Cycle Length Queries in a Tree

Cycle Length Queries in a Tree

What is this problem about?

The Cycle Length Queries in a Tree interview question presents you with a full binary tree where nodes are numbered from 1 to 2^30 - 1. You are given a series of queries, each consisting of two nodes (a, b). For each query, imagine adding an edge between a and b, which creates a cycle. You need to find the length of this cycle. This Cycle Length Queries in a Tree coding problem is a test of your ability to navigate perfectly structured trees using mathematical relationships.

Why is this asked in interviews?

Arcesium and other high-frequency trading or system companies ask this to test your understanding of Lowest Common Ancestor (LCA) in a specialized context. Since the tree is a full binary tree with 2^30 nodes, you cannot build the tree in memory. You must use the mathematical property that the parent of node x is floor(x / 2). It evaluates your ability to work with large-scale data through mathematical abstraction.

Algorithmic pattern used

This utilize the Array, Binary Tree, Tree interview pattern, specifically focusing on finding the LCA using binary paths.

1. LCA Logic: For nodes a and b, continuously move the larger node to its parent (x = x / 2) until both nodes meet at the same ID. This intersection is the LCA.
2. Distance Calculation: Track the number of steps taken to move a to LCA (d1) and b to LCA (d2).
3. Cycle Length: The length of the path from a to b through the tree is d1 + d2. Adding the new edge (a, b) completes the cycle, making the total length d1 + d2 + 1.

Example explanation

Query: Nodes 5 and 3.

1. Node 5's parent is 5/2 = 2.
2. Node 3's parent is 3/2 = 1.
3. Compare 2 and 3. Move 3 to parent (1). Steps: 1.
4. Compare 2 and 1. Move 2 to parent (1). Steps: 1 + 1 = 2.
5. Now both are at 1 (the LCA).
6. Total tree path length = 2 steps.
7. Cycle length = 2 + 1 = 3.

Common mistakes candidates make

• Trying to build the tree: With 2^30 nodes, any attempt to instantiate a tree structure will result in a Memory Limit Exceeded error.
• Incorrect parent formula: Forgetting that in a 1-indexed binary tree, the parent is always x // 2.
• Complexity: Performing an O(N) search instead of an O(log N) path climb.

Interview preparation tip

Practice finding the LCA in different types of trees. For balanced or perfect binary trees, the "climb to the root" method is the most efficient and requires the least amount of boilerplate code.