What is the time complexity of the space-optimized bottom-up dynamic programming solution for the Equal Partition problem, where n is the number of elements and target is half the total sum?

medium🪤 Complexity Trap Q13 of Q15

Dynamic Programming: Knapsack - Equal Partition (Partition Equal Subset Sum)

AO(2^n) because the problem explores all subsets in worst case

BO(n * n) because we iterate over all elements and all sums up to n

CO(target^2) because the dp array size is target and we update it multiple times

DO(n * target) because for each element we iterate over sums up to target

Step-by-Step Solution

Solution:

Step 1: Identify loops in the code
The outer loop runs n times (for each element), and the inner loop runs up to target times (half the total sum).
Step 2: Calculate total time complexity
Multiplying these gives O(n * target). The dp array updates are constant time per iteration.
Final Answer:
Option D -> Option D
Quick Check:
Time depends on number of elements and target sum, not n squared or exponential [OK]

Quick Trick: Time depends on n and target sum, not just n [OK]

Common Mistakes:

MISTAKES

Confusing n with target or assuming quadratic in n
Forgetting target can be large

Trap Explanation:

PITFALL

Option A looks plausible since n appears twice, but inner loop depends on target, not n

Interviewer Note:

CONTEXT

Tests understanding of DP complexity and difference between input size and sum constraints

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More Dynamic Programming: Knapsack Quizzes

What is the time complexity of the space-optimized bottom-up dynamic programming solution for the Equal Partition problem, where n is the number of elements and target is half the total sum?

Step 1: Identify loops in the code

Step 2: Calculate total time complexity

Final Answer:

Quick Check:

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