leetcode396

C++的实现：

class Solution {
public:
    int maxRotateFunction(vector<int>& A) {
        long N = A.size();
        long S = 0;
        long t = 0;
        for (int i = 0; i < N; ++i) {
            S += A[i];
            t += i * A[i];
        }
        long res = t;
        for (int i = N - 1; i >= 0; --i) {
            // F(k+1) = F(k) + S - n * Bk[n-1]
            t += S - N * (long)A[i];
            res = max(res, t);
        }
        return res;
    }
};

python的实现：

class Solution:
    def maxRotateFunction(self, A: 'List[int]') -> int:
        N,S,t = len(A),0,0
        for i in range(N):
            S += A[i]#原始数组之和
            t += i * A[i]#旋转数组之和
        res = t
        for i in range(N-1,-1,-1):
            t += S - N * A[i]
            res = max(res,t)
        return res

算法思路：数学题。

根据题目找规律，错位相减，以提高算法执行速度。

推导过程：
（1）F(k) = 0 * Bk[0] + 1 * Bk[1] + ... + (n-2) * Bk[n-2] + (n-1) * Bk[n-1]
（2）F(k+1) = 0 * Bk[n-1] + 1 * Bk[0] + 2 * Bk[2] + ... + (n-1) * Bk[n-2]
（2）-（1）得：F(k+1) - F(k) = (Bk[0] + Bk[1] + ... + Bk[n-2]) - (n-1)*Bk[n-1]
可得：F(k+1) - F(k) = (Bk[0] + Bk[1] + ... + Bk[n-2] + Bk[n-1]) - n*Bk[n-1]
令S=Sum{Bk}
有：F(k+1) = F(k) + S - n * Bk[n-1]

参考地址：https://leetcode-cn.com/problems/rotate-function/solution/c-cuo-wei-xiang-jian-fa-by-da-li-wang/