<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>

class Solution:
    def longestIncreasingPath(self, matrix: List[List[int]]) -> int:
        
        
        # Create graph
        m,n = len(matrix), len(matrix[0])
        graph = {}
        
        for i in range(m):
            for j in range(n):
                source = (i,j)
                graph[source] = [] 
                if i > 0:
                    if matrix[i][j] < matrix[i-1][j]:
                        graph[source].append((i-1,j))
                if i < m - 1:
                    if matrix[i][j] < matrix[i+1][j]:
                        graph[source].append((i+1,j))
                if j > 0:
                    if matrix[i][j] < matrix[i][j-1]:
                        graph[source].append((i,j-1))
                if j < n - 1:
                    if matrix[i][j] < matrix[i][j+1]:
                        graph[source].append((i,j+1))
        
                
        def dfs(start):
            nonlocal graph, cache

            if start in cache:
                return cache[start]
            
            max_length = 1
            
            for neighbor in graph[start]:
                max_length = max(max_length, 1 + dfs(neighbor))
            
            cache[start] = max_length
            
            return max_length
        
        # Apply dfs on each square, caching results along the way
        cache = {}
        longest_path = 0
        for start in graph.keys():
            longest_path = max(longest_path, dfs(start))           

        return longest_path
                
                
        
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