<p>Given a <code>m x n</code> <code>grid</code> filled with non-negative numbers, find a path from top left to bottom right, which minimizes the sum of all numbers along its path.</p>

<p><strong>Note:</strong> You can only move either down or right at any point in time.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2020/11/05/minpath.jpg" style="width: 242px; height: 242px;" />
<pre>
<strong>Input:</strong> grid = [[1,3,1],[1,5,1],[4,2,1]]
<strong>Output:</strong> 7
<strong>Explanation:</strong> Because the path 1 &rarr; 3 &rarr; 1 &rarr; 1 &rarr; 1 minimizes the sum.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> grid = [[1,2,3],[4,5,6]]
<strong>Output:</strong> 12
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>m == grid.length</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 200</code></li>
	<li><code>0 &lt;= grid[i][j] &lt;= 200</code></li>
</ul>

class Solution:
    def minPathSum(self, grid: List[List[int]]) -> int:
        # dfs to navigate
        # O(m*n) in time
        # O(m*n) in space
        rows, cols = len(grid), len(grid[0])
        memo = {}
        def dfs(r, c):
            dirs = [[1,0],[0,1]] # down right
            if r == rows -1 and c == cols -1:
                return grid[r][c] # valid path
            if (r,c) in memo:
                return memo[(r,c)]
            minSum = float('inf')
            for dx, dy in dirs:
                move_x = r + dx
                move_y = c + dy 
                if (move_x == rows or move_y == cols):
                    continue
                pathSum =  grid[r][c] + dfs(move_x, move_y) # currsum + nextSum
                minSum = min(minSum, pathSum)
            memo[(r,c)] = minSum
            return minSum
        ans = dfs(0,0)
        return ans

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