Tree of Coprimes

Tree of Coprimes

What is this problem about?

The "Tree of Coprimes coding problem" is a challenging combination of tree traversal and number theory. You are given a tree where each node has a value. For each node, you need to find its nearest ancestor such that the node's value and the ancestor's value are "coprime" (their greatest common divisor is 1). If no such ancestor exists, return -1. With up to 50 nodes having values between 1 and 50, but the tree itself being large, efficiency is key.

Why is this asked in interviews?

This "Tree of Coprimes interview question" is a sophisticated problem that tests a candidate's ability to optimize a search using constraints. While there might be thousands of nodes, the values are small (1-50). This "pigeonhole principle" or "small value range" optimization is a common theme in advanced interviews at companies like Google. It assesses if you can avoid redundant work by storing state for only the relevant small set of values.

Algorithmic pattern used

The "Array, Math, Number Theory, Depth-First Search, Tree interview pattern" is used here. We perform a DFS to traverse the tree. While traversing, we maintain a data structure (like an array or hash map) that stores the depth and ID of the most recent ancestor for each value from 1 to 50. For the current node, we check all values from 1 to 50 that are coprime to the node's value, and pick the one with the maximum depth among the stored ancestors.

Example explanation

1. Precompute all coprime pairs for values 1-50.
2. DFS starts.
3. Node A (val=3) at depth 2. Store: ancestors[3] = {depth: 2, id: A}.
4. Move to child Node B (val=5) at depth 3.
5. Check all values coprime to 5 (1, 2, 3, 4, 6...).
6. We see ancestors[3] is available.
7. Nearest coprime ancestor for Node B is Node A.
8. When backtracking from A, remove it from ancestors[3].

Common mistakes candidates make

A major error in the "Tree of Coprimes coding problem" is trying to check every single ancestor for every node, which results in $O(N^2)$ and is too slow. Another mistake is not correctly managing the state during DFS backtracking—forgetting to restore the previous ancestor for a value after visiting a subtree. Candidates might also forget that 1 is coprime to every number.

Interview preparation tip

When you see a problem where the "number of nodes" is large but the "possible values" are very small, look for an optimization that iterates over the values rather than the nodes. This is a common pattern in competitive programming and advanced system design.