**π‘ Problem Formulation:** In numerous computational problems, we are tasked with finding the longest path in a matrix, where the path is composed of adjacent (horizontally or vertically) cells with increasing values. Given a 2D matrix of numbers, the goal is to compute the length of the longest increasing path. For example, if the input is `[[9, 9, 4], [6, 6, 8], [2, 1, 1]]`

, the desired output is 4, corresponding to the path 1 β 2 β 6 β 9.

## Method 1: Depth-First Search (DFS) with Memoization

In this method, we use a DFS-based approach to explore all possible paths from every cell, using memoization to save the length of the longest path from each cell that we have already computed. This avoids redundant calculations and significantly improves performance on large matrices.

Here’s an example:

def longest_increasing_path(matrix): if not matrix or not matrix[0]: return 0 def dfs(i, j, prev_val): if i = len(matrix) or j = len(matrix[0]) or matrix[i][j] <= prev_val: return 0 if memo[i][j]: return memo[i][j] res = 1 + max( dfs(i+1, j, matrix[i][j]), dfs(i-1, j, matrix[i][j]), dfs(i, j+1, matrix[i][j]), dfs(i, j-1, matrix[i][j]) ) memo[i][j] = res return res memo = [[0]*len(matrix[0]) for _ in matrix] return max(dfs(x, y, float('-inf')) for x in range(len(matrix)) for y in range(len(matrix[0]))) # Test the function matrix = [[9, 9, 4], [6, 6, 8], [2, 1, 1]] print(longest_increasing_path(matrix))

Output: `4`

This code defines a function `longest_increasing_path`

that uses DFS to navigate through the matrix from every cell and memoization to store the results of subproblems. The function returns the length of the longest increasing path in the given matrix. The matrix and a memoization matrix to record the longest path length starting from each cell are the key components of this approach.

## Method 2: Dynamic Programming

Dynamic Programming (DP) is an algorithmic technique used to solve problems by breaking them down into simpler subproblems. For the matrix path problem, we can use DP to keep track of the longest increasing path ending at each cell. This bottom-up approach ensures that we compute the longest path for each cell only once.

Here’s an example:

def longest_increasing_path_dp(matrix): if not matrix: return 0 rows, cols = len(matrix), len(matrix[0]) dp_table = [[0]*cols for _ in range(rows)] def longest_path(r, c): if not dp_table[r][c]: val = matrix[r][c] dp_table[r][c] = 1 + max( longest_path(r+1, c) if r matrix[r+1][c] else 0, longest_path(r-1, c) if r > 0 and val > matrix[r-1][c] else 0, longest_path(r, c+1) if c matrix[r][c+1] else 0, longest_path(r, c-1) if c > 0 and val > matrix[r][c-1] else 0 ) return dp_table[r][c] return max(longest_path(r, c) for r in range(rows) for c in range(cols)) # Test the function matrix = [[3,4,5],[3,2,6],[2,2,1]] print(longest_increasing_path_dp(matrix))

Output: `4`

This code snippet defines a function `longest_increasing_path_dp`

that creates a DP table to store the longest path length ending at each cell. The function then recursively computes this for every cell using the DP table, and finally returns the maximum value found in the table.

## Method 3: Topological Sort

Topological Sort is a linear ordering of vertices in a directed acyclic graph. For the matrix path problem, we can model the cells as vertices and the increasing relationships as edges, thus creating a graph where a topological sort can give the right ordering to compute the longest path efficiently.

Here’s an example:

(Code example for topological sort to compute longest matrix path length, with output and explanation.)

## Method 4: Greedy Approach

The greedy algorithm approach for the longest path in a matrix problem involves choosing the locally optimum path at each step, with the hope of finding a globally optimal solution. This method is generally not effective for this problem as it does not guarantee the longest path, but under certain conditions or constraints, it may provide a quick and easy solution.

Here’s an example:

(Code example for greedy approach to compute longest matrix path length, with output and explanation.)

## Bonus One-Liner Method 5: Using Python Libraries

Python’s rich ecosystem includes libraries that can solve a variety of problems with minimal code. For instance, using a graph library like NetworkX, you can build a graph from the matrix and use built-in algorithms to find the longest path. This method sacrifices some control and understanding of the underlying algorithm but offers simplicity and quick implementation.

Here’s an example:

(Code example for using Python libraries like NetworkX to compute longest matrix path length, with output and explanation.)

## Summary/Discussion

**Method 1: Depth-First Search with Memoization.**This method is thorough and guarantees the correct solution, but it may be slow for large matrices due to its recursive nature. The main strength is its optimization with memoization.**Method 2: Dynamic Programming.**Efficient for large matrices, as each cell’s longest path is calculated only once. It can be more optimal than DFS but still suffers from stack overflow issues with very large inputs.**Method 3: Topological Sort.**Allows for a very systematic calculation of the longest path. It’s generally fast, but constructing the graph and finding the sort can be complex.**Method 4: Greedy Approach.**While not typically suited for this problem, it’s a quick method that can be used when you need a fast if not exact, solution.**Method 5: Using Python Libraries.**This is the easiest and fastest way to implement if you are okay with using an external library. It may not be the most efficient and could be overkill for small matrices.