 'B', 10), ('jane', 'B', 12), ('john', 'A', 15)]

The Timsort algorithm used in Python does multiple sorts efficiently
because it can take advantage of any ordering already present in a
dataset.


Decorate-Sort-Undecorate
========================

This idiom is called Decorate-Sort-Undecorate after its three steps:

* First, the initial list is decorated with new values that control
  the sort order.

* Second, the decorated list is sorted.

* Finally, the decorations are removed, creating a list that contains
  only the initial values in the new order.

For example, to sort the student data by *grade* using the DSU
approach:

   >>> decorated = [(student.grade, i, student) for i, student in enumerate(student_objects)]
   >>> decorated.sort()
   >>> [student for grade, i, student in decorated]               # undecorate
   [('john', 'A', 15), ('jane', 'B', 12), ('dave', 'B', 10)]

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:

* The sort is stable -- if two items have the same key, their order
  will be preserved in the sorted list.

* 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.

Another name for this idiom is Schwartzian transform, after Randal L.
Schwartz, who popularized it among Perl programmers.

Now that Python sorting provides key-functions, this technique is not
often needed.


Comparison Functions
====================

Unlike key functions that return an absolute value for sorting, a
comparison function computes the relative ordering for two inputs.

For example, a balance scale compares two samples giving a relative
ordering: lighter, equal, or heavier. Likewise, a comparison function
such as "cmp(a, b)" 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,
"locale.strcoll()" is a comparison function.

To accommodate those situations, Python provides
"functools.cmp_to_key" to wrap the comparison function to make it
usable as a key function:

   sorted(words, key=cmp_to_key(strcoll))  # locale-aware sort order


Strategies For Unorderable Types and Values
===========================================

A number of type and value issues can arise when sorting. Here are
some strategies that can help:

* Convert non-comparable input types to strings prior to sorting:

   >>> data = ['twelve', '11', 10]
   >>> sorted(map(str, data))
   ['10', '11', 'twelve']

This is needed because most cross-type comparisons raise a
"TypeError".

* Remove special values prior to sorting:

   >>> from math import isnan
   >>> from itertools import filterfalse
   >>> data = [3.3, float('nan'), 1.1, 2.2]
   >>> sorted(filterfalse(isnan, data))
   [1.1, 2.2, 3.3]

This is needed because the IEEE-754 standard specifies that, "Every
NaN shall compare unordered with everything, including itself."

Likewise, "None" can be stripped from datasets as well:

   >>> data = [3.3, None, 1.1, 2.2]
   >>> sorted(x for x in data if x is not None)
   [1.1, 2.2, 3.3]

This is needed because "None" is not comparable to other types.

* Convert mapping types into sorted item lists before sorting:

   >>> data = [{'a': 1}, {'b': 2}]
   >>> sorted(data, key=lambda d: sorted(d.items()))
   [{'a': 1}, {'b': 2}]

This is needed because dict-to-dict comparisons raise a "TypeError".

* Convert set types into sorted lists before sorting:

   >>> data = [{'a', 'b', 'c'}, {'b', 'c', 'd'}]
   >>> sorted(map(sorted, data))
   [['a', 'b', 'c'], ['b', 'c', 'd']]

This is needed because the elements contained in set types do not have
a deterministic order.  For example, "list({'a', 'b'})" may produce
either "['a', 'b']" or "['b', 'a']".


Odds and Ends
=============

* For locale aware sorting, use "locale.strxfrm()" for a key function
  or "locale.strcoll()" for a comparison function.  This is necessary
  because "alphabetical" sort orderings can vary across cultures even
  if the underlying alphabet is the same.

* 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 "reversed()" function twice:

     >>> data = [('red', 1), ('blue', 1), ('red', 2), ('blue', 2)]
     >>> standard_way = sorted(data, key=itemgetter(0), reverse=True)
     >>> double_reversed = list(reversed(sorted(reversed(data), key=itemgetter(0))))
     >>> assert standard_way == double_reversed
     >>> standard_way
     [('red', 1), ('red', 2), ('blue', 1), ('blue', 2)]

* The sort routines use "<" when making comparisons between two
  objects. So, it is easy to add a standard sort order to a class by
  defining an "__lt__()" method:

     >>> Student.__lt__ = lambda self, other: self.age < other.age
     >>> sorted(student_objects)
     [('dave', 'B', 10), ('jane', 'B', 12), ('john', 'A', 15)]

  However, note that "<" can fall back to using "__gt__()" if
  "__lt__()" is not implemented (see "object.__lt__()" for details on
  the mechanics).  To avoid surprises, **PEP 8** recommends that all
  six comparison methods be implemented. The "total_ordering()"
  decorator is provided to make that task easier.

* 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:

     >>> students = ['dave', 'john', 'jane']
     >>> newgrades = {'john': 'F', 'jane':'A', 'dave': 'C'}
     >>> sorted(students, key=newgrades.__getitem__)
     ['jane', 'dave', 'john']


Partial Sorts
=============

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:

* "min()" and "max()" return the smallest and largest values,
  respectively.  These functions make a single pass over the input
  data and require almost no auxiliary memory.

* "heapq.nsmallest()" and "heapq.nlargest()" 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.

* "heapq.heappush()" and "heapq.heappop()" create and maintain a
  partially sorted arrangement of data that keeps the smallest element
  at position "0".  These functions are suitable for implementing
  priority queues which are commonly used for task scheduling.
