<p>You are given an <code>m x n</code> integer matrix <code>grid</code>, where you can move from a cell to any adjacent cell in all <code>4</code> directions.</p>

<p>Return <em>the number of <strong>strictly</strong> <strong>increasing</strong> paths in the grid such that you can start from <strong>any</strong> cell and end at <strong>any</strong> cell. </em>Since the answer may be very large, return it <strong>modulo</strong> <code>10<sup>9</sup> + 7</code>.</p>

<p>Two paths are considered different if they do not have exactly the same sequence of visited cells.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2022/05/10/griddrawio-4.png" style="width: 181px; height: 121px;" />
<pre>
<strong>Input:</strong> grid = [[1,1],[3,4]]
<strong>Output:</strong> 8
<strong>Explanation:</strong> The strictly increasing paths are:
- Paths with length 1: [1], [1], [3], [4].
- Paths with length 2: [1 -&gt; 3], [1 -&gt; 4], [3 -&gt; 4].
- Paths with length 3: [1 -&gt; 3 -&gt; 4].
The total number of paths is 4 + 3 + 1 = 8.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> grid = [[1],[2]]
<strong>Output:</strong> 3
<strong>Explanation:</strong> The strictly increasing paths are:
- Paths with length 1: [1], [2].
- Paths with length 2: [1 -&gt; 2].
The total number of paths is 2 + 1 = 3.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>m == grid.length</code></li>
	<li><code>n == grid[i].length</code></li>
	<li><code>1 &lt;= m, n &lt;= 1000</code></li>
	<li><code>1 &lt;= m * n &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= grid[i][j] &lt;= 10<sup>5</sup></code></li>
</ul>

class Solution:
    count = 0
    def helper(self, graph, cache, vertex):
        if vertex in cache:
            return cache[vertex]
        if vertex not in graph:
            self.count += 1
            cache[vertex] = 1
            return cache[vertex]
        
        total = 0
        for node in graph[vertex]:
            total += self.helper(graph, cache, node)
        cache[vertex] = total + 1
        self.count += cache[vertex]
        return cache[vertex]

    def countPaths(self, grid: List[List[int]]) -> int:
        graph = collections.defaultdict(list)
        m = len(grid)
        n = len(grid[0])
        moves = [[-1, 0], [1, 0], [0,1], [0,-1]]
        inCount = collections.defaultdict(int)
        for i in range(m):
            for j in range(n):
                vertex = i * n + j

                for move in moves:
                    x, y = i + move[0], j + move[1]
                    if x >= 0 and x < m and y >= 0 and y < n and grid[x][y] > grid[i][j]:
                        v1 = x * n + y
                        graph[vertex].append(v1)
                        inCount[v1] += 1

        cache = {}
        for i in range(m * n):
            if inCount[i] == 0:
                self.helper(graph, cache, i)
        return self.count % 1000000007
        
        
1816