2509. Cycle Length Queries in a Tree

Description

You are given an integer n. There is a complete binary tree with 2ⁿ - 1 nodes. The root of that tree is the node with the value 1, and every node with a value val in the range [1, 2^{n - 1} - 1] 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 queries[i] = [a_i, b_i]. For each query, solve the following problem:

Add an edge between the nodes with values a_i and b_i.
Find the length of the cycle in the graph.
Remove the added edge between nodes with values a_i and b_i.

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 i^th 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
1 <= m <= 10⁵
queries[i].length == 2
1 <= a_i, b_i <= 2ⁿ - 1
a_i != b_i

Solution

cycle-length-queries-in-a-tree.py

class Solution:
    def cycleLengthQueries(self, n: int, queries: List[List[int]]) -> List[int]:
        res = []

        for a, b in queries:
            count = 1

            while a != b:
                if a > b:
                    a //= 2
                else:
                    b //= 2

                count += 1

            res.append(count)

        return res