<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 > > memoi;
    Solution() {
        const int N = 205;
        memoi = vector< vector< int > >(N, vector< int >(N, -1));
    }

    int findLIP(vector<vector<int>>& matrix, int x, int y) {
        int n = matrix.size();
        int m = matrix[0].size();
        int xx[] = {-1, 1, 0, 0};
        int yy[] = {0, 0, -1, 1};

        if (memoi[x][y] != -1) {
            return memoi[x][y];
        }

        int maxLIP = 1;

        for (int d = 0; d < 4; d++) {
            int newX = x + xx[d];
            int newY = y + yy[d];

            if (newX >= n || newX < 0 || newY >= m || newY < 0) {
                // out bound
                continue;
            }

            if (matrix[newX][newY] <= matrix[x][y]) {
                continue;
            }

            // e.g 3 -> 4
            maxLIP = max(maxLIP, findLIP(matrix, newX, newY) + 1);
        }

        return memoi[x][y] = maxLIP;
    }

    int longestIncreasingPath(vector<vector<int>>& matrix) {
        // 1. define the state: LIP starts at (x, y)
        // 2. define the output: LIP
        // 3. define transition: (x, y) -> adjection cells
        int n = matrix.size();
        int m = matrix[0].size();

        int ans = 1;
        for (int x = 0; x < n; x++) {
            for (int y = 0; y < m; y++) {
                ans = max(ans, findLIP(matrix, x, y));
            }
        }       

        return ans;
    }
};
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