:mod:`bisect` --- Array bisection algorithm
===========================================

.. module:: bisect
   :synopsis: Array bisection algorithms for binary searching.
.. sectionauthor:: Fred L. Drake, Jr. <fdrake@acm.org>
.. sectionauthor:: Raymond Hettinger <python at rcn.com>
.. example based on the PyModules FAQ entry by Aaron Watters <arw@pythonpros.com>

**Source code:** :source:`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 :mod:`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:


.. function:: 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 :term:`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).

   .. versionchanged:: 3.10
      Added the *key* parameter.


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

   Similar to :func:`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 :term:`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).

   .. versionchanged:: 3.10
      Added the *key* parameter.


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

   Insert *x* in *a* in sorted order.

   *key* specifies a :term:`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 :func:`bisect_left` to locate an insertion point.
   Next, it runs the :meth:`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.

   .. versionchanged:: 3.10
      Added the *key* parameter.


.. function:: insort_right(a, x, lo=0, hi=len(a), *, key=None)
              insort(a, x, lo=0, hi=len(a), *, key=None)

   Similar to :func:`insort_left`, but inserting *x* in *a* after any existing
   entries of *x*.

   *key* specifies a :term:`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 :func:`bisect_right` to locate an insertion point.
   Next, it runs the :meth:`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.

   .. versionchanged:: 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
  :func:`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).

.. seealso::

   * `Sorted Collections
     <http://www.grantjenks.com/docs/sortedcollections/>`_ is a high performance
     module that uses *bisect* to managed sorted collections of data.

   * The `SortedCollection recipe
     <https://code.activestate.com/recipes/577197-sortedcollection/>`_ 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 :func:`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
--------

.. _bisect-example:

The :func:`bisect` function can be useful for numeric table lookups. This
example uses :func:`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)

