<p>Given an <code>m x n</code> integers <code>matrix</code>, return <em>the length of the longest increasing path in </em><code>matrix</code>.</p>

<p>From each cell, you can either move in four directions: left, right, up, or down. You <strong>may not</strong> move <strong>diagonally</strong> or move <strong>outside the boundary</strong> (i.e., wrap-around is not allowed).</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/01/05/grid1.jpg" style="width: 242px; height: 242px;" />
<pre>
<strong>Input:</strong> matrix = [[9,9,4],[6,6,8],[2,1,1]]
<strong>Output:</strong> 4
<strong>Explanation:</strong> The longest increasing path is <code>[1, 2, 6, 9]</code>.
</pre>

<p><strong class="example">Example 2:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2021/01/27/tmp-grid.jpg" style="width: 253px; height: 253px;" />
<pre>
<strong>Input:</strong> matrix = [[3,4,5],[3,2,6],[2,2,1]]
<strong>Output:</strong> 4
<strong>Explanation: </strong>The longest increasing path is <code>[3, 4, 5, 6]</code>. Moving diagonally is not allowed.
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> matrix = [[1]]
<strong>Output:</strong> 1
</pre>

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

<ul>
	<li><code>m == matrix.length</code></li>
	<li><code>n == matrix[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 200</code></li>
	<li><code>0 &lt;= matrix[i][j] &lt;= 2<sup>31</sup> - 1</code></li>
</ul>

'''
At any (r, c), we have 4 (technically 3) allowed moves:
Go right: c + 1.
Go left: c - 1
Go up: r + 1.
Go down: r - 1.

Before each move, verify the following:
Next move's value HAS TO BE greater than current (also ensures we don't go back)
Next move within bounds
'''

class Solution:
    def longestIncreasingPath(self, matrix: List[List[int]]) -> int:

        @cache
        def longestIncreasingPathStartingFrom(r: int, c: int) -> int:
            left = 0
            right = 0
            up = 0
            down = 0

            if c > 0 and matrix[r][c - 1] > matrix[r][c]:
                left = longestIncreasingPathStartingFrom(r, c - 1)

            if c < len(matrix[0]) - 1 and matrix[r][c + 1] > matrix[r][c]:
                right = longestIncreasingPathStartingFrom(r, c + 1)

            if r > 0 and matrix[r - 1][c] > matrix[r][c]:
                up = longestIncreasingPathStartingFrom(r - 1, c)

            if r < len(matrix) - 1 and matrix[r + 1][c] > matrix[r][c]:
                down = longestIncreasingPathStartingFrom(r + 1, c)


            return 1 + max(left, right, up, down)

        curMax = 1

        for r in range(len(matrix)):
            for c in range(len(matrix[0])):
                curMax = max(curMax, longestIncreasingPathStartingFrom(r, c))

        return curMax
        
26005