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.


Solutions

找到从左上角到右下角所有路径和中最小的，移动操作只能是向下或者向右。

1. 2D-DP

# Time: O(mn)
# Space: O(mn)
class Solution:
def minPathSum(self, grid: List[List[int]]) -> int:
if not grid or len(grid) == 0:
return 0

r, c = len(grid), len(grid[0])
dp = [[0 for _ in range(c)] for _ in range(r)]
dp[0][0] = grid[0][0]
for i in range(1, r):
dp[i][0] = dp[i-1][0] + grid[i][0]
for i in range(1, c):
dp[0][i] = dp[0][i-1] + grid[0][i]

for i in range(1, r):
for j in range(1, c):
dp[i][j] = min(dp[i-1][j], dp[i][j-1]) + grid[i][j]
return dp[-1][-1]
# Runtime: 100 ms, faster than 73.26%
# Memory Usage: 14.4 MB, less than 75.44%