
``bisect`` --- Array bisection algorithm
****************************************

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))

   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.

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

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

   Insert *x* in *a* in sorted order.  This is equivalent to
   ``a.insert(bisect.bisect_left(a, x, lo, hi), x)`` assuming that *a*
   is already sorted.  Keep in mind that the O(log n) search is
   dominated by the slow O(n) insertion step.

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

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

See also:

   SortedCollection recipe that 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


Other 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']

Unlike the ``sorted()`` function, it does not make sense for the
``bisect()`` functions to have *key* or *reversed* arguments because
that would lead to an inefficient design (successive calls to bisect
functions would not "remember" all of the previous key lookups).

Instead, it is better to search a list of precomputed keys to find the
index of the record in question:

   >>> data = [('red', 5), ('blue', 1), ('yellow', 8), ('black', 0)]
   >>> data.sort(key=lambda r: r[1])
   >>> keys = [r[1] for r in data]         # precomputed 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)
