1000. Minimum Cost to Merge Stones
There are
N
piles of stones arranged in a row. Thei
-th pile hasstones[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 theseK
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
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