There are only 2 directions to reach a given cell[i][j], either from above it or from left,

dp[i][j  1] (min path sum from its left)

dp[i  1][j] (min path sum from its above)
Min path = min(min path sum from its left, min path sum from its above) + Current Value
min{ dp[i][j  1], dp[i  1][j] } + grid[i][j]
;
class Solution {
public:
int minPathSum(vector<vector<int>>& grid) {
if(grid.empty()) return 0;
int m = grid.size(), 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] = grid[i][0] + dp[i  1][0];
for(int i = 1; i < n; i++) dp[0][i] = grid[0][i] + dp[0][i  1];
for(int i = 1; i < m; i++)
for(int j = 1; j < n; j++)
dp[i][j] = min(dp[i][j  1], dp[i  1][j]) + grid[i][j];
return dp[m  1][n  1];
}
};