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"""Bisection algorithms."""

def insort_right(a, x, lo=0, hi=None, *, key=None):

"""Insert item x in list a, and keep it sorted assuming a is sorted.

If x is already in a, insert it to the right of the rightmost x.

Optional args lo (default 0) and hi (default len(a)) bound the

slice of a to be searched.

"""

if key is None:

lo = bisect_right(a, x, lo, hi)

else:

lo = bisect_right(a, key(x), lo, hi, key=key)

a.insert(lo, x)

def bisect_right(a, x, lo=0, hi=None, *, key=None):

"""Return the index where to insert item x in list a, assuming a is sorted.

The return value i is such that all e in a[:i] have e <= x, and all e in

a[i:] have e > x. So if x already appears in the list, a.insert(i, x) will

insert just after the rightmost x already there.

Optional args lo (default 0) and hi (default len(a)) bound the

slice of a to be searched.

"""

if lo < 0:

raise ValueError('lo must be non-negative')

if hi is None:

hi = len(a)

# Note, the comparison uses "<" to match the

# __lt__() logic in list.sort() and in heapq.

if key is None:

while lo < hi:

mid = (lo + hi) // 2

if x < a[mid]:

hi = mid

else:

lo = mid + 1

else:

while lo < hi:

mid = (lo + hi) // 2

if x < key(a[mid]):

hi = mid

else:

lo = mid + 1

return lo

def insort_left(a, x, lo=0, hi=None, *, key=None):

"""Insert item x in list a, and keep it sorted assuming a is sorted.

If x is already in a, insert it to the left of the leftmost x.

Optional args lo (default 0) and hi (default len(a)) bound the

slice of a to be searched.

"""

if key is None:

lo = bisect_left(a, x, lo, hi)

else:

lo = bisect_left(a, key(x), lo, hi, key=key)

a.insert(lo, x)

def bisect_left(a, x, lo=0, hi=None, *, key=None):

"""Return the index where to insert item x in list a, assuming a is sorted.

The return value i is such that all e in a[:i] have e < x, and all e in

a[i:] have e >= x. So if x already appears in the list, a.insert(i, x) will

insert just before the leftmost x already there.

Optional args lo (default 0) and hi (default len(a)) bound the

slice of a to be searched.

"""

if lo < 0:

raise ValueError('lo must be non-negative')

if hi is None:

hi = len(a)

# Note, the comparison uses "<" to match the

# __lt__() logic in list.sort() and in heapq.

if key is None:

while lo < hi:

mid = (lo + hi) // 2

if a[mid] < x:

lo = mid + 1

else:

hi = mid

else:

while lo < hi:

mid = (lo + hi) // 2

if key(a[mid]) < x:

lo = mid + 1

else:

hi = mid

return lo

# Overwrite above definitions with a fast C implementation

try:

from _bisect import *

except ImportError:

pass

# Create aliases

bisect = bisect_right

insort = insort_right

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