Longest Special Path

Start by traversing the tree from the root using DFS, keeping track of node values in a hash set to ensure uniqueness. Update the maximum path length whenever a longer unique sequence is found. Return both the maximum path length and the minimum number of nodes along any path of that length for complete results.

Problem Statement

You are given an undirected tree rooted at node 0 with n nodes numbered from 0 to n - 1. The tree is represented as a 2D array edges where edges[i] = [ui, vi, lengthi] indicates an edge between nodes ui and vi with length lengthi. Each node has a value given in the array nums where nums[i] is the value at node i.

Define a special path as a downward path from an ancestor node to a descendant node where all node values are unique. Paths may start and end at the same node. Your task is to compute the length of the longest special path and the minimum number of nodes along any longest special path.

Examples

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]

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]

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.

Solution Approach

DFS Traversal with Hash Set

Perform a depth-first search from the root while maintaining a hash set of node values along the current path. For each node, skip adding it to the path if its value already exists to maintain uniqueness.

Track Path Lengths

Accumulate the sum of edge lengths along the DFS path. Whenever a longer special path is found, update both the maximum length and record the number of nodes along that path. Use careful array indexing to map edges to child nodes efficiently.

Return Maximum and Minimum Nodes

After DFS completes, return a pair: the maximum path length and the minimum number of nodes among all paths of that length. This ensures both correctness and handles edge cases where multiple paths share the same length.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

Time and space complexity depend on DFS traversal. Each node is visited once, and the hash set tracks unique values along paths, making worst-case space O(n) and time O(n) if each path is unique.

What Interviewers Usually Probe

• Do you maintain uniqueness along the DFS path using a hash structure?
• Can you efficiently track path lengths and node counts simultaneously?
• Have you considered how array scanning and edge mapping affect traversal speed?

Common Pitfalls or Variants

Common pitfalls

• Failing to reset the hash set when backtracking leads to incorrect uniqueness checks.
• Not updating the minimum node count for multiple longest paths.
• Incorrectly mapping edges to children in the tree array structure.

Follow-up variants

• Compute longest special path for trees with weighted or unweighted edges.
• Find the longest downward path where node values satisfy a specific condition, not just uniqueness.
• Determine the longest path starting at any node, not necessarily the root.

FAQ

What is a longest special path in this problem?

It is a downward path from an ancestor to a descendant where all node values are unique.

Which data structure ensures node value uniqueness?

A hash set is used to track node values along the current DFS path.

Does the path have to start at the root?

No, but the standard solution initiates DFS from the root and considers all downward paths.

What pattern does this problem follow?

It follows an array scanning plus hash lookup pattern over tree structures.

How do I handle multiple longest special paths?

Track both the maximum length and the minimum number of nodes among all paths of that length to return the correct pair.