rotateFunction.py

# Given an array of integers A and let n to be its length.
#
# Assume Bk to be an array obtained by rotating the array A k positions clock-wise, we define a "rotation function" F on A as follow:
#
# F(k) = 0 * Bk[0] + 1 * Bk[1] + ... + (n-1) * Bk[n-1].
#
# Calculate the maximum value of F(0), F(1), ..., F(n-1).
#
# Note:
# n is guaranteed to be less than 105.
#
# Example:
#
# A = [4, 3, 2, 6]
#
# F(0) = (0 * 4) + (1 * 3) + (2 * 2) + (3 * 6) = 0 + 3 + 4 + 18 = 25
# F(1) = (0 * 6) + (1 * 4) + (2 * 3) + (3 * 2) = 0 + 4 + 6 + 6 = 16
# F(2) = (0 * 2) + (1 * 6) + (2 * 4) + (3 * 3) = 0 + 6 + 8 + 9 = 23
# F(3) = (0 * 3) + (1 * 2) + (2 * 6) + (3 * 4) = 0 + 2 + 12 + 12 = 26
#
# So the maximum value of F(0), F(1), F(2), F(3) is F(3) = 26.

# F(1) = 1 * 4 + 2 * 3 + 3 * 2 + 0 * 6
# F(1) - F (0) = 1 * 4 + 1 * 3 + 1 * 2 - 3 * 6
# F(2) = 2 * 4 + 3 * 3 + 0 * 2 + 1 * 6
# F(2) - F(1) = 1 * 4 + 1 * 3 - 3 * 2 + 1 * 6

# F(k) - F(k-1) = sum (A) - n * A_(n-k)


class Solution(object):
    def maxRotateFunction(self, A):
        """
        :type A: List[int]
        :rtype: int
        """
        n = len(A)
        if n == 0 :
            return 0
        sumA = sum(A)
        Fk = [0] * n
        Fk[0] = sum(i * j for (i,j) in zip(range(n), A))
        for i in range(1,n):
            Fk[i] = Fk[i-1] + sumA - n * A[n-i]
        return max(Fk)