# Minimum Path Sum

## Description

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.

**Example:**

```
Input:
[
  [1,3,1],
  [1,5,1],
  [4,2,1]
]
Output: 7
Explanation: Because the path 1→3→1→1→1 minimizes the sum.
```

## Solution

Let `dp[i, j]` be the minimum path sum from the top-left to the grid at row `i` and column `j`. Then `dp[i, j] = min(dp[i-1, j], dp[i, j-1]) + grid[i][j]`.

```cpp
class Solution {
public:
    int minPathSum(vector<vector<int>>& grid) {
        int m = grid.size();
        int n = grid[0].size();
        vector<vector<int>> dp(m, vector<int>(n));
        dp[0][0] = grid[0][0];
        for(int i = 1; i < m; ++i){
            dp[i][0] = dp[i - 1][0] + grid[i][0];
        }
        for(int j = 1; j < n; ++j){
            dp[0][j] = dp[0][j - 1] + grid[0][j];
        }
        for(int i = 1; i < m; ++i){
            for(int j = 1; j < n; ++j){
                dp[i][j] = min(dp[i - 1][j], dp[i][j - 1]) + grid[i][j];
            }
        }
        return dp[m-1][n-1];
    }
};
```
