2509 - Cycle Length Queries in a Tree\.

Hard

You are given an integer n. There is a complete binary tree with <code>2ⁿ - 1</code> nodes. The root of that tree is the node with the value 1, and every node with a value val in the range <code>1, 2^{n - 1} - 1</code> has two children where:

• The left node has the value 2 * val, and

• The right node has the value 2 * val + 1.

You are also given a 2D integer array queries of length m, where <code>queriesi = a_i, b_i</code>. For each query, solve the following problem:

• Add an edge between the nodes with values <code>a_i</code> and <code>b_i</code>.

• Find the length of the cycle in the graph.

• Remove the added edge between nodes with values <code>a_i</code> and <code>b_i</code>.

Note that:

• A cycle is a path that starts and ends at the same node, and each edge in the path is visited only once.

• The length of a cycle is the number of edges visited in the cycle.

• There could be multiple edges between two nodes in the tree after adding the edge of the query.

Return an array answer of length m where answer[i] is the answer to the <code>i^th</code> query.

Example 1:

Input: n = 3, queries = \[\[5,3],4,7,2,3]

Output: 4,5,3

Explanation: The diagrams above show the tree of 2³ - 1 nodes. Nodes colored in red describe the nodes in the cycle after adding the edge.

• After adding the edge between nodes 3 and 5, the graph contains a cycle of nodes 5,2,1,3. Thus answer to the first query is 4. We delete the added edge and process the next query.

• After adding the edge between nodes 4 and 7, the graph contains a cycle of nodes 4,2,1,3,7. Thus answer to the second query is 5. We delete the added edge and process the next query.

• After adding the edge between nodes 2 and 3, the graph contains a cycle of nodes 2,1,3. Thus answer to the third query is 3. We delete the added edge.

Example 2:

Input: n = 2, queries = \[\[1,2]]

Output: 2

Explanation: The diagram above shows the tree of 2² - 1 nodes. Nodes colored in red describe the nodes in the cycle after adding the edge.

• After adding the edge between nodes 1 and 2, the graph contains a cycle of nodes 2,1. Thus answer for the first query is 2. We delete the added edge.

Constraints:

• 2 <= n <= 30

• m == queries.length

• <code>1 <= m <= 10⁵</code>

• queries[i].length == 2

• <code>1 <= a_i, b_i<= 2ⁿ - 1</code>

• <code>a_i != b_i</code>