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329. Longest Increasing Path in a Matrix

hard

Return the length of the longest path of strictly increasing values in a matrix, where moves are to 4-directional neighbors. The strict-increase rule makes the path graph acyclic — that's what unlocks memoized DFS over the grid.

By Alex Chen, Founder, InterviewChamp.AI · Last verified

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).

Constraints

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

Examples

Example 1

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

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.

Example 3

Input
matrix = [[1]]
Output
1

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Output

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Hints

Progressive — try the first before opening the next.

Hint 1

Strict increase means edges only go from smaller to larger — no cycles. Every cell has a well-defined longest path starting from it.

Hint 2

Memoize dfs(i, j) = 1 + max over strictly-larger neighbors of dfs(neighbor).

Hint 3

Iterate every cell, return the global max. Each cell is computed once.

Hint 4

Alternative: topological sort by cell value, then relax distances on the DAG — same big-O.

Solution approach

Reveal approach

Memoized DFS. dp[i][j] caches the length of the longest strictly-increasing path starting at (i, j). For each cell, recursively explore its four neighbors; recurse only into neighbors whose value is strictly greater. dp[i][j] = 1 + max over valid neighbors of dp[neighbor]. Iterate every starting cell and return the global max. Because strict increase forbids cycles, each cell is visited and memoized exactly once. An alternative is topological-order DP: sort all cells by value ascending, then relax each cell's two smaller-or-equal neighbors' dp into the cell. Both run in O(m * n) time and O(m * n) space.

Complexity

Time
O(m * n)
Space
O(m * n)

Related patterns

  • dynamic-programming
  • memoization
  • grid-dp
  • dfs

Related problems

Asked at

Companies reported asking this problem (sourced from public Glassdoor, Blind, and Levels.fyi interview posts).

  • Amazon
  • Google
  • Apple

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