Leetcode 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. It can only move either down or right at any point in time.

Algorithm

Using dynamic programming to solve:
$$\{\begin{array}{llc}sum(i,j)=sum(i-1,j)+node(i,j)& if& sum(i-1,j)\le sum(i,j-1)\\ sum(i,j)=sum(i,j-1)+node(i,j)& if& sum(i-1,j)>sum(i,j-1)\end{array}$$

Code

class Solution:
    def minPathSum(self, grid: List[List[int]]) -> int:
        if not grid:
            return 0
        row = len(grid)
        if not row:
            return 0
        col = len(grid[0])
        if not col:
            return 0
        step = [[0] * col for _ in range(row)]
        
        step[0][0] = grid[0][0]
        for i in range(row):
            for j in range(col):
                flag = False
                if i and flag == False:
                    step[i][j] = step[i-1][j] + grid[i][j]
                    flag = True
                if j and (flag == False or step[i][j] > step[i][j-1] + grid[i][j]):
                    step[i][j] = step[i][j-1] + grid[i][j]
        
        return step[row-1][col-1]