Kth Ancestor of a Tree Node

Kth Ancestor of a Tree Node

1. What is this problem about?

The Kth Ancestor of a Tree Node interview question asks you to design a system that can quickly find the $k^{th}$ ancestor of any node in a tree. The tree structure is fixed, but you need to answer many queries efficiently. An ancestor is any node on the path from the current node to the root.

2. Why is this asked in interviews?

Companies like Microsoft and Amazon use the Kth Ancestor coding problem to test a candidate's knowledge of Binary Lifting. A naive jump-by-jump search is $O(K)$, which is too slow for thousands of queries. It evaluations your ability to use pre-calculation to reduce query time to $O(\log K)$. This is an advanced Tree interview pattern.

3. Algorithmic pattern used

This problem is solved using the Binary Lifting technique.

1. Pre-processing: Create a 2D table up[node][j] which stores the $2^j$-th ancestor of node.
   • up[node][0] is the direct parent.
   • up[node][j] = up[up[node][j-1]][j-1]. (The $2^j$ ancestor is the $2^{j-1}$ ancestor of the $2^{j-1}$ ancestor).
2. Query: To find the $k^{th}$ ancestor, use the binary representation of $k$.
   • If the $i^{th}$ bit of $k$ is 1, jump to node = up[node][i].
   • This reduces $k$ jumps into at most $\log_2(k)$ leaps.

4. Example explanation

Find $5^{th}$ ancestor of Node X. $5$ in binary is 101 ($2^2 + 2^0$).

1. Leap 1: Jump to the $2^0 = 1^{st}$ ancestor of X.
2. Leap 2: From that node, jump to its $2^2 = 4^{th}$ ancestor. Total jump distance: 5.

5. Common mistakes candidates make

• Linear queries: Not pre-calculating jumps and doing a standard traversal for every query ($O(N \cdot Q)$).
• Indexing errors: Mismanaging the powers of 2 in the DP table.
• Handling the root: Forgetting to return -1 if a jump goes beyond the root of the tree.

6. Interview preparation tip

Binary Lifting is the standard way to optimize ancestor and distance queries in trees. If you see "many queries" and "tree path," mention LCA (Lowest Common Ancestor) and Binary Lifting. These are top-tier Graph interview pattern concepts.