# `bisect` --- 配列二分法アルゴリズム

ソースコード: 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 linear searches or frequent resorting.

The module is called `bisect` because it uses a basic bisection algorithm to do its work. Unlike other bisection tools that search for a specific value, the functions in this module are designed to locate an insertion point. Accordingly, the functions never call an `__eq__()` method to determine whether a value has been found. Instead, the functions only call the `__lt__()` method and will return an insertion point between values in an array.

次の関数が用意されています:

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

  ソートされた順序を保ったまま x を a に挿入できる点を探し当てます。リストの中から検索する部分集合を指定するには、パラメータの lo と hi を使います。デフォルトでは、リスト全体が使われます。x がすでに a に含まれている場合、挿入点は既存のどのエントリーよりも前(左)になります。戻り値は、`list.insert()` の第一引数として使うのに適しています。a はすでにソートされているものとします。

  The returned insertion point ip partitions the array a into two slices such that `all(elem < x for elem in a[lo : ip])` is true for the left slice and `all(elem >= x for elem in a[ip : hi])` is true for the right slice.

  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 and no key function is called.

  バージョン 3.10 で変更: key パラメータが追加されました。

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 ip partitions the array a into two slices such that `all(elem <= x for elem in a[lo : ip])` is true for the left slice and `all(elem > x for elem in a[ip : hi])` is true for the right slice.

  バージョン 3.10 で変更: key パラメータが追加されました。

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

  x を a にソート順で挿入します。

  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.

  バージョン 3.10 で変更: key パラメータが追加されました。

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.

  バージョン 3.10 で変更: key パラメータが追加されました。

## パフォーマンスに関するメモ

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).

参考

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

- bisect を利用して、直接の探索ができ、キー関数をサポートする、完全な機能を持つコレクションクラスを組み立てる SortedCollection recipe。キーは、探索中に不必要な呼び出しをさせないために、予め計算しておきます。

## ソート済みリストの探索

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

## 使用例

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']

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, 'Spielberg'),
    ...     Movie('Titanic', 1997, 'Cameron'),
    ...     Movie('The Birds', 1963, 'Hitchcock'),
    ...     Movie('Aliens', 1986, 'Cameron')
    ... ]

    >>> # Find the first movie released 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='Spielberg'),
     Movie(name='Aliens', released=1986, director='Cameron'),
     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)