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What is the time complexity of the optimal bottom-up dynamic programming solution for the Coin Change II problem with n coins and target amount amount?

medium🪤 Complexity Trap Q13 of Q15

Dynamic Programming: Knapsack - Coin Change II (Count Ways)

What is the time complexity of the optimal bottom-up dynamic programming solution for the Coin Change II problem with n coins and target amount amount?

AO(amount^2)

BO(n + amount)

CO(2^amount)

DO(n * amount)

Step-by-Step Solution

Solution:

Step 1: Identify loops in code
Outer loop iterates over n coins, inner loop iterates over amount from coin to amount.
Step 2: Calculate complexity
Each dp[w] update is O(1), total updates are roughly n * amount, so time complexity is O(n * amount).
Final Answer:
Option D -> Option D
Quick Check:
Nested loops over coins and amounts confirm O(n * amount) [OK]

Quick Trick: Count nested loops and their ranges [OK]

Common Mistakes:

MISTAKES

Confusing amount with n
Forgetting recursion stack space
Assuming quadratic in amount

Trap Explanation:

PITFALL

O(amount^2) looks plausible if one thinks inner loop is amount^2, but it's amount * n instead.

Interviewer Note:

CONTEXT

Checks understanding of DP complexity and loop structure.

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