"""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 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, i points just
    beyond the rightmost x already there.

    Optional args lo (default 0) and hi (default len(a)) bound the
    slice of a to be searched.
    """

    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 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, i points 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 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