bisect — Array bisection algorithm

Source code: Lib/bisect.py

This module provides support for maintaining a list in sorted order without having to sort the list after each insertion. For long lists of items with expensive comparison operations, this can be an improvement over the more common approach. The module is called bisect because it uses a basic bisection algorithm to do its work. The source code may be most useful as a working example of the algorithm (the boundary conditions are already right!).

The following functions are provided:

bisect.bisect_left(a, x, lo=0, hi=len(a), *, key=None)

Locate the insertion point for x in a to maintain sorted order. The parameters lo and hi may be used to specify a subset of the list which should be considered; by default the entire list is used. If x is already present in a, the insertion point will be before (to the left of) any existing entries. The return value is suitable for use as the first parameter to list.insert() assuming that a is already sorted.

The returned insertion point i partitions the array a into two halves so that all(val < x for val in a[lo : i]) for the left side and all(val >= x for val in a[i : hi]) for the right side.

key specifies a key function of one argument that is used to extract a comparison key from each input element. The default value is None (compare the elements directly).

Changed in version 3.10: Added the key parameter.

bisect.bisect_right(a, x, lo=0, hi=len(a), *, key=None)

bisect.bisect(a, x, lo=0, hi=len(a))

Similar to bisect_left(), but returns an insertion point which comes after (to the right of) any existing entries of x in a.

The returned insertion point i partitions the array a into two halves so that all(val <= x for val in a[lo : i]) for the left side and all(val > x for val in a[i : hi]) for the right side.

key specifies a key function of one argument that is used to extract a comparison key from each input element. The default value is None (compare the elements directly).

Changed in version 3.10: Added the key parameter.

bisect.insort_left(a, x, lo=0, hi=len(a), *, key=None)

Insert x in a in sorted order.

key specifies a key function of one argument that is used to extract a comparison key from each input element. The default value is None (compare the elements directly).

This function first runs bisect_left() to locate an insertion point. Next, it runs the insert() method on a to insert x at the appropriate position to maintain sort order.

Keep in mind that the O(log n) search is dominated by the slow O(n) insertion step.

Changed in version 3.10: Added the key parameter.

bisect.insort_right(a, x, lo=0, hi=len(a), *, key=None)

bisect.insort(a, x, lo=0, hi=len(a))

Similar to insort_left(), but inserting x in a after any existing entries of x.

key specifies a key function of one argument that is used to extract a comparison key from each input element. The default value is None (compare the elements directly).

This function first runs bisect_right() to locate an insertion point. Next, it runs the insert() method on a to insert x at the appropriate position to maintain sort order.

Keep in mind that the O(log n) search is dominated by the slow O(n) insertion step.

Changed in version 3.10: Added the key parameter.

Performance Notes

When writing time sensitive code using bisect() and insort(), keep these thoughts in mind:

• Bisection is effective for searching ranges of values. For locating specific values, dictionaries are more performant.
• The insort() functions are O(n) because the logarithmic search step is dominated by the linear time insertion step.
• The search functions are stateless and discard key function results after they are used. Consequently, if the search functions are used in a loop, the key function may be called again and again on the same array elements. If the key function isn’t fast, consider wrapping it with functools.cache() to avoid duplicate computations. Alternatively, consider searching an array of precomputed keys to locate the insertion point (as shown in the examples section below).

Searching Sorted Lists

The above bisect() functions are useful for finding insertion points but can be tricky or awkward to use for common searching tasks. The following five functions show how to transform them into the standard lookups for sorted lists:

def index(a, x):
    'Locate the leftmost value exactly equal to x'
    i = bisect_left(a, x)
    if i != len(a) and a[i] == x:
        return i
    raise ValueError

def find_lt(a, x):
    'Find rightmost value less than x'
    i = bisect_left(a, x)
    if i:
        return a[i-1]
    raise ValueError

def find_le(a, x):
    'Find rightmost value less than or equal to x'
    i = bisect_right(a, x)
    if i:
        return a[i-1]
    raise ValueError

def find_gt(a, x):
    'Find leftmost value greater than x'
    i = bisect_right(a, x)
    if i != len(a):
        return a[i]
    raise ValueError

def find_ge(a, x):
    'Find leftmost item greater than or equal to x'
    i = bisect_left(a, x)
    if i != len(a):
        return a[i]
    raise ValueError

Examples

The bisect() function can be useful for numeric table lookups. This example uses bisect() to look up a letter grade for an exam score (say) based on a set of ordered numeric breakpoints: 90 and up is an ‘A’, 80 to 89 is a ‘B’, and so on:

>>> def grade(score, breakpoints=[60, 70, 80, 90], grades='FDCBA'):
...     i = bisect(breakpoints, score)
...     return grades[i]
...
>>> [grade(score) for score in [33, 99, 77, 70, 89, 90, 100]]
['F', 'A', 'C', 'C', 'B', 'A', 'A']

One technique to avoid repeated calls to a key function is to search a list of precomputed keys to find the index of a record:

>>> data = [('red', 5), ('blue', 1), ('yellow', 8), ('black', 0)]
>>> data.sort(key=lambda r: r[1])       # Or use operator.itemgetter(1).
>>> keys = [r[1] for r in data]         # Precompute a list of keys.
>>> data[bisect_left(keys, 0)]
('black', 0)
>>> data[bisect_left(keys, 1)]
('blue', 1)
>>> data[bisect_left(keys, 5)]
('red', 5)
>>> data[bisect_left(keys, 8)]
('yellow', 8)