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Which approach guarantees an optimal solution in terms of time complexity?

easy🔍 Pattern Recognition Q11 of Q15
Tree: Depth-First Search - Path Sum III (Any Path)
Given a binary tree and a target sum, you need to count all paths where the sum of the node values along the path equals the target. The path can start and end at any node but must go downwards (parent to child). Which approach guarantees an optimal solution in terms of time complexity?
AUse a brute force DFS starting from every node, checking all downward paths for the target sum.
BUse dynamic programming to store sums of subtrees and combine results bottom-up.
CUse a DFS with a prefix sum hash map to track cumulative sums and count valid paths in one traversal.
DUse a greedy approach to prune paths that exceed the target sum early.
Step-by-Step Solution
Solution:

  1. Step 1: Understand problem constraints and path definition

    The path can start and end anywhere but must go downwards, so checking all paths naively is expensive.

  2. Step 2: Recognize prefix sum with DFS pattern

    Using a prefix sum hash map during DFS allows counting all valid paths in O(n) time by tracking cumulative sums and their frequencies.

  3. Final Answer:

    Option C -> Option C

  4. Quick Check:

    Prefix sum + DFS solves in linear time [OK]
Quick Trick: Prefix sums track cumulative sums efficiently [OK]
Common Mistakes:
MISTAKES
  • Assuming paths must start at root or leaf
  • Using brute force DFS from every node without pruning
  • Trying greedy pruning which fails on negative values
Trap Explanation:
PITFALL
  • Brute force looks straightforward but is O(n²); DP bottom-up doesn't fit path-anywhere pattern; greedy fails with negative values.
Interviewer Note:
CONTEXT
  • Tests if candidate can identify the optimal pattern beyond brute force or naive DP.
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