<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 {
public:
    vector<vector<int>> dp;
    vector<vector<int>> matrix;
    int n, m;

    bool isValid(int i, int j) {
        return i >= 0 && j >= 0 && i < n && j < m;
    }

    int dfs(int i, int j) {
        if (dp[i][j] != 0) return dp[i][j]; // Return cached result

        vector<int> dx = {1, -1, 0, 0};
        vector<int> dy = {0, 0, -1, 1};
        int maxPath = 1; // The node itself counts as a path

        for (int k = 0; k < 4; ++k) {
            int r = i + dx[k];
            int c = j + dy[k];
            if (isValid(r, c) && matrix[r][c] > matrix[i][j]) {
                maxPath = max(maxPath, 1 + dfs(r, c)); // Recursively calculate the longest path
            }
        }

        dp[i][j] = maxPath; // Memoize the result
        return maxPath;
    }

    int longestIncreasingPath(vector<vector<int>>& matrix) {
        if (matrix.empty()) return 0;

        this->n = matrix.size();
        this->m = matrix[0].size();
        this->matrix = matrix;
        dp = vector<vector<int>>(n, vector<int>(m, 0)); // Initialize memoization array with 0

        int ans = 0;
        for (int i = 0; i < n; ++i) {
            for (int j = 0; j < m; ++j) {
                ans = max(ans, dfs(i, j)); // Find the longest path starting from each cell
            }
        }

        return ans;
    }
};

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