Longest Increasing Path In A Matrix Leetcode Problem 329 [Python]

If you're a coding enthusiast or a beginner looking to tackle challenging programming problems, you've come to the right place.

In this blog post, we're going to dive into the Longest Increasing Path In A Matrix problem, which is featured on LeetCode as Problem 329. We'll explore the problem, its constraints, a brute-force approach, and an efficient Python solution.

By the end of this guide, you'll have a solid understanding of how to approach this problem.

Problem Overview

The Challenge

You are given an m x n integer matrix, and your task is to return the length of the longest increasing path in the matrix.

An "increasing path" means that you can move in four directions: left, right, up, or down.

However, moving diagonally or outside the matrix boundaries is not allowed.

Example:

Let's take a look at a sample scenario to better understand the problem:

Input:

matrix = [[9,9,4],
[6,6,8],
[2,1,1]]


Output: 4

Explanation:
The longest increasing path in the given matrix is [1, 2, 6, 9].

Understanding the Constraints

Before we dive into the solutions, it's essential to understand the constraints of the problem:

• m is the number of rows in the matrix.
• n is the number of columns in the matrix.
• The dimensions of the matrix must satisfy: 1 ≤ m, n ≤ 200.
• The matrix elements are integers ranging from 0 to 2^31 – 1. Now, let's explore the brute-force and efficient approaches to solving this problem.

Example 1:

Let's take a simple example to understand how the algorithm works.

Consider the following matrix:

Input:

matrix = [[9,9,4],
[6,6,8],
[2,1,1]]


Output: 4

Explanation:
The longest increasing path in the given matrix is [1, 2, 6, 9].

Problem Overview

The Longest Increasing Path In A Matrix problem is a classic example of a dynamic programming problem.

Given an m x n matrix, we need to find the length of the longest increasing path within the matrix while adhering to certain constraints.

To solve this problem efficiently, we'll use a depth-first search (DFS) algorithm and dynamic programming to store and reuse computed results.

Constraints

Before we dive into the solution, let's clarify the constraints:

• The matrix is represented as an m x n grid, where m is the number of rows and n is the number of columns.
• The matrix contains integers, and each integer can range from 0 to 2^31 – 1.
• You can move in four directions: up, down, left, or right, but not diagonally.
• The goal is to find the length of the longest increasing path in the matrix.

Example 1

Let's start with an example to better understand the problem:

Input:

matrix = [[9,9,4],
[6,6,8],
[2,1,1]]


Output: 4

Explanation: The longest increasing path is [1, 2, 6, 9].

Now that we have a clear understanding of the problem, let's explore the steps to solve it efficiently.

Efficient Python Code Solution

To efficiently solve the Longest Increasing Path In A Matrix problem, we'll use a depth-first search (DFS) algorithm and dynamic programming.

The main idea is to traverse the matrix using DFS and store the results of each computed path in a cache to avoid redundant calculations.

Let's dive into the code solution:

def longestIncreasingPath(matrix):
    # Dimensions of the matrix
    ROWS, COLS = len(matrix), len(matrix[0])

    # Cache to store computed results (row, col) -> Longest Increasing Path (LIP)
    dp = {}

    def dfs(r, c, prevVal):
        # Check if we are out of bounds or if the path is not increasing
        if r < 0 or r == ROWS or c < 0 or c == COLS or matrix[r][c] <= prevVal:
            return 0

        # Check if we have already computed the LIP for this position
        if (r, c) in dp:
            return dp[(r, c)]

        # Initialize the result with 1 (the minimum path length)
        res = 1

        # Explore all four possible directions
        res = max(res, 1 + dfs(r + 1, c, matrix[r][c]))
        res = max(res, 1 + dfs(r - 1, c, matrix[r][c]))
        res = max(res, 1 + dfs(r, c + 1, matrix[r][c]))
        res = max(res, 1 + dfs(r, c - 1, matrix[r][c]))

        # Store the result in the cache
        dp[(r, c)] = res

        return res

    # Iterate through all positions in the matrix and run DFS
    for r in range(ROWS):
        for c in range(COLS):
            dfs(r, c, -1)

    # Find the maximum LIP from all positions and return it
    return max(dp.values())

In this Python solution, we first define the dimensions of the matrix and initialize a cache (dp) to store the results of computed paths.

We then implement a depth-first search (DFS) function (dfs) to explore the matrix while adhering to the constraints.

The DFS function takes three arguments: the current row (r), the current column (c), and the previous value (prevVal) in the path.

It checks if the path is out of bounds or not increasing, and if so, it returns 0. We also check if we've already computed the LIP for this position using the cache (dp).

If the position is valid and we haven't computed the LIP for it, we start by initializing the result (res) to 1, as any position can have a minimum path length of 1. We then explore all four possible directions (up, down, left, and right), considering the maximum path length.

After computing the result for the current position, we store it in the cache (dp).

Finally, we iterate through all positions in the matrix, running the DFS for each of them and passing -1 as the initial prevVal.

Once we've computed the LIP for all positions, we find the maximum LIP from the cache and return it as the final result.

Time and Space Complexity

The time complexity of this solution is O(m * n), where m is the number of rows and n is the number of columns in the matrix.

This is because we perform a depth-first search (DFS) starting from each position in the matrix.

The DFS complexity is O(m * n), and we have to perform it for each position.

The space complexity is also O(m * n) due to the dynamic programming cache (dp) used to store the results of computed paths.

Reasoning Behind Our Approach

The key to solving the Longest Increasing Path In A Matrix problem efficiently lies in understanding the power of dynamic programming and using depth-first search (DFS) to traverse the matrix while storing and reusing computed results.

Dynamic Programming (DP)

Dynamic programming is a powerful technique that involves breaking down a complex problem into simpler subproblems and storing their solutions to avoid redundant calculations.

In this problem, we use DP to store the results of the longest increasing paths (LIP) for each position in the matrix.

This enables us to compute the LIP for any position once and reuse it when needed, significantly improving the algorithm's efficiency.

Depth-First Search (DFS)

DFS is a graph traversal algorithm that explores as far as possible along each branch before backtracking.

It's a natural choice for this problem, as we need to navigate through the matrix while considering all possible paths.

DFS helps us determine the longest increasing path for a given position by exploring the matrix in a depth-first manner.

Caching Results

By caching the results of the LIP for each position in the matrix, we avoid redundant calculations.

This ensures that the time complexity remains efficient, even for large matrices.

When we revisit a position during DFS, we can simply look up the LIP in the cache, saving valuable time.

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Conclusion

In this blog post, we've explored the Longest Increasing Path In A Matrix problem featured on LeetCode.

We've covered the problem's constraints, provided an efficient Python solution using dynamic programming and depth-first search (DFS), and explained the reasoning behind our approach.

This problem is a great example of how dynamic programming and graph traversal algorithms can be combined to solve challenging programming tasks.

By understanding the problem's constraints and applying the concepts discussed here, you can approach similar problems with confidence.

If you found this guide helpful, please consider sharing it with your fellow coders and leaving a comment with any questions or suggestions.

Coding challenges are a fantastic way to enhance your problem-solving skills, and understanding the principles behind efficient solutions is key to becoming a proficient programmer.

Now, it's your turn to tackle the Longest Increasing Path In A Matrix problem on LeetCode and apply the concepts you've learned in this guide.

Happy coding!

Question Link: Longest Increasing Path In a Matrix – LeetCode