1:mod:`bisect` --- Array bisection algorithm
2===========================================
3
4.. module:: bisect
5   :synopsis: Array bisection algorithms for binary searching.
6.. sectionauthor:: Fred L. Drake, Jr. <fdrake@acm.org>
7.. sectionauthor:: Raymond Hettinger <python at rcn.com>
8.. example based on the PyModules FAQ entry by Aaron Watters <arw@pythonpros.com>
9
10.. versionadded:: 2.1
11
12**Source code:** :source:`Lib/bisect.py`
13
14--------------
15
16This module provides support for maintaining a list in sorted order without
17having to sort the list after each insertion.  For long lists of items with
18expensive comparison operations, this can be an improvement over the more common
19approach.  The module is called :mod:`bisect` because it uses a basic bisection
20algorithm to do its work.  The source code may be most useful as a working
21example of the algorithm (the boundary conditions are already right!).
22
23The following functions are provided:
24
25
26.. function:: bisect_left(a, x, lo=0, hi=len(a))
27
28   Locate the insertion point for *x* in *a* to maintain sorted order.
29   The parameters *lo* and *hi* may be used to specify a subset of the list
30   which should be considered; by default the entire list is used.  If *x* is
31   already present in *a*, the insertion point will be before (to the left of)
32   any existing entries.  The return value is suitable for use as the first
33   parameter to ``list.insert()`` assuming that *a* is already sorted.
34
35   The returned insertion point *i* partitions the array *a* into two halves so
36   that ``all(val < x for val in a[lo:i])`` for the left side and
37   ``all(val >= x for val in a[i:hi])`` for the right side.
38
39.. function:: bisect_right(a, x, lo=0, hi=len(a))
40              bisect(a, x, lo=0, hi=len(a))
41
42   Similar to :func:`bisect_left`, but returns an insertion point which comes
43   after (to the right of) any existing entries of *x* in *a*.
44
45   The returned insertion point *i* partitions the array *a* into two halves so
46   that ``all(val <= x for val in a[lo:i])`` for the left side and
47   ``all(val > x for val in a[i:hi])`` for the right side.
48
49.. function:: insort_left(a, x, lo=0, hi=len(a))
50
51   Insert *x* in *a* in sorted order.  This is equivalent to
52   ``a.insert(bisect.bisect_left(a, x, lo, hi), x)`` assuming that *a* is
53   already sorted.  Keep in mind that the O(log n) search is dominated by
54   the slow O(n) insertion step.
55
56.. function:: insort_right(a, x, lo=0, hi=len(a))
57              insort(a, x, lo=0, hi=len(a))
58
59   Similar to :func:`insort_left`, but inserting *x* in *a* after any existing
60   entries of *x*.
61
62.. seealso::
63
64   `SortedCollection recipe
65   <https://code.activestate.com/recipes/577197-sortedcollection/>`_ that uses
66   bisect to build a full-featured collection class with straight-forward search
67   methods and support for a key-function.  The keys are precomputed to save
68   unnecessary calls to the key function during searches.
69
70
71Searching Sorted Lists
72----------------------
73
74The above :func:`bisect` functions are useful for finding insertion points but
75can be tricky or awkward to use for common searching tasks. The following five
76functions show how to transform them into the standard lookups for sorted
77lists::
78
79    def index(a, x):
80        'Locate the leftmost value exactly equal to x'
81        i = bisect_left(a, x)
82        if i != len(a) and a[i] == x:
83            return i
84        raise ValueError
85
86    def find_lt(a, x):
87        'Find rightmost value less than x'
88        i = bisect_left(a, x)
89        if i:
90            return a[i-1]
91        raise ValueError
92
93    def find_le(a, x):
94        'Find rightmost value less than or equal to x'
95        i = bisect_right(a, x)
96        if i:
97            return a[i-1]
98        raise ValueError
99
100    def find_gt(a, x):
101        'Find leftmost value greater than x'
102        i = bisect_right(a, x)
103        if i != len(a):
104            return a[i]
105        raise ValueError
106
107    def find_ge(a, x):
108        'Find leftmost item greater than or equal to x'
109        i = bisect_left(a, x)
110        if i != len(a):
111            return a[i]
112        raise ValueError
113
114
115Other Examples
116--------------
117
118.. _bisect-example:
119
120The :func:`bisect` function can be useful for numeric table lookups. This
121example uses :func:`bisect` to look up a letter grade for an exam score (say)
122based on a set of ordered numeric breakpoints: 90 and up is an 'A', 80 to 89 is
123a 'B', and so on::
124
125   >>> def grade(score, breakpoints=[60, 70, 80, 90], grades='FDCBA'):
126           i = bisect(breakpoints, score)
127           return grades[i]
128
129   >>> [grade(score) for score in [33, 99, 77, 70, 89, 90, 100]]
130   ['F', 'A', 'C', 'C', 'B', 'A', 'A']
131
132Unlike the :func:`sorted` function, it does not make sense for the :func:`bisect`
133functions to have *key* or *reversed* arguments because that would lead to an
134inefficient design (successive calls to bisect functions would not "remember"
135all of the previous key lookups).
136
137Instead, it is better to search a list of precomputed keys to find the index
138of the record in question::
139
140    >>> data = [('red', 5), ('blue', 1), ('yellow', 8), ('black', 0)]
141    >>> data.sort(key=lambda r: r[1])
142    >>> keys = [r[1] for r in data]         # precomputed list of keys
143    >>> data[bisect_left(keys, 0)]
144    ('black', 0)
145    >>> data[bisect_left(keys, 1)]
146    ('blue', 1)
147    >>> data[bisect_left(keys, 5)]
148    ('red', 5)
149    >>> data[bisect_left(keys, 8)]
150    ('yellow', 8)
151
152