<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>

from typing import List

class Solution:
    def __init__(self):
        self.dirs = ((0, 1), (0, -1), (1, 0), (-1, 0))
        self.n = 0
        self.m = 0
        self.cache = []

    def longestIncreasingPath(self, matrix: List[List[int]]) -> int:
        if not matrix or not matrix[0]:
            return 0

        self.n, self.m = len(matrix), len(matrix[0])
        self.cache = [[0] * self.m for _ in range(self.n)]

        ans = 0

        for i in range(self.n):
            for j in range(self.m):
                ans = max(ans, self.dfs(matrix, i, j))
        return ans

    def dfs(self, matrix: List[List[int]], i: int, j: int) -> int:
        if self.cache[i][j] != 0:
            return self.cache[i][j]
        
        max_length = 1
        for dir in self.dirs:
            x, y = dir[0] + i, dir[1] + j
            if 0 <= x < self.n and 0 <= y < self.m and matrix[x][y] > matrix[i][j]:
                max_length = max(max_length, 1 + self.dfs(matrix, x, y))
        
        self.cache[i][j] = max_length
        return self.cache[i][j]

18725