"""Bisection algorithms."""

def insort_right(a, x, lo=0, hi=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 lo < 0:

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

    if hi is None:

        hi = len(a)

    while lo < hi:

        mid = (lo+hi)//2

        if x < a[mid]: hi = mid

        else: lo = mid+1

    a.insert(lo, x)

insort = insort_right   # backward compatibility

def bisect_right(a, x, lo=0, hi=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(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)

    while lo < hi:

        mid = (lo+hi)//2

        if x < a[mid]: hi = mid

        else: lo = mid+1

    return lo

bisect = bisect_right   # backward compatibility

def insort_left(a, x, lo=0, hi=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 lo < 0:

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

    if hi is None:

        hi = len(a)

    while lo < hi:

        mid = (lo+hi)//2

        if a[mid] < x: lo = mid+1

        else: hi = mid

    a.insert(lo, x)

def bisect_left(a, x, lo=0, hi=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(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)

    while lo < hi:

        mid = (lo+hi)//2

        if a[mid] < x: lo = mid+1

        else: hi = mid

    return lo

# Overwrite above definitions with a fast C implementation

try:

    from _bisect import bisect_right, bisect_left, insort_left, insort_right, insort, bisect

except ImportError:

    pass