Question
Formatted question description: https://leetcode.ca/all/396.html
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.
@tag-array
Algorithm
Abstract the concrete numbers as A, B, C, D, then we can get:
F(0) = 0A + 1B + 2C +3D
F(1) = 0D + 1A + 2B +3C
F(2) = 0C + 1D + 2A +3B
F(3) = 0B + 1C + 2D +3A
Then, through careful observation, we can draw the following law:
sum = 1A + 1B + 1C + 1D
F(1) = F(0) + sum-4D
F(2) = F(1) + sum-4C
F(3) = F(2) + sum-4B
Then we have found the rule, F(i) = F(i-1) + sum-n*A[n-i]
Code
Java
-
public class Rotate_Function { class Solution { public int maxRotateFunction(int[] A) { int prevValue = 0; int sum = 0; int n = A.length; for (int i = 0; i < n; ++i) { sum += A[i]; // get: 1A+1B+1C+1D+... prevValue += i * A[i]; // get: F(0) first } int result = prevValue; for (int i = 1; i < n; i++) { // start from index=1 prevValue = prevValue + sum - n * A[n - i]; result = Math.max(result, prevValue); } return result; } } }
-
// OJ: https://leetcode.com/problems/rotate-function/ // Time: O(N) // Space: O(1) class Solution { public: int maxRotateFunction(vector<int>& A) { if (A.empty()) return 0; long long f = 0, ans = INT_MIN, N = A.size(), sum = accumulate(A.begin(), A.end(), (long long)0); for (int i = 0; i < N; ++i) f += i * A[i]; for (int i = N - 1; i >= 0; --i) ans = max(ans, f += (sum - N * A[i])); return ans; } };
-
class Solution(object): def maxRotateFunction(self, A): """ :type A: List[int] :rtype: int """ if not A: return 0 sumA = sum(A) fk = 0 n = len(A) for i, num in enumerate(A): fk += i * num idx = n - 1 ans = float("-inf") for _ in range(n): fk += sumA - n * A[idx] ans = max(ans, fk) idx -= 1 return ans