<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:
        m, n = len(matrix), len(matrix[0])

        adj = defaultdict(list)
        ins = [0] * (m * n)

        dir = [[0, -1], [0, 1], [1, 0], [-1, 0]]

        for i in range(m):
            for j in range(n):
                value = matrix[i][j]
                currIdx = i * n + j

                for d in dir:
                    x, y = i + d[0], j + d[1]

                    if self.isValid(x, y, m, n) and value < matrix[x][y]:
                        newIdx = x * n + y

                        adj[currIdx].append(newIdx)
                        ins[newIdx] += 1
                    
        q = deque()
        for i in range(m*n):
            if ins[i] == 0:
                q.append((i, 1))
        
        ans = 0

        while q:
            node, dst = q.popleft()
            ans = max(ans, dst)

            for u in adj[node]:
                ins[u] -= 1

                if ins[u] == 0:
                    q.append((u, dst + 1))
            
        return ans

    
    def isValid(self, x, y, m, n):
        return 0 <= x < m and 0 <= y < n
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