Find the Maximum Sum of Node Values

Problem Statement

In the context of the problem, you are working with a tree structure consisting of n nodes, each labeled from 0 to n - 1. The tree is defined using a list of edges, where each edge binds two nodes together, creating an undirected connection. Alongside this structural information, each node holds a numerical value specified in the array nums. You are also given an integer k which plays a crucial role in a type of operation you can perform on this tree.

The core task is to maximize the sum of the values of all nodes. You can perform a specific operation to achieve this, which involves choosing any edge, and for the two nodes that the edge connects, you toggle their values using the XOR operation with k. The challenge is to determine the highest possible sum of all node values in the tree after performing this operation any number of times, including the possibility of not performing it at all.

Examples

Example 1

Input:

nums = [1,2,1], k = 3, edges = [[0,1],[0,2]]

Output:

6

Explanation:

Alice can achieve the maximum sum of 6 using a single operation:
- Choose the edge [0,2]. nums[0] and nums[2] become: 1 XOR 3 = 2, and the array nums becomes: [1,2,1] -> [2,2,2].
The total sum of values is 2 + 2 + 2 = 6.
It can be shown that 6 is the maximum achievable sum of values.

Example 2

Input:

nums = [2,3], k = 7, edges = [[0,1]]

Output:

9

Explanation:

Alice can achieve the maximum sum of 9 using a single operation:
- Choose the edge [0,1]. nums[0] becomes: 2 XOR 7 = 5 and nums[1] become: 3 XOR 7 = 4, and the array nums becomes: [2,3] -> [5,4].
The total sum of values is 5 + 4 = 9.
It can be shown that 9 is the maximum achievable sum of values.

Example 3

Input:

nums = [7,7,7,7,7,7], k = 3, edges = [[0,1],[0,2],[0,3],[0,4],[0,5]]

Output:

42

Explanation:

The maximum achievable sum is 42 which can be achieved by Alice performing no operations.

Constraints

• 2 <= n == nums.length <= 2 * 104
• 1 <= k <= 109
• 0 <= nums[i] <= 109
• edges.length == n - 1
• edges[i].length == 2
• 0 <= edges[i][0], edges[i][1] <= n - 1
• The input is generated such that edges represent a valid tree.

Approach and Intuition

To grasp the solution approach towards achieving the maximum possible sum of the nodes' values, let's break down what happens during the operation and how this influences the sum:

1. Understand XOR Operation: The XOR operation introduces a toggle effect. If num XOR k results in a higher value than num, then performing the XOR operation is advantageous in increasing the summed value of the tree's nodes.

2. Observe Patterns and Implications:

   • If toggling a node’s value using the XOR operation yields a higher individual value, you would ideally want to toggle such a node.
   • However, the operation must be performed on pairs of nodes that share an edge. This implies that while you may increase one node's value, you might simultaneously decrease the connected node's value.

3. Strategy Extraction from Examples:

   • Example 1: The optimal approach involved finding the right edge where the XOR operation maximizes the values of both connected nodes.
   • Example 2: A direct application of the XOR operation on the only available edge provided the result.
   • Example 3: It highlighted a scenario where performing no operations yields the maximum sum, demonstrating the necessity of evaluating the initial configuration before attempting operations.

4. Optimal Strategy:

   • Begin by evaluating the initial sum of all node values.
   • For each edge, simulate the effect of applying the XOR operation to see if it provides a better sum.
   • Keep track of the highest sum observed.

This problem challenges you to apply both the breadth of algorithmic knowledge, regarding tree traversal and manipulation, and the depth of understanding bitwise operations and their effects on data. The solution would likely need to efficiently traverse the tree while keeping a keen eye on the potential value changes introduced by operations on any edge. The constraint of performing operations any number of times implies that some nodes could be toggled multiple times, either directly or indirectly, through different paths and sequences of operations.

Solutions

class Solution {
public:
    long long maxValSum(vector<int>& vals, int xor_key, vector<vector<int>>& connections) {
        long long totalSum = 0;
        int evenPosChanges = 0, smallestPosDiff = INT_MAX, largestNegDiff = INT_MIN;
            
        for (int val : vals) {
            int newVal = val ^ xor_key;
            totalSum += val;
            int delta = newVal - val;
                
            if (delta > 0) {
                smallestPosDiff = min(smallestPosDiff, delta);
                totalSum += delta;
                evenPosChanges++;
            } else {
                largestNegDiff = max(largestNegDiff, delta);
            }
        }
            
        if (evenPosChanges % 2 == 0)
            return totalSum;
            
        return max(totalSum - smallestPosDiff, totalSum + largestNegDiff);
    }
};