Longest Special Path II

This problem asks for the longest path in a tree where node values are unique except for one allowed repeat. Use DFS to scan arrays while maintaining a hash table of current node values for fast lookup. Track path length and minimum nodes dynamically to return both required metrics efficiently.

Problem Statement

Given an undirected tree with n nodes numbered 0 to n - 1, represented by edges where edges[i] = [ui, vi, lengthi], find a downward path from ancestor to descendant such that all node values are distinct except at most one repeated value. You are also given an array nums where nums[i] is the value of node i.

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 among all longest special paths. A special path allows one repeated value but otherwise only distinct nodes.

Examples

Example 1

Input: edges = [[0,1,1],[1,2,3],[1,3,1],[2,4,6],[4,7,2],[3,5,2],[3,6,5],[6,8,3]], nums = [1,1,0,3,1,2,1,1,0]

Output: [9,3]

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

The longest special paths are 1 -> 2 -> 4 and 1 -> 3 -> 6 -> 8 , both having a length of 9. The minimum number of nodes across all longest special paths is 3.

Example 2

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

Output: [5,2]

The longest path is 0 -> 3 consisting of 2 nodes with a length of 5.

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 with Hash Table

Perform depth-first search from the root, maintaining a hash table of node values along the path. Check at each step if adding a child violates the special path rule. Update maximum length and minimum node count when valid paths are found.

Dynamic Path Tracking

Track the current path length and node count dynamically. Use the hash table to quickly identify duplicates and allow at most one repeated value. Backtrack properly to remove nodes from the hash table when returning from recursion.

Array Scanning for Edge Weights

Scan the edges array while DFS traverses the tree. Accumulate edge lengths for valid paths to compute the longest special path. Combine edge scanning with hash lookup to ensure both correctness and efficiency.

Complexity Analysis

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

Time complexity depends on visiting each node and checking hash table entries along the path, roughly O(n). Space complexity depends on recursion depth and hash table usage, also O(n) in the worst case.

What Interviewers Usually Probe

• They may hint at hash table use for tracking duplicates in the path.
• Expect to explain how DFS ensures downward traversal without revisiting ancestors.
• Discuss trade-offs of maintaining path length vs. number of nodes dynamically.

Common Pitfalls or Variants

Common pitfalls

• Failing to allow exactly one repeated value while rejecting additional duplicates.
• Incorrectly calculating the minimum number of nodes when multiple paths tie for maximum length.
• Not backtracking hash table entries properly, causing false duplicate detection.

Follow-up variants

• Compute the longest special path allowing up to k repeated values instead of just one.
• Return all distinct longest special paths instead of only length and minimum nodes.
• Consider weighted node values where the path's sum must be maximized under the same uniqueness rules.

FAQ

What is a special path in Longest Special Path II?

A special path is a downward path where all node values are distinct except at most one value can appear twice.

How does array scanning plus hash lookup apply here?

Array scanning lets you traverse edges while hash lookup quickly detects duplicates in the current path for validation.

Can the root node be counted twice?

No, the root node counts once; only one node value anywhere along the path can repeat.

What if multiple paths tie for longest length?

Return the minimum number of nodes among all paths that achieve the maximum length.

What is the recommended approach for large trees?

Use DFS with a hash table for value tracking and dynamic path length updates to handle up to 5*10^4 nodes efficiently.