Number of Increasing Paths in a Grid

class Solution {
    int[][] memoization;
    int[][] moves = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}};
    int modulus = 1_000_000_007;
        
    int depthFirstSearch(int[][] matrix, int x, int y) {
        if (memoization[x][y] != -1)
            return memoization[x][y];
    
        int pathCount = 1;
        for (int[] move : moves) {
            int newX = x + move[0], newY = y + move[1];
            if (newX >= 0 && newX < matrix.length && newY >= 0 && 
                newY < matrix[0].length && matrix[newX][newY] < matrix[x][y]) {
                pathCount = (pathCount + depthFirstSearch(matrix, newX, newY)) % modulus;
            }
        }
    
        memoization[x][y] = pathCount;
        return pathCount;
    }
        
    public int totalPaths(int[][] matrix) {
        int rows = matrix.length, cols = matrix[0].length;
        memoization = new int[rows][cols];
            
        for (int i = 0; i < rows; ++i) {
            for (int j = 0; j < cols; ++j) {
                memoization[i][j] = -1;
            }
        }
    
        int totalPathCount = 0;
        for (int i = 0; i < rows; ++i) {
            for (int j = 0; j < cols; ++j) {
                totalPathCount = (totalPathCount + depthFirstSearch(matrix, i, j)) % modulus;
            }
        }
    
        return totalPathCount;
    }
}

In the provided Java solution, the objective is to calculate the number of increasing paths in a grid using a depth-first search (DFS) approach combined with memoization. Here’s an outline and explanation of how this operation is structurally performed:

• Initialization of Data Structures:

  • memoization: A 2D array to store the number of paths from each cell, initialized to -1 which indicates that the number of paths has not yet been calculated for that cell.
  • moves: An array representing the four possible movements from a cell (right, down, left, up).

• Depth-First Search (DFS) Function:

  • The function depthFirstSearch checks if a cell's path count is already computed by seeing if memoization[x][y] is not -1. If not cached, it computes the paths by exploring all valid adjacent cells that have a greater value than the current cell, recursively calculating the number of paths from these cells.
  • Each cell starts with a path count of 1 (the cell itself).
  • The recursive nature, combined with memoization, ensures each cell's paths are computed only once, improving efficiency.

• Calculation of Total Paths:

  • The totalPaths function initializes the memoization matrix and iterates through each cell of the grid.
  • It invokes depthFirstSearch for each cell to compute the total number of paths starting from that cell and aggregates these into totalPathCount.
  • Modular arithmetic is used (modulus 1,000,000,007) to prevent overflow and manage large numbers which is common in combinatorial problems.

• Modular Arithmetic:

  • The use of modulus (1_000_000_007), a large prime number, is a common technique to keep numbers within integer bounds and is particularly useful in interview settings and competitive programming.

By initializing and strategically caching results, this approach efficiently handles the computation of increasing paths in potentially large matrices. The granularity and strategic checks prevent unnecessary recalculations, deeming this solution optimal for performance-sensitive environments.

class Solution:
    def calculatePaths(self, matrix: List[List[int]]) -> int:
        rows, cols = len(matrix), len(matrix[0])
        modulus = 1000000007
        movements = [[0, 1], [1, 0], [0, -1], [-1, 0]]
            
        memory = [[-1] * cols for _ in range(rows)]
            
        def explore(x, y):
            if memory[x][y] != -1:
                return memory[x][y]
                
            path_count = 1
            for dx, dy in movements:
                nx, ny = x + dx, y + dy
                if 0 <= nx < rows and 0 <= ny < cols and matrix[nx][ny] < matrix[x][y]:
                    path_count += explore(nx, ny) % modulus
                
            memory[x][y] = path_count
            return path_count
            
        total_paths = sum(explore(r, c) for r in range(rows) for c in range(cols)) % modulus
        return total_paths

In the provided Python solution, you calculate the number of increasing paths within a grid using depth-first search (DFS) and dynamic programming. The solution employs a memorization technique to store previously computed values, optimizing the performance by avoiding redundant calculations.

• Define the number of rows and columns from the input matrix.
• Initialize a modulus to handle potential integer overflow issues by taking results modulo 10^9 + 7.
• Assign the possible movements in the grid (right, down, left, up).
• Create a 2D list memory initialized with -1, used to store the number of paths from any cell (x, y).

For each cell in the grid, the explore function is used to calculate the paths recursively:

• If the value in memory for the current cell is not -1, return the memorized result to avoid recalculating.
• Initialize path_count to 1, representing the path of standing still.
• Traverse through all possible movements, and for each valid movement (staying within bounds and ensuring that the path is increasing), recursively calculate the number of paths from the next cell.
• The count of paths from each direction is added to path_count.
• Store path_count in memory[x][y] and return this value.

Finally, compute the total number of paths across all cells in the grid by iterating through each cell, invoking explore, and summing up results. Apply modulo operation to ensure the result is within the required bounds before returning it.