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What is the time complexity of the optimal iterative DFS solution using a prefix sum hash map for the Path Sum III problem on a binary tree with n nodes?

medium🪤 Complexity Trap Q13 of Q15
Tree: Depth-First Search - Path Sum III (Any Path)
What is the time complexity of the optimal iterative DFS solution using a prefix sum hash map for the Path Sum III problem on a binary tree with n nodes?
AO(n) because each node is visited once and prefix sum operations are O(1) on average
BO(n log n) because each node's prefix sum is updated and queried in a balanced tree
CO(n^2) because prefix sums are recalculated for each path starting at every node
DO(n) but with O(n^2) space due to storing all prefix sums for all paths
Step-by-Step Solution
Solution:

  1. Step 1: Identify number of node visits

    Each node is pushed and popped from the stack once, so O(n) visits.

  2. Step 2: Analyze prefix sum operations

    Hash map lookups and updates are O(1) average, so total O(n) time.

  3. Final Answer:

    Option A -> Option B

  4. Quick Check:

    Linear time due to single DFS traversal and constant-time prefix sum ops [OK]
Quick Trick: Prefix sums with hash map yield O(n) time [OK]
Common Mistakes:
MISTAKES
  • Confusing with brute force O(n^2) complexity
  • Assuming prefix sums require recomputation for each path
  • Overestimating space as O(n^2)
Trap Explanation:
PITFALL
  • O(n^2) looks plausible if one thinks prefix sums are recomputed per path; hash map operations are constant time on average.
Interviewer Note:
CONTEXT
  • Checks if candidate understands why prefix sums reduce complexity from naive solutions.
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