# Minimum Path Sum Problem

## Description

LeetCode Problem 64.

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 1:

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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:

``````1
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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

## Sample C++ Code

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class Solution {
public:
int minPathSum(vector<vector<int>>& grid) {
if (grid.size() == 0) return 0;

int m = grid.size();
int n = grid.size();

vector<vector<int>> dp(m, vector<int>(n, 0));

for (int i = 0; i < m; i ++) {
for (int j = 0; j < n; j ++) {
if (i == 0 && j == 0) {
dp[i][j] = grid[i][j];
} else if (i == 0) {
dp[i][j] = dp[i][j-1] + grid[i][j];
} else if (j == 0) {
dp[i][j] = dp[i-1][j] + grid[i][j];
} else {
dp[i][j] = min(dp[i-1][j], dp[i][j-1]) + grid[i][j];
}
}
}

return dp[m-1][n-1];
}
};
``````