minimum-one-bit-operations-to-make-integers-zero.py

# Time:  O(logn)
# Space: O(1)

# reference: https://en.wikipedia.org/wiki/Gray_code
class Solution(object):
    def minimumOneBitOperations(self, n):
        """
        :type n: int
        :rtype: int
        """
        def gray_to_binary(n):
            result = 0
            while n:
                result ^= n
                n >>= 1
            return result
        
        # [observation]
        # n    f(n)
        # 000    0
        # 001    1
        # 011    2
        # 010    3
        # 110    4
        # 111    5
        # 101    6
        # 100    7
	# f(0XX...X) + f(1XX...X) = f(100...0) implies n is a gray code
        # => f(n) is actually the inverse of gray code
        return gray_to_binary(n)


# Time:  O(logn)
# Space: O(1)
class Solution2(object):
    def minimumOneBitOperations(self, n):
        """
        :type n: int
        :rtype: int
        """
        # [observation1]:
        # f(1) = 1
        # f(10) = 2 * f(1) + 1 = 3
        # f(100) = 2 * f(10) + 1 = 7
        # by mathematical induction
        # => f(2^k) = 2^(k+1)-1
        #
        # [observation2]
        # n    f(n)
        # 000    0
        # 001    1
        # 011    2
        # 010    3
        # 110    4
        # 111    5
        # 101    6
        # 100    7
        # let pos be an array of positions where the bit is 1 in ascending order:
	# f(0XX...X) + f(1XX...X) = f(100...0)
        # f(1XX...X) = f(100...0) - f(0XX...X)
        #            = (2^(pos[k-1]+1)-1) - f(0XX...X)
        # by mathematical induction
        # => f(n) = (2^(pos[k-1]+1)-1) - (2^(pos[k-2])+1) + ... + (-1)^(k-1) * (2^(pos[0]+1)-1)
        result = 0
        while n:
            result = -result - (n^(n-1))  # 2^(pos[i]+1)-1
            n &= n-1
        return abs(result)