<p>Given an <code>n x n</code> array of integers <code>matrix</code>, return <em>the <strong>minimum sum</strong> of any <strong>falling path</strong> through</em> <code>matrix</code>.</p>

<p>A <strong>falling path</strong> starts at any element in the first row and chooses the element in the next row that is either directly below or diagonally left/right. Specifically, the next element from position <code>(row, col)</code> will be <code>(row + 1, col - 1)</code>, <code>(row + 1, col)</code>, or <code>(row + 1, col + 1)</code>.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/11/03/failing1-grid.jpg" style="width: 499px; height: 500px;" />
<pre>
<strong>Input:</strong> matrix = [[2,1,3],[6,5,4],[7,8,9]]
<strong>Output:</strong> 13
<strong>Explanation:</strong> There are two falling paths with a minimum sum as shown.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/11/03/failing2-grid.jpg" style="width: 164px; height: 365px;" />
<pre>
<strong>Input:</strong> matrix = [[-19,57],[-40,-5]]
<strong>Output:</strong> -59
<strong>Explanation:</strong> The falling path with a minimum sum is shown.
</pre>

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

<ul>
	<li><code>n == matrix.length == matrix[i].length</code></li>
	<li><code>1 &lt;= n &lt;= 100</code></li>
	<li><code>-100 &lt;= matrix[i][j] &lt;= 100</code></li>
</ul>

class Solution:
    def minFallingPathSum(self, matrix: List[List[int]]) -> int:
        
        def get_nbrs(pos):
            nbrs = []
            if pos[0] < len(matrix) - 1:
                if pos[1] > 0:
                    nbrs.append((pos[0] + 1, pos[1] - 1))
                if pos[1] < len(matrix[0]) - 1:
                    nbrs.append((pos[0] + 1, pos[1] + 1))
                nbrs.append((pos[0] + 1, pos[1]))
            return nbrs
        

        def dfs(pos):
            if pos in cache:
                return cache[pos]
            
            nbrs = get_nbrs(pos)
            min_sum = []
            for e in nbrs:
                min_sum.append(dfs(e))
            
            if len(min_sum) == 0:
                cache[pos] = matrix[pos[0]][pos[1]]
                return matrix[pos[0]][pos[1]]
            else:
                cache[pos] = matrix[pos[0]][pos[1]] + min(min_sum)
                return matrix[pos[0]][pos[1]] + min(min_sum)
        
        cache = {}
        out = []
        for i in range(len(matrix[0])):
            out.append(dfs((0, i)))
        
        return min(out)

25900