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Longest Special Path - Solution & Explanation

HardArrayHash TableTreeDepth-First Search4 min read
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Problem Statement

You are given an undirected tree rooted at node 0 with n nodes numbered from 0 to n - 1, represented by a 2D array edges of length n - 1, where edges[i] = [ui, vi, lengthi] indicates an edge between nodes ui and vi with length lengthi. You are also given an integer array nums, where nums[i] represents the value at node i.

A special path is defined as a downward path from an ancestor node to a descendant node such that all the values of the nodes in that path are unique.

Note that a path may start and end at the same node.

Return an array result of size 2, where result[0] is the length of the longest special path, and result[1] is the minimum number of nodes in all possible longest special paths.

 

Example 1:

Input: edges = [[0,1,2],[1,2,3],[1,3,5],[1,4,4],[2,5,6]], nums = [2,1,2,1,3,1]

Output: [6,2]

Explanation:

In the image below, nodes are colored by their corresponding values in nums

The longest special paths are 2 -> 5 and 0 -> 1 -> 4, both having a length of 6. The minimum number of nodes across all longest special paths is 2.

Example 2:

Input: edges = [[1,0,8]], nums = [2,2]

Output: [0,1]

Explanation:

The longest special paths are 0 and 1, both having a length of 0. The minimum number of nodes across all longest special paths is 1.

 

Constraints:

  • 2 <= n <= 5 * 104
  • edges.length == n - 1
  • edges[i].length == 3
  • 0 <= ui, vi < n
  • 1 <= lengthi <= 103
  • nums.length == n
  • 0 <= nums[i] <= 5 * 104
  • The input is generated such that edges represents a valid tree.

Approach Overview

Problem Overview: You are given a tree where each node has a value. A path is considered special if no value appears more than once along that path. The task is to compute the maximum length path that satisfies this constraint.

Approach 1: Brute Force DFS from Every Node (O(n^2) time, O(n) space)

Start a DFS from every node and explore all possible downward paths while tracking the values seen so far using a hash set. If the next node introduces a duplicate value, stop exploring that branch. During traversal, track the maximum path length encountered. This approach works because a tree has no cycles, but it redundantly explores many overlapping paths, leading to quadratic complexity in the worst case.

Approach 2: DFS with Hash Map + Prefix Distance (O(n) time, O(n) space)

The optimal approach performs a single DFS from the root while maintaining a sliding-window style constraint on the current path. Use a hash map to store the most recent depth where each value appeared. Also maintain prefix distances so you can compute the length of any valid segment quickly. When a duplicate value appears, shift the left boundary of the valid path to the position after the previous occurrence. This technique mirrors the classic "longest substring without repeating characters" pattern but applied along a tree traversal using Depth-First Search. Hash lookups ensure duplicates are detected in O(1).

During DFS, update the current window boundary based on the last occurrence of the node value. The prefix distance array allows constant-time computation of the path length between two depths. Backtracking restores the previous state of the hash map so other branches are evaluated independently. Combining Hash Table tracking with Tree traversal ensures each node is processed once.

Recommended for interviews: The brute force DFS demonstrates understanding of tree traversal and path validation but is rarely acceptable for large inputs. Interviewers expect the optimized DFS with a hash map and prefix distance tracking because it reduces repeated work and runs in linear time. Recognizing the similarity to the "longest unique subarray" sliding window pattern applied to a DFS path is the key insight.

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Detailed Complexity Analysis

ApproachTimeSpaceWhen to Use
Brute Force DFS from Every NodeO(n^2)O(n)Useful for understanding the problem or when constraints are very small
DFS with Hash Map + Prefix DistanceO(n)O(n)Optimal solution for large trees where duplicate values must be tracked efficiently

Video Solution

3425. Longest Special Path | Sliding Window on Trees | Crash Course | Backtracking | Sliding Window • Aryan Mittal • 4,930 views views

Watch 4 more video solutions →

Frequently Asked Questions

Is Longest Special Path easy or hard?
Longest Special Path is categorized as a Hard problem because it combines multiple concepts: tree traversal, hash map tracking, prefix calculations, and sliding window logic applied to DFS paths. Recognizing this pattern and managing state during backtracking makes the implementation challenging.
Longest Special Path Python/Java solution
The solution in Python or Java typically builds an adjacency list for the tree and runs a recursive DFS. A hash map stores the last depth of each value and prefix distances track cumulative path length. The same algorithm works across Python, Java, C++, and Go with O(n) complexity.
How to solve Longest Special Path in O(n)?
Perform a DFS while tracking the current path depth and the most recent depth of each value in a hash map. Maintain a sliding window boundary representing the earliest valid node in the path. Using prefix distance arrays allows quick computation of path length, and backtracking restores state for sibling branches.
What is the best approach for Longest Special Path?
The most efficient approach uses Depth-First Search with a hash map that stores the last occurrence depth of each value along the current path. By maintaining prefix distances and adjusting the left boundary when duplicates appear, the algorithm processes each node once. This results in O(n) time and O(n) space complexity.
Is Longest Special Path asked at Google/Amazon/Meta?
Tree DFS problems combined with hash maps and prefix techniques appear frequently in interviews at companies like Google, Amazon, and Meta. Variants of longest unique path or longest substring patterns mapped onto trees are common in high-level coding rounds.
What data structure is used in Longest Special Path?
The core data structures are a hash table for tracking the last occurrence of node values, adjacency lists for representing the tree, and arrays for prefix distances or depths. These structures allow efficient DFS traversal and constant-time duplicate detection.
What is the time complexity of Longest Special Path?
The optimal solution runs in O(n) time where n is the number of nodes in the tree. Each node is visited once during DFS and hash map operations take constant time. Space complexity is O(n) due to recursion depth, prefix storage, and the value-to-depth map.

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