<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:
        n = len(grid)
        m = len(grid[0])
        distances = [[inf for i in range(m)] for j in range(n)]
        distances[0][0] = grid[0][0]

        pq = []
        heappush(pq, (grid[0][0], 0, 0))

        while pq:
            cost, x, y = heappop(pq)
            if distances[x][y] < cost:
                continue
            
            for i, j in ((0,1), (1,0)):
                if 0 <= x+i < n and 0 <= y+j < m and distances[x+i][y+j] > cost+grid[x+i][y+j]:
                    distances[x+i][y+j] = cost + grid[x+i][y+j]
                    heappush(pq, (distances[x+i][y+j], x+i, y+j))
        
        return distances[-1][-1]
        
18500