> ## Documentation Index
> Fetch the complete documentation index at: https://leetcode-py.wisl.dev/llms.txt
> Use this file to discover all available pages before exploring further.

> ## Agent Instructions
> leetcode-py is a Python LeetCode practice environment generator with one CLI: lcpy. It is not a service or platform.
> Each problem is a directory under leetcode/ with README.md, solution.py, test_solution.py, helpers.py, and playground.ipynb. lcpy gen creates them from JSON templates bundled with the package.
> Examples are backed by tests; copy them verbatim.

# Longest Increasing Path in a Matrix

> Tested Python solution for LeetCode 329 with 14 pytest cases. Generate a practice environment with lcpy.

LeetCode 329, Hard. Topics: Array, Dynamic Programming, Depth-First Search, Breadth-First Search, Graph Theory, Topological Sort, Memoization, Matrix. [View on LeetCode](https://leetcode.com/problems/longest-increasing-path-in-a-matrix/description/).

Generate this problem as a practice environment: tested reference solution, 14 [parametrized pytest cases](/practice/testing), and a playground notebook:

```bash theme={"theme":{"light":"github-light","dark":"github-dark"}}
lcpy gen -n 329   # by problem number
lcpy gen -s longest_increasing_path_in_a_matrix   # by problem name
```

## Problem

Given an `m x n` integers `matrix`, return *the length of the longest increasing path in* `matrix`.

From each cell, you can either move in four directions: left, right, up, or down. You **may not** move **diagonally** or move **outside the boundary** (i.e., wrap-around is not allowed).

### Examples

![Example 1](https://assets.leetcode.com/uploads/2021/01/05/grid1.jpg)

```
Input: matrix = [[9,9,4],[6,6,8],[2,1,1]]
Output: 4
```

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

![Example 2](https://assets.leetcode.com/uploads/2021/01/27/tmp-grid.jpg)

```
Input: matrix = [[3,4,5],[3,2,6],[2,2,1]]
Output: 4
```

**Explanation:** The longest increasing path is `[3, 4, 5, 6]`. Moving diagonally is not allowed.

```
Input: matrix = [[1]]
Output: 1
```

### Constraints

* m == matrix.length
* n == matrix\[i].length
* 1 \<= m, n \<= 200
* 0 \<= matrix\[i]\[j] \<= 2^31 - 1

## Solution

Reference implementation from [solution.py on GitHub](https://github.com/wislertt/leetcode-py/blob/main/leetcode/longest_increasing_path_in_a_matrix/solution.py), full suite in [test\_solution.py](https://github.com/wislertt/leetcode-py/blob/main/leetcode/longest_increasing_path_in_a_matrix/test_solution.py):

```python theme={"theme":{"light":"github-light","dark":"github-dark"}}
from functools import cache


class Solution:
    # Time: O(m * n)
    # Space: O(m * n)
    def longest_increasing_path(self, matrix: list[list[int]]) -> int:
        if not matrix or not matrix[0]:
            return 0

        rows, cols = len(matrix), len(matrix[0])
        directions = [(0, 1), (1, 0), (0, -1), (-1, 0)]

        @cache
        def dfs(r: int, c: int) -> int:
            max_length = 1
            for dr, dc in directions:
                nr, nc = r + dr, c + dc
                if 0 <= nr < rows and 0 <= nc < cols and matrix[nr][nc] > matrix[r][c]:
                    max_length = max(max_length, 1 + dfs(nr, nc))
            return max_length

        result = 0
        for i in range(rows):
            for j in range(cols):
                result = max(result, dfs(i, j))

        return result
```

## Complexity

| Time      | Space     |
| --------- | --------- |
| O(m \* n) | O(m \* n) |

## Tags

[Grind](/catalog/grind), [NeetCode 150](/catalog/neetcode-150), [NeetCode 250](/catalog/neetcode-250), [NeetCode All](/catalog/neetcode), [AlgoMaster 75](/catalog/algo-master-75).