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What is the time complexity of the parent pointer + ancestor set approach for finding the Lowest Common Ancestor in a binary tree with n nodes and height h?

medium🪤 Complexity Trap Q13 of Q15
Tree: Depth-First Search - Lowest Common Ancestor of Binary Tree
What is the time complexity of the parent pointer + ancestor set approach for finding the Lowest Common Ancestor in a binary tree with n nodes and height h?
AO(h^2) because ancestor set lookups take O(h) each and we do up to h lookups
BO(n^2) because each node's parent is checked multiple times
CO(n) to build parent map plus O(h) to find LCA, total O(n)
DO(n log n) due to sorting ancestors before comparison
Step-by-Step Solution
Solution:

  1. Step 1: Identify parent map construction cost

    Building the parent map requires visiting each node once, O(n).

  2. Step 2: Analyze ancestor set and LCA search

    Ancestor set insertion and lookup are O(1) average. Traversing up to height h for p and q is O(h). Total is O(n) + O(h) ≈ O(n).

  3. Final Answer:

    Option C -> Option C

  4. Quick Check:

    Parent map O(n) + ancestor lookup O(h) = O(n) total [OK]
Quick Trick: Parent map build dominates; ancestor lookup is O(h) [OK]
Common Mistakes:
MISTAKES
  • Confusing ancestor set lookup as O(h)
  • Assuming sorting ancestors needed
  • Overestimating repeated parent checks
Trap Explanation:
PITFALL
  • Option A looks plausible if one thinks ancestor set lookups are O(h), but they are O(1) average.
Interviewer Note:
CONTEXT
  • Tests understanding of complexity combining tree traversal and set operations.
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