Min Sum Path in Matrix
Logical Breakdown
- [ ] Subproblem Identification:
- [ ] Optimal Substructure:
- [ ] Constraint Handling: (e.g., Modulo \(10^9 + 7\))
- [ ] Optimization: (Matrix Exponentiation if 'A' is huge)
Mathematical Rigor
State Definition
Let \(dp[i]\) be the state for the \(i\)-th subproblem.
Recurrence Relation
Visualization
Complexity Analysis
| Approach | Time Complexity | Space Complexity |
|---|---|---|
| Iterative DP | \(O(N)\) | \(O(1)\) |
Code Reference
package com.dsa.dynamic_programming;
import java.util.*;
/ * Problem: Min Sum Path in Matrix * Group: 11. Dynamic Programming */ / * ============================================================================================ * PROBLEM: Min Sum Path in Matrix * ============================================================================================ * * --------------------- * 1. Problem Description * --------------------- * Given a N x M grid of non-negative integers. * Find a path from top-left to bottom-right which minimizes the sum of all numbers along its path. * Movement: Only DOWN or RIGHT. * * --------------------- * 2. Logical Breakdown * --------------------- * Let dp[i][j] be the minimum path sum to reach cell (i, j). * * Recurrence: * To arrive at (i, j) with minimal cost, we should choose the cheaper path between * coming from Top (i-1, j) or Left (i, j-1). * * dp[i][j] = grid[i][j] + min(dp[i-1][j], dp[i][j-1]) * * Base Cases: * - dp[0][0] = grid[0][0] * - First Row: dp[0][j] = dp[0][j-1] + grid[0][j] (Can only come from left). * - First Col: dp[i][0] = dp[i-1][0] + grid[i][0] (Can only come from top). * * --------------------- * 3. Complexity * --------------------- * Time: O(N * M) * Space: O(1) (If we modify the input grid in-place) or O(N * M). * * ============================================================================================ */ public class MinSumPathInMatrix { public int solve(int A) { // TODO: Implement solution return 0; }
public int minPathSum(int[][] grid2) {
return 1;
}
}