# `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 element in the array. To support searching complex records, the key function is not applied to the x value.

If key is `None`, the elements are compared directly with no intervening function call.

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), *, key=None)

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 element in the array. To support searching complex records, the key function is not applied to the x value.

If key is `None`, the elements are compared directly with no intervening function call.

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.

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.

To support inserting records in a table, the key function (if any) is applied to x for the search step but not for the insertion step.

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), *, key=None)

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

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.

To support inserting records in a table, the key function (if any) is applied to x for the search step but not for the insertion step.

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).

• Sorted Collections is a high performance module that uses bisect to managed sorted collections of data.

• The SortedCollection recipe uses bisect to build a full-featured collection class with straight-forward search methods and support for a key-function. The keys are precomputed to save unnecessary calls to the key function during searches.

## 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)
...
>>> [grade(score) for score in [33, 99, 77, 70, 89, 90, 100]]
['F', 'A', 'C', 'C', 'B', 'A', 'A']
```

The `bisect()` and `insort()` functions also work with lists of tuples. The key argument can serve to extract the field used for ordering records in a table:

```>>> from collections import namedtuple
>>> from operator import attrgetter
>>> from bisect import bisect, insort
>>> from pprint import pprint

>>> Movie = namedtuple('Movie', ('name', 'released', 'director'))

>>> movies = [
...     Movie('Jaws', 1975, 'Speilberg'),
...     Movie('Titanic', 1997, 'Cameron'),
...     Movie('The Birds', 1963, 'Hitchcock'),
...     Movie('Aliens', 1986, 'Scott')
... ]

>>> # Find the first movie released on or after 1960
>>> by_year = attrgetter('released')
>>> movies.sort(key=by_year)
>>> movies[bisect(movies, 1960, key=by_year)]
Movie(name='The Birds', released=1963, director='Hitchcock')

>>> # Insert a movie while maintaining sort order
>>> romance = Movie('Love Story', 1970, 'Hiller')
>>> insort(movies, romance, key=by_year)
>>> pprint(movies)
[Movie(name='The Birds', released=1963, director='Hitchcock'),
Movie(name='Love Story', released=1970, director='Hiller'),
Movie(name='Jaws', released=1975, director='Speilberg'),
Movie(name='Aliens', released=1986, director='Scott'),
Movie(name='Titanic', released=1997, director='Cameron')]
```

If the key function is expensive, it is possible to avoid repeated function calls by searching 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)
```