Leetcode 410. Split Array Largest Sum

Given an array which consists of non-negative integers and an integer m, you can split the array into m non-empty continuous subarrays. Write an algorithm to minimize the largest sum among these m subarrays.

Note:
If n is the length of array, assume the following constraints are satisfied:

• 1 ≤ n ≤ 1000
• 1 ≤ m ≤ min(50, n)

Examples:

Input:
nums = [7,2,5,10,8]
m = 2

Output:
18

Explanation:
There are four ways to split nums into two subarrays.
The best way is to split it into [7,2,5] and [10,8],
where the largest sum among the two subarrays is only 18.

Analysis:
A sequence DP problem. 
Define dp[n + 1][m + 1], where dp[i][k] means for the first i elements, divide into number of k subarrays, dp[i][k] is the minimum largeset sum among them. 

Code (Java):

class Solution {
    public int splitArray(int[] nums, int m) {
        int[] preSum = new int[nums.length + 1];
        
        for (int i = 1; i < preSum.length; i++) {
            preSum[i] = preSum[i - 1] + nums[i - 1];
        }
        
        int[][] dp = new int[nums.length + 1][m + 1];
        for (int i = 0; i < dp.length; i++) {
            for (int j = 0; j < dp[0].length; j++) {
                dp[i][j] = Integer.MAX_VALUE;
            }
        }
        
        dp[0][0] = 0;
        
        for (int i = 1; i <= nums.length; i++) {
            for (int j = 0; j < i; j++) {
                for (int k = 1; k <= m; k++) {
                    dp[i][k] = Math.min(dp[i][k], Math.max(dp[j][k - 1], preSum[i] - preSum[j]));
                }
            }
        }
        
        return dp[nums.length][m];
    }
}