python_utils.py

def is_even(n):
    """uses bitwise operator to check if number is even or odd
    Args: 
        n (int):
    Returns:
        answer (bool): True if even, False otherwise
    """
    return  n & 1 == 0


def is_even2(n):
    """
    Args: 
        n (int):
    Returns:
        answer (bool): True if even, False otherwise
    """
    return  n % 2 == 0



def exponentiation_by_squaring_recursively(x, n):
    """The following recursive algorithm computes x^n for integer values of n:
    https://en.wikipedia.org/wiki/Exponentiation_by_squaring
    Time and space complexity: O(log N)
    """
    if n < 0:
        return exponentiation_by_squaring_recursively(1.0 / x, -n)
    elif n == 0:
        return 1
    elif n == 1:
        return x
    elif is_even(n):
        return exponentiation_by_squaring_recursively(x * x,  n / 2)
    elif not is_even(n):
        return x * exponentiation_by_squaring_recursively(x * x, (n - 1) / 2)

def exponentiation_by_squaring_recursively2(x, n):
    """ https://eli.thegreenplace.net/2009/03/21/efficient-integer-exponentiation-algorithms/
    """
    if n < 0:
        return exponentiation_by_squaring2(1.0/x, -n)
    elif n == 0:
        return 1
    elif n % 2 == 1:
        return x * exponentiation_by_squaring2(x, n - 1)
    else:
        p = exponentiation_by_squaring2(x, n / 2)
        return  p * p

#raymondh - In Python3.6 the fastest way to eliminate hashable duplicates while retaining insertion order:
list(dict.fromkeys('uppdattemoma'))