<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:
        cache = {}
        ROW, COL = len(matrix), len(matrix[0])
        
        def dfs(r,c, prev):
            if r < 0 or r >= ROW or c < 0 or c >= COL:
                return 0
            
            if matrix[r][c] <= prev:
                return 0
            
            if (r,c) in cache:
                return cache[(r,c)]
            
            cache[(r,c)] =  1 + max(dfs(r+1, c, matrix[r][c]), dfs(r-1, c, matrix[r][c]), dfs(r, c+1, matrix[r][c]), dfs(r, c-1, matrix[r][c]))
            return cache[(r,c)]
        
        max_lip = 0
        origin = (-1,-1)
        for r in range(ROW):
            for c in range(COL):
                if (r,c) not in cache:
                    dfs(r,c,-1)
                
                if cache[(r,c)] > max_lip:
                    max_lip = cache[(r,c)]
                    origin = (r,c)

        def findPath(r, c, count, path):
            if len(path) == max_lip:
                return path
            
            if r < 0 or r >= ROW or c < 0 or c >= COL or cache[(r,c)] != count:
                return []
            
            path.append(matrix[r][c])

            up = findPath(r-1, c, count-1, path)
            down = findPath(r+1, c, count-1, path)
            left = findPath(r, c-1, count-1, path)
            right = findPath(r, c+1, count-1, path)

            if up: return up
            if down: return down
            if left: return left
            if right: return right

        print(findPath(origin[0], origin[1], max_lip, []))            
        return max_lip


        # follow up: print the path
23665