Find the Maximum Sum of Node Values

arraydynamic programminggreedy

Problem Statement

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 , for which Alice 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 .

Examples

Example 1

Input: nums = [1,2,1], k = 3, edges = [[0,1],[0,2]]
Output: 6
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
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
The maximum achievable sum is 42 which can be achieved by Alice performing no operations.

Constraints

2 <= n == nums.length <= 2 * 10 ^4

1 <= k <= 10 ^9

0 <= nums[i] <= 10 ^9

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.

😤

Hard

Difficulty

Topic Info

ModuleDSA

CategoryArray

Sub-topicDynamic Programming