Number of Increasing Paths in a Grid - Solution & Explanation

This approach uses Depth First Search (DFS) combined with memoization to count the number of increasing paths starting from each cell. The main idea is to recursively explore each cell's neighbors, moving to the next cell only if its value is higher, and storing intermediate results to avoid redundant calculations.

The Python solution implements DFS with memoization to explore each cell's increasing paths. A 2D list 'dp' stores the number of paths starting from each cell and is initialized to -1. The function 'dfs' calculates paths from the starting cell using recursion and updating the 'dp' table to avoid redundant calculations. Results are computed modulo 10^9 + 7 to handle large path counts.

This approach involves sorting the cells of the grid based on their values and then using dynamic programming to calculate the number of increasing paths from each cell. By processing cells in increasing order, we ensure that when calculating the number of paths for a cell, all smaller cells have been processed.

The C++ solution performs topological sorting based on grid values. It initially sets up each cell with a path count of 1. Cells are processed in value order using a vector of pairs, sorted by their value. For each cell, we explore adjacent cells that contain larger values, updating path counts as we go, ensuring earlier cells' paths are calculated before later ones, thus respecting dependencies.