Find the Maximum Sum of Node Values

Problem Description

There exists an undirected tree with n nodes numbered 0 to n - 1. You are given a 0-indexed 2D integer array edges of length n - 1, where edges[i] = [u_i, v_i] indicates that there is an edge between nodes u_i and v_i in the tree. You are also given a positive integer k, and a 0-indexed array of non-negative integers nums of length n, where nums[i] represents the value of the node numbered i.

Alice wants the sum of values of tree nodes to be maximum. She can perform the following operation any number of times (including zero) on the tree:

• Choose any edge [u, v] connecting the nodes u and v, and update their values as follows:
  • nums[u] = nums[u] XOR k
  • nums[v] = nums[v] XOR k

Return the maximum possible sum of the values Alice can achieve by performing the operation any number of times.

Key Insights

• Each node's value can be toggled between nums[i] and nums[i] XOR k.
• The goal is to maximize the sum of all node values, which can be computed by determining the best value for each node that can be either nums[i] or nums[i] XOR k.
• The operations can be applied to any connected nodes, meaning the values can be altered in a way that may benefit the overall sum.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(1)

Solution

The solution involves iterating through each node's value and computing two potential sums: one using the original values and one using the values after applying the XOR operation with k. The maximum sum can then be derived by taking the maximum value of these two sums for each node.

1. Initialize two sums: original_sum and xor_sum.
2. For each node in the nums array, add its value to original_sum.
3. Simultaneously, compute the XOR of the node value with k and add it to xor_sum.
4. The result will be the maximum of original_sum and xor_sum.

Code Solutions