#2509

Hard

auto_awesomeBinary-tree traversal and state tracking

LeetCode Problem Workspace

Cycle Length Queries in a Tree

Solve the problem of determining cycle lengths in a binary tree with added edges through queries.

Problem Statement

You are given an integer n, representing a complete binary tree with 2n - 1 nodes. The root of the tree is node 1, and every node val in the range [1, 2n - 1 - 1] has two children. Each query in a given array contains two nodes, and for each query, you need to determine the cycle length after adding an edge between those two nodes.

After processing each query, the added edge is removed, and the next query is processed. The goal is to return the cycle length for each query, which involves finding the distance between nodes 'a' and 'b' in the tree and calculating the cycle formed after adding the edge.

Examples

Example 1

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

Output: [4,5,3]

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]

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 <= ai, bi <= 2n - 1
ai != bi

Solution Approach

Binary Tree Traversal

The problem involves using binary tree traversal techniques to find the shortest path between any two given nodes. Efficient traversal can help find cycles that form after adding edges between nodes.

Union-Find Data Structure

Union-Find is essential for efficiently managing the dynamic tree structure as we add edges and track connected components. It helps track which nodes belong to the same component and efficiently detect cycles.

Cycle Detection and Length Calculation

Once the edge is added, cycle detection is crucial for determining the cycle length. A method to trace the cycle back to the root of the tree and compute its length is required.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on the method used to detect the cycle and handle the dynamic structure. With a Union-Find approach, the amortized time complexity for each query is nearly constant, making the algorithm efficient enough for the given constraints.

What Interviewers Usually Probe

Look for an understanding of tree traversal and cycle detection techniques.
Assess the ability to apply Union-Find for dynamic connectivity management.
Evaluate the candidate's approach to handling edge cases like tree structure updates after each query.

Common Pitfalls or Variants

Common pitfalls

Failure to manage dynamic tree structure efficiently leads to poor performance.
Not accounting for the removal of the edge after each query could result in incorrect results.
Inefficient cycle detection can lead to excessive computation time, especially for large inputs.

Follow-up variants

Modify the problem to consider non-binary trees.
Handle a scenario where queries are static and no edges are removed between queries.
Optimize for scenarios with large n and more queries.

FAQ

How do I calculate the cycle length in a binary tree?

The cycle length is determined by the distance between the two nodes connected by the added edge, including the path back to the root. Efficient traversal methods like DFS or BFS can be used to compute this.

What is the role of Union-Find in solving this problem?

Union-Find is used to manage dynamic connectivity in the tree. It helps efficiently find and merge components, which is essential for detecting cycles as edges are added.

Can this problem be solved without a Union-Find approach?

While it's possible to solve without Union-Find, it would require more inefficient methods for cycle detection and pathfinding, making it less optimal for large inputs.

What is the time complexity of the solution?

The time complexity per query is nearly constant due to the amortized time of Union-Find operations, but the overall complexity depends on the number of queries and the tree structure.

How does GhostInterview help with the Cycle Length Queries in a Tree problem?

GhostInterview helps by guiding you through Union-Find concepts, offering solutions for tree traversal, and optimizing for performance with large input sizes.

terminal

Solution

Solution 1: Finding the Lowest Common Ancestor

For each query, we find the lowest common ancestor of the two nodes $a$ and $b$, and record the number of steps taken upwards. The answer to the query is the number of steps plus one.

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class Solution:
    def cycleLengthQueries(self, n: int, queries: List[List[int]]) -> List[int]:
        ans = []
        for a, b in queries:
            t = 1
            while a != b:
                if a > b:
                    a >>= 1
                else:
                    b >>= 1
                t += 1
            ans.append(t)
        return ans

Continue Practicing

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