## DEV Community

Abhishek Chaudhary

Posted on

# Minimum Path Sum

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 1:

Input: grid = [[1,3,1],[1,5,1],[4,2,1]]
Output: 7
Explanation: Because the path 1 → 3 → 1 → 1 → 1 minimizes the sum.

Example 2:

Input: grid = [[1,2,3],[4,5,6]]
Output: 12

Constraints:

• `m == grid.length`
• `n == grid[i].length`
• `1 <= m, n <= 200`
• `0 <= grid[i][j] <= 100`

SOLUTION:

``````class Solution:
def calcMin(self, grid, x, y, m, n):
if (x, y) in self.cache:
return self.cache[(x, y)]
if (x, y) == (m - 1, n - 1):
self.cache[(x, y)] = grid[m - 1][n - 1]
return self.cache[(x, y)]
right = float('inf')
bottom = float('inf')
if y < n - 1:
right = self.calcMin(grid, x, y + 1, m, n)
if x < m - 1:
bottom = self.calcMin(grid, x + 1, y, m, n)
currMin = min(right, bottom)
self.cache[(x, y)] = grid[x][y] + currMin
return self.cache[(x, y)]

def minPathSum(self, grid: List[List[int]]) -> int:
self.cache = {}
return self.calcMin(grid, 0, 0, len(grid), len(grid[0]))
``````