**source:**https://leetcode.com/problems/minimum-path-sum/**C/C++**

**Solution to LeetCode**problem

**64**.

**Minimum Path Sum**.

## Problem

Given a `m x n`

grid filled with non-negative numbers, find a path from top left to bottom right, which minimizes the sum of all numbers along its path.

**Note:** You can only move either down or right at any point in time.

## Examples

**Example 1:**

Input:grid = [[1,3,1],[1,5,1],[4,2,1]]

Output:7

Explanation:Because the path 1 → 3 → 1 → 1 → 1 minimizes the sum.

**Example 2:**

Input:grid = [[1,2,3],[4,5,6]]

Output:12

## Constraints

`m == grid.length`

`n == grid[i].length`

`1 <= m, n <= 200`

`0 <= grid[i][j] <= 100`

## Solution

We use **dynamic programming**:

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class Solution {
private:
vector<vector<int>> dp;
int minPath(vector<vector<int>>& grid, int r, int c) {
if (r+1 == grid.size() && c+1 == grid[0].size())
return grid[r][c];
if (r == grid.size() || c == grid[0].size())
return INT_MAX;
if (dp[r][c] != -1)
return dp[r][c];
dp[r][c] = grid[r][c] + min(minPath(grid, r+1, c), minPath(grid, r, c+1));
return dp[r][c];
}
public:
int minPathSum(vector<vector<int>>& grid) {
dp = vector(grid.size(), vector<int>(grid[0].size(), -1));
return minPath(grid, 0, 0);
}
};