2322 - Minimum Score After Removals on a Tree.

Hard

There is an undirected connected tree with n nodes labeled from 0 to n - 1 and n - 1 edges.

You are given a 0-indexed integer array nums of length n where nums[i] represents the value of the <code>i^th</code> node. You are also given a 2D integer array edges of length n - 1 where <code>edgesi = a_i, b_i</code> indicates that there is an edge between nodes <code>a_i</code> and <code>b_i</code> in the tree.

Remove two distinct edges of the tree to form three connected components. For a pair of removed edges, the following steps are defined:

• Get the XOR of all the values of the nodes for each of the three components respectively.

• The difference between the largest XOR value and the smallest XOR value is the score of the pair.

• For example, say the three components have the node values: [4,5,7], [1,9], and [3,3,3]. The three XOR values are <code>4 ^ 5 ^ 7 = <ins> 6 </ins></code>, <code>1 ^ 9 = <ins> 8 </ins></code>, and <code>3 ^ 3 ^ 3 = <ins> 3 </ins></code>. The largest XOR value is 8 and the smallest XOR value is 3. The score is then 8 - 3 = 5.

Return the minimum score of any possible pair of edge removals on the given tree.

Example 1:

Input: nums = 1,5,5,4,11, edges = \[\[0,1],1,2,1,3,3,4]

Output: 9

Explanation: The diagram above shows a way to make a pair of removals.

• The 1^st component has nodes 1,3,4 with values 5,4,11. Its XOR value is 5 ^ 4 ^ 11 = 10.

• The 2^nd component has node 0 with value 1. Its XOR value is 1 = 1.

• The 3^rd component has node 2 with value 5. Its XOR value is 5 = 5.

The score is the difference between the largest and smallest XOR value which is 10 - 1 = 9.

It can be shown that no other pair of removals will obtain a smaller score than 9.

Example 2:

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

Output: 0

Explanation: The diagram above shows a way to make a pair of removals.

• The 1^st component has nodes 3,4 with values 4,4. Its XOR value is 4 ^ 4 = 0.

• The 2^nd component has nodes 1,0 with values 5,5. Its XOR value is 5 ^ 5 = 0.

• The 3^rd component has nodes 2,5 with values 2,2. Its XOR value is 2 ^ 2 = 0.

The score is the difference between the largest and smallest XOR value which is 0 - 0 = 0.

We cannot obtain a smaller score than 0.

Constraints:

• n == nums.length

• 3 <= n <= 1000

• <code>1 <= numsi<= 10⁸</code>

• edges.length == n - 1

• edges[i].length == 2

• <code>0 <= a_i, b_i< n</code>

• <code>a_i != b_i</code>

• edges represents a valid tree.