<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 collections import deque

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

        def get_neighbors(matrix, node):
            m, n = len(matrix), len(matrix[0])
            i, j = node
            neighbors = []
            if (i+1 <  m) and (matrix[i+1][j] > matrix[i][j]):
                neighbors.append((i+1, j))

            if (i-1 >= 0) and (matrix[i-1][j] > matrix[i][j]):
                neighbors.append((i-1, j))

            if (j+1 <  n) and (matrix[i][j+1] > matrix[i][j]):
                neighbors.append((i, j+1))

            if (j-1 >= 0) and (matrix[i][j-1] > matrix[i][j]):
                neighbors.append((i, j-1))
    
            return neighbors


        def dfs(source, finishing_time, visited):
            
            visited.add(source)

            for neighbor in get_neighbors(matrix, source):
                if neighbor not in visited:
                    dfs(neighbor, finishing_time, visited)

            finishing_time.append(source)


        def find_topological_order(matrix):
            m, n = len(matrix), len(matrix[0])
            visited = set()
            finishing_time = []
            
            for i in range(m):
                for j in range(n):
                    if (i,j) not in visited:
                        dfs((i,j), finishing_time, visited)

            return list(reversed(finishing_time))



        # Find The Longest Path in DAG
        topological_order = find_topological_order(matrix)

        m, n = len(matrix), len(matrix[0])
        longest_path = {}
        for i in range(m):
            for j in range(n):
                longest_path[(i,j)] = 1

        for u in topological_order:
            for v in get_neighbors(matrix, u):
                if longest_path[v] < 1 + longest_path[u]:
                    longest_path[v] = 1 + longest_path[u]
        
        return max(longest_path.values())











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