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211 lines
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ReStructuredText
211 lines
7.5 KiB
ReStructuredText
:mod:`bisect` --- Array bisection algorithm
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===========================================
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.. module:: bisect
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:synopsis: Array bisection algorithms for binary searching.
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.. sectionauthor:: Fred L. Drake, Jr. <fdrake@acm.org>
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.. sectionauthor:: Raymond Hettinger <python at rcn.com>
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.. example based on the PyModules FAQ entry by Aaron Watters <arw@pythonpros.com>
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**Source code:** :source:`Lib/bisect.py`
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--------------
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This module provides support for maintaining a list in sorted order without
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having to sort the list after each insertion. For long lists of items with
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expensive comparison operations, this can be an improvement over the more common
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approach. The module is called :mod:`bisect` because it uses a basic bisection
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algorithm to do its work. The source code may be most useful as a working
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example of the algorithm (the boundary conditions are already right!).
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The following functions are provided:
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.. function:: bisect_left(a, x, lo=0, hi=len(a), *, key=None)
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Locate the insertion point for *x* in *a* to maintain sorted order.
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The parameters *lo* and *hi* may be used to specify a subset of the list
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which should be considered; by default the entire list is used. If *x* is
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already present in *a*, the insertion point will be before (to the left of)
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any existing entries. The return value is suitable for use as the first
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parameter to ``list.insert()`` assuming that *a* is already sorted.
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The returned insertion point *i* partitions the array *a* into two halves so
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that ``all(val < x for val in a[lo : i])`` for the left side and
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``all(val >= x for val in a[i : hi])`` for the right side.
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*key* specifies a :term:`key function` of one argument that is used to
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extract a comparison key from each input element. The default value is
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``None`` (compare the elements directly).
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.. versionchanged:: 3.10
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Added the *key* parameter.
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.. function:: bisect_right(a, x, lo=0, hi=len(a), *, key=None)
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bisect(a, x, lo=0, hi=len(a))
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Similar to :func:`bisect_left`, but returns an insertion point which comes
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after (to the right of) any existing entries of *x* in *a*.
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The returned insertion point *i* partitions the array *a* into two halves so
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that ``all(val <= x for val in a[lo : i])`` for the left side and
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``all(val > x for val in a[i : hi])`` for the right side.
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*key* specifies a :term:`key function` of one argument that is used to
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extract a comparison key from each input element. The default value is
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``None`` (compare the elements directly).
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.. versionchanged:: 3.10
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Added the *key* parameter.
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.. function:: insort_left(a, x, lo=0, hi=len(a), *, key=None)
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Insert *x* in *a* in sorted order.
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*key* specifies a :term:`key function` of one argument that is used to
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extract a comparison key from each input element. The default value is
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``None`` (compare the elements directly).
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This function first runs :func:`bisect_left` to locate an insertion point.
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Next, it runs the :meth:`insert` method on *a* to insert *x* at the
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appropriate position to maintain sort order.
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Keep in mind that the ``O(log n)`` search is dominated by the slow O(n)
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insertion step.
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.. versionchanged:: 3.10
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Added the *key* parameter.
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.. function:: insort_right(a, x, lo=0, hi=len(a), *, key=None)
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insort(a, x, lo=0, hi=len(a))
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Similar to :func:`insort_left`, but inserting *x* in *a* after any existing
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entries of *x*.
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*key* specifies a :term:`key function` of one argument that is used to
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extract a comparison key from each input element. The default value is
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``None`` (compare the elements directly).
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This function first runs :func:`bisect_right` to locate an insertion point.
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Next, it runs the :meth:`insert` method on *a* to insert *x* at the
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appropriate position to maintain sort order.
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Keep in mind that the ``O(log n)`` search is dominated by the slow O(n)
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insertion step.
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.. versionchanged:: 3.10
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Added the *key* parameter.
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Performance Notes
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-----------------
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When writing time sensitive code using *bisect()* and *insort()*, keep these
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thoughts in mind:
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* Bisection is effective for searching ranges of values.
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For locating specific values, dictionaries are more performant.
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* The *insort()* functions are ``O(n)`` because the logarithmic search step
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is dominated by the linear time insertion step.
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* The search functions are stateless and discard key function results after
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they are used. Consequently, if the search functions are used in a loop,
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the key function may be called again and again on the same array elements.
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If the key function isn't fast, consider wrapping it with
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:func:`functools.cache` to avoid duplicate computations. Alternatively,
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consider searching an array of precomputed keys to locate the insertion
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point (as shown in the examples section below).
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.. seealso::
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* `Sorted Collections
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<http://www.grantjenks.com/docs/sortedcollections/>`_ is a high performance
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module that uses *bisect* to managed sorted collections of data.
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* The `SortedCollection recipe
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<https://code.activestate.com/recipes/577197-sortedcollection/>`_ uses
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bisect to build a full-featured collection class with straight-forward search
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methods and support for a key-function. The keys are precomputed to save
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unnecessary calls to the key function during searches.
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Searching Sorted Lists
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----------------------
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The above :func:`bisect` functions are useful for finding insertion points but
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can be tricky or awkward to use for common searching tasks. The following five
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functions show how to transform them into the standard lookups for sorted
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lists::
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def index(a, x):
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'Locate the leftmost value exactly equal to x'
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i = bisect_left(a, x)
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if i != len(a) and a[i] == x:
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return i
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raise ValueError
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def find_lt(a, x):
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'Find rightmost value less than x'
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i = bisect_left(a, x)
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if i:
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return a[i-1]
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raise ValueError
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def find_le(a, x):
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'Find rightmost value less than or equal to x'
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i = bisect_right(a, x)
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if i:
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return a[i-1]
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raise ValueError
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def find_gt(a, x):
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'Find leftmost value greater than x'
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i = bisect_right(a, x)
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if i != len(a):
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return a[i]
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raise ValueError
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def find_ge(a, x):
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'Find leftmost item greater than or equal to x'
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i = bisect_left(a, x)
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if i != len(a):
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return a[i]
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raise ValueError
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Examples
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--------
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.. _bisect-example:
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The :func:`bisect` function can be useful for numeric table lookups. This
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example uses :func:`bisect` to look up a letter grade for an exam score (say)
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based on a set of ordered numeric breakpoints: 90 and up is an 'A', 80 to 89 is
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a 'B', and so on::
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>>> def grade(score, breakpoints=[60, 70, 80, 90], grades='FDCBA'):
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... i = bisect(breakpoints, score)
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... return grades[i]
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...
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>>> [grade(score) for score in [33, 99, 77, 70, 89, 90, 100]]
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['F', 'A', 'C', 'C', 'B', 'A', 'A']
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One technique to avoid repeated calls to a key function is to search a list of
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precomputed keys to find the index of a record::
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>>> data = [('red', 5), ('blue', 1), ('yellow', 8), ('black', 0)]
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>>> data.sort(key=lambda r: r[1]) # Or use operator.itemgetter(1).
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>>> keys = [r[1] for r in data] # Precompute a list of keys.
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>>> data[bisect_left(keys, 0)]
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('black', 0)
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>>> data[bisect_left(keys, 1)]
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('blue', 1)
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>>> data[bisect_left(keys, 5)]
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('red', 5)
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>>> data[bisect_left(keys, 8)]
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('yellow', 8)
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