class Solution(object):def minPathSum(self, grid):""":type grid: List[List[int]]:rtype: int"""m = len(grid)if m > 0:n = len(grid[0])else:return 0dp = [[0] * n for _ in range(m)]dp[0][0] = grid[0][0]for i in range(1, m):dp[i][0] = grid[i][0] + dp[i-1][0]for j in range(1, n):dp[0][j] = grid[0][j] + dp[0][j-1]for i in range(1, m):for j in range(1, n):dp[i][j] = grid[i][j] + min(dp[i-1][j], dp[i][j-1])return dp[m-1][n-1]

We will initialize a 2D array named `dp`

with zeros as always. The path sum for position (0,0) will obviously be `grid[0][0]`

. For the first row the sum to position `dp[0][j]`

will be the sum of all `dp[0][k]`

where `k < j`

. Similarly for the first column the sum to position dp[i][0] will be the sum of all `dp[k][0] `

where `k < i`

.

After that we simply have to iterate through the array and make sure each time to update the path sum by adding `grid[i][j]`

to the minimum of `dp[i-1][j]`

and `dp[i][j-1]`

and store it in `dp[i][j]`