Number of Increasing Paths in a Grid

Problem Description

You are given an m x n integer matrix grid, where you can move from a cell to any adjacent cell in all 4 directions. Return the number of strictly increasing paths in the grid such that you can start from any cell and end at any cell. Since the answer may be very large, return it modulo 10^9 + 7. Two paths are considered different if they do not have exactly the same sequence of visited cells.

Key Insights

• The problem requires counting paths that are strictly increasing in a grid.
• Paths can start and end at any cell, and can move in all four directions (up, down, left, right).
• A dynamic programming approach can be utilized to count paths by leveraging memoization to avoid recalculating paths from previously visited cells.
• The problem can also be viewed as a graph where cells represent nodes and edges exist between adjacent cells that satisfy the increasing condition.

Space and Time Complexity

Time Complexity: O(m * n)
Space Complexity: O(m * n)

Solution

To solve this problem, we can use Depth-First Search (DFS) combined with memoization. We traverse each cell in the grid and explore all possible paths from that cell, storing the results of already computed paths in a memoization table to avoid redundant calculations. The algorithm will:

1. Initialize a memoization table of the same size as the grid to store the number of increasing paths from each cell.
2. For each cell, use DFS to explore all adjacent cells that have a greater value.
3. Count the paths recursively and update the memoization table.
4. Return the sum of all paths starting from each cell.

Code Solutions