Maximum Genetic Difference Query

This problem asks you to compute the maximum genetic difference between a given value and any ancestor along a node's path to the root. Use a combination of array scanning to traverse the tree, hash tables to track active paths, and a trie structure to efficiently calculate XOR values. Carefully managing the trie while traversing ensures optimal time complexity for large trees and query sets.

Problem Statement

Given a rooted tree with n nodes numbered from 0 to n - 1, each node's number represents its genetic value. The tree structure is provided by an array parents where parents[i] is the parent of node i, and parents[x] == -1 indicates the root node. The genetic difference between two nodes is defined as the bitwise XOR of their genetic values.

You are also given an array queries where queries[i] = [nodei, vali]. For each query, find the maximum XOR value between vali and the genetic value of any node along the path from nodei to the root. Return an array ans where ans[i] is the maximum genetic difference for the ith query.

Examples

Example 1

Input: parents = [-1,0,1,1], queries = [[0,2],[3,2],[2,5]]

Output: [2,3,7]

The queries are processed as follows:

• [0,2]: The node with the maximum genetic difference is 0, with a difference of 2 XOR 0 = 2.
• [3,2]: The node with the maximum genetic difference is 1, with a difference of 2 XOR 1 = 3.
• [2,5]: The node with the maximum genetic difference is 2, with a difference of 5 XOR 2 = 7.

Example 2

Input: parents = [3,7,-1,2,0,7,0,2], queries = [[4,6],[1,15],[0,5]]

Output: [6,14,7]

The queries are processed as follows:

• [4,6]: The node with the maximum genetic difference is 0, with a difference of 6 XOR 0 = 6.
• [1,15]: The node with the maximum genetic difference is 1, with a difference of 15 XOR 1 = 14.
• [0,5]: The node with the maximum genetic difference is 2, with a difference of 5 XOR 2 = 7.

Constraints

• 2 <= parents.length <= 105
• 0 <= parents[i] <= parents.length - 1 for every node i that is not the root.
• parents[root] == -1
• 1 <= queries.length <= 3 * 104
• 0 <= nodei <= parents.length - 1
• 0 <= vali <= 2 * 105

Solution Approach

Build the tree and preprocess paths

Create an adjacency list from the parents array to represent the tree. Identify the root and prepare a data structure to track all ancestors along a node's path for efficient lookup during queries.

Use a trie for XOR maximization

Traverse the tree using DFS. For each node on the path, insert its genetic value into a binary trie. To answer a query for a node, compute the maximum XOR of the query value with values stored in the trie, which represents all ancestors along the path to the root.

Backtrack to maintain active path

After visiting child nodes, remove the current node's value from the trie before backtracking. This ensures the trie only contains the active path to correctly compute maximum XOR for subsequent queries without interference from unrelated branches.

Complexity Analysis

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

Time complexity is roughly O(N * log M + Q * log M), where N is the number of nodes, Q is the number of queries, and M is the maximum genetic value, due to trie insertions and XOR lookups. Space complexity is O(N * log M) for the trie storing active path values.

What Interviewers Usually Probe

• Looking for understanding of XOR properties and trie usage in path queries.
• Expect the candidate to manage insertion and removal in the trie to maintain correct path context.
• Interested in whether the solution handles large trees and query sets efficiently with DFS and hashing.

Common Pitfalls or Variants

Common pitfalls

• Failing to remove nodes from the trie during backtracking, which produces incorrect XOR results.
• Attempting to compute maximum XOR without a trie, leading to O(N*Q) complexity.
• Mismanaging the mapping between node indices and their genetic values when scanning paths.

Follow-up variants

• Queries limited to leaf nodes instead of arbitrary nodes.
• Return the minimum genetic difference instead of the maximum.
• Allow dynamic updates to node genetic values, requiring persistent trie structures.

FAQ

What is the main idea behind Maximum Genetic Difference Query?

It computes the maximum XOR between a query value and any ancestor along a node's path to the root using array scanning plus a trie for efficiency.

Why do we need a trie in this problem?

The trie allows fast computation of the maximum XOR along the current path without scanning all ancestors for every query.

Can we solve it without a trie?

Yes, but scanning all ancestors for each query results in O(N*Q) time, which is inefficient for large trees.

How does backtracking affect the solution?

Backtracking ensures only active path nodes remain in the trie, preventing interference from unrelated branches when computing XOR.

What is a common mistake when implementing this solution?

A frequent error is forgetting to remove nodes from the trie during DFS backtracking, leading to incorrect maximum XOR results.