1000. Minimum Cost to Merge Stones

There are N piles of stones arranged in a row.  The i-th pile has stones[i] stones.

A move consists of merging exactly K consecutive piles into one pile, and the cost of this move is equal to the total number of stones in these K piles.

Find the minimum cost to merge all piles of stones into one pile.  If it is impossible, return -1.

Example 1:

Input: stones = [3,2,4,1], K = 2
Output: 20
Explanation: 
We start with [3, 2, 4, 1].
We merge [3, 2] for a cost of 5, and we are left with [5, 4, 1].
We merge [4, 1] for a cost of 5, and we are left with [5, 5].
We merge [5, 5] for a cost of 10, and we are left with [10].
The total cost was 20, and this is the minimum possible.

Example 2:

Input: stones = [3,2,4,1], K = 3
Output: -1
Explanation: After any merge operation, there are 2 piles left, and we can't merge anymore.  So the task is impossible.

Example 3:

Input: stones = [3,5,1,2,6], K = 3
Output: 25
Explanation: 
We start with [3, 5, 1, 2, 6].
We merge [5, 1, 2] for a cost of 8, and we are left with [3, 8, 6].
We merge [3, 8, 6] for a cost of 17, and we are left with [17].
The total cost was 25, and this is the minimum possible.

Note:

• 1 <= stones.length <= 30
• 2 <= K <= 30
• 1 <= stones[i] <= 100

Approach #1: DP. [C++]

class Solution {
public:
    int mergeStones(vector<int>& stones, int K) {
        int n = stones.size();
        if ((n-1) % (K-1)) return -1;
        
        const int kInf = 1e9 + 7;
        
        vector<int> sum(n+1, 0);
        for (int i = 0; i < n; ++i)
            sum[i+1] = sum[i] + stones[i];
            
        vector<vector<vector<int>>> dp(n+1, vector<vector<int>>(n+1, vector<int>(K+1, kInf)));
        
        for (int i = 0; i < n; ++i)
            dp[i][i][1] = 0;
        
        for (int l = 2; l <= n; ++l) {
            for (int i = 0; i <= n - l; ++i) {
                int j = i + l - 1;
                for (int k = 2; k <= K; ++k) {
                    for (int m = i; m < j; ++m) {
                        dp[i][j][k] = min(dp[i][j][k], dp[i][m][1] + dp[m+1][j][k-1]);
                    }
                }
                dp[i][j][1] = dp[i][j][K] + sum[j+1] - sum[i]; 
            }
        }
        
        return dp[0][n-1][1];
    }
};

　　

Analysis:

Non-overlapping subproblems: min cost of merging subarray A[i]~A[j] into k piles.

dp[i][j][k] : min cost to merge A[i]~A[j] into k piles

Init: dp[i][i][1] = 0 : no cost to merge one into one

Transition:

1. dp[i][j][k] = min(dp[i][m][1] + dp[i][m+1][k-1]}, i <= m < j, 2 <= k <= K

2. dp[i][j][1] = dp[i][j][K] + sum(A[i] ~ A[j])

ans : dp[0][n-1][1] # merge he whole array into one.

Approach #2: DP + Optimization. [Java]

class Solution {
    public int mergeStones(int[] stones, int K) {
        int n = stones.length;
        if ((n-1) % (K-1) != 0) return -1;
        int[] prefix = new int[n+1];
        for (int i = 0; i < n; ++i)
            prefix[i+1] = prefix[i] + stones[i];
        
        int[][] dp = new int[n][n];
        
        for (int l = 2; l <= n; ++l) {
            for (int i = 0; i <= n-l; ++i) {
                int j = i + l - 1;
                dp[i][j] = Integer.MAX_VALUE;
                for (int m = i; m < j; m += K - 1)
                    dp[i][j] = Math.min(dp[i][j], dp[i][m] + dp[m+1][j]);
                if ((j-i) % (K-1) == 0)
                    dp[i][j] += prefix[j+1] - prefix[i];
            }
        }
        
        return dp[0][n-1];
    }
}

　　

Analysis:

Optimization 1: In order to merge left part into 1 pile, (len - 1) % (K - 1) == 0

++m => m += K-1;

Optimization 2: The number of piles the right part can be merge into can be determined, (len - 1) % (K - 1) + 1. And we need to merge them first before the final merge of left and right.

dp[i][j] : min cost to merge A[i] ~ A[j] to (j-i) % (K-1) + 1 piles

Init: dp[i][i] = 0

dp[i][j] = min(dp[i][m] + dp[m+1][j]} + sum(A[i] ~ A[j]) if (j - i) % (K - 1) == 0

Reference:

https://zxi.mytechroad.com/blog/dynamic-programming/leetcode-1000-minimum-cost-to-merge-stones/

https://leetcode.com/problems/minimum-cost-to-merge-stones/discuss/247567/JavaC%2B%2BPython-DP