1770

1770. Maximum Score from Performing Multiplication Operations (Medium)

You are given two integer arrays nums and multipliers of size n and m respectively, where n >= m. The arrays are 1-indexed.

You begin with a score of 0. You want to perform exactly m operations. On the ith operation (1-indexed), you will:

• Choose one integer x from either the start or the end of the array nums.
• Add multipliers[i] * x to your score.
• Remove x from the array nums.

Return the maximum score after performing m operations.

 

Example 1:

Input: nums = [1,2,3], multipliers = [3,2,1]
Output: 14
Explanation: An optimal solution is as follows:
- Choose from the end, [1,2,3], adding 3 * 3 = 9 to the score.
- Choose from the end, [1,2], adding 2 * 2 = 4 to the score.
- Choose from the end, [1], adding 1 * 1 = 1 to the score.
The total score is 9 + 4 + 1 = 14.

Example 2:

Input: nums = [-5,-3,-3,-2,7,1], multipliers = [-10,-5,3,4,6]
Output: 102
Explanation: An optimal solution is as follows:
- Choose from the start, [-5,-3,-3,-2,7,1], adding -5 * -10 = 50 to the score.
- Choose from the start, [-3,-3,-2,7,1], adding -3 * -5 = 15 to the score.
- Choose from the start, [-3,-2,7,1], adding -3 * 3 = -9 to the score.
- Choose from the end, [-2,7,1], adding 1 * 4 = 4 to the score.
- Choose from the end, [-2,7], adding 7 * 6 = 42 to the score. 
The total score is 50 + 15 - 9 + 4 + 42 = 102.

 

Constraints:

• n == nums.length
• m == multipliers.length
• 1 <= m <= 103
• m <= n <= 105
• -1000 <= nums[i], multipliers[i] <= 1000

Related Topics:
Dynamic Programming

Similar Questions:

• Stone Game VII (Medium)

Solution 1. DP

This is a search problem and there are overlapping subproblems, so we can use DP to solve it.

Let dp[j][i] be the maximum score we can get using M[0..j] and a subarray of A that starts at i and has j elements less than A (because they are already selected). For the previous j steps, we already selected j elements out of A, and since there are i selected elements in front of the subarray, there are j - i selected elements after the subarray. So the index of the last element of the subarray is r = A.size() - j + i - 1.

dp[j][i] = max(
                A[i] * M[j] + dp[i + 1][j + 1], // select A[i]
                A[r] * M[j] + dp[i][j + 1], // select A[r]
)
where r = A.size() - j + i - 1

Note:

1. The range of i is [0, M.size() - 1] so we just need M instead of N for the second dimension of dp.
2. that for simplicity, we used 0 as unvisited number here. But since 0 is a valid score, when the memo[j][i] is indeed 0, we will recompute the value and thus waste time. We can initialize memo[j][i] to INT_MIN instead.

// OJ: https://leetcode.com/problems/maximum-score-from-performing-multiplication-operations/
// Author: github.com/lzl124631x
// Time: O(M^2)
// Space: O(M^2)
class Solution {
    int memo[1001][1001] = {};
    int dfs(vector<int> &A, vector<int> &M, int i, int j) {
        if (j == M.size()) return 0;
        if (memo[j][i]) return memo[j][i];
        return memo[j][i] = max(A[i] * M[j] + dfs(A, M, i + 1, j + 1), A[A.size() - j + i - 1] * M[j] + dfs(A, M, i, j + 1));
    }
public:
    int maximumScore(vector<int>& A, vector<int>& M) {
        return dfs(A, M, 0, 0);
    }
};