<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:
        visited = {}
        visited[(0,0)] = grid[0][0]
        h = [(grid[0][0], 0, 0)]
        while len(h) > 0:
            dist, x, y = heappop(h)
            if x == len(grid) - 1 and y == len(grid[0]) - 1:
                return dist
            for direction in [(0, 1), (1, 0)]:
                new_x = x + direction[0]
                new_y = y + direction[1]
                if new_x <= len(grid) - 1 and new_y <= len(grid[0]) - 1:
                    new_dist = dist + grid[new_x][new_y]
                    if (new_x, new_y) not in visited or visited[(new_x, new_y)] > new_dist:
                        visited[(new_x, new_y)] = new_dist
                        heappush(h, (new_dist, new_x, new_y))

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