64. Minimum Path Sum

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本文介绍了一个算法问题：在一个非负数构成的矩阵中寻找从左上角到右下角的路径，使得路径上的数字之和最小。文章提供了一种递归加动态规划的方法，并附带详细的Python代码实现。

https://leetcode.com/problems/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.

class Solution(object):
    def minSum(self, grid, row, col):
        m, n = len(grid), len(grid[0])
        if row == m - 1:
            res = 0
            for i in range(col + 1, n):
                res += grid[row][i]
            return res
        elif col == n - 1:
            res = 0
            for i in range(row + 1, m):
                res += grid[i][col]
            return res
        if self.dp[row][col + 1] < 0:
            self.dp[row][col + 1] = self.minSum(grid, row, col + 1)
        if self.dp[row + 1][col] < 0:
            self.dp[row + 1][col] = self.minSum(grid, row + 1, col)
        return min(grid[row][col + 1] + self.dp[row][col + 1], grid[row + 1][col] + self.dp[row + 1][col])
    def minPathSum(self, grid):
        """
        :type grid: List[List[int]]
        :rtype: int
        """
        m, n = len(grid), len(grid[0])
        self.dp = [[-1] * n for _ in range(m)]
        return grid[0][0] + self.minSum(grid, 0, 0)