What is the worst-case time complexity of the optimal backtracking solution with sum bound pruning for Combination Sum III (k numbers from 1 to 9 summing to n)?

medium🪤 Complexity Trap Q13 of Q15

Subsets & Combinations - Combination Sum III (K Numbers to N)

AO(k!) because permutations of k numbers are explored

BO(k * n) because we explore all sums up to n for k elements

CO(n^k) because we try all sequences of length k with sum n

DO(2^9) because each number 1-9 can be either included or excluded, pruning reduces calls but worst case remains exponential

Step-by-Step Solution

Step 1: Identify search space size
Numbers 1 to 9 can be included or excluded, so 2^9 subsets possible.
Step 2: Consider pruning effect
Pruning reduces calls but worst-case complexity remains O(2^9) since all subsets might be explored.
Final Answer:
Option D -> Option D
Quick Check:
Backtracking worst case is exponential in number of candidates [OK]

Quick Trick: Backtracking subsets of 9 elements is O(2^9) worst case [OK]

Common Mistakes:

MISTAKES

Confusing with DP complexity O(k*n)
Thinking permutations cause factorial complexity

Trap Explanation:

PITFALL

O(k*n) looks plausible but pruning doesn't reduce exponential worst case; permutations are not generated.

Interviewer Note:

CONTEXT

Tests if candidate understands exponential nature of backtracking despite pruning.

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What is the worst-case time complexity of the optimal backtracking solution with sum bound pruning for Combination Sum III (k numbers from 1 to 9 summing to n)?

Step 1: Identify search space size

Step 2: Consider pruning effect

Final Answer:

Quick Check:

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