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What is the worst-case time complexity of the bottom-up dynamic programming solution for the Coin Change II problem with n coin denominations and target amount amount?

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

What is the worst-case time complexity of the bottom-up dynamic programming solution for the Coin Change II problem with n coin denominations and target amount amount?

AO(n + amount)

BO(amount^n)

CO(n * amount)

DO(n^2 * amount)

Step-by-Step Solution

Solution:

Step 1: Understand DP table dimensions
The DP table has dimensions n (coins) by amount.
Step 2: Analyze nested loops
For each coin, we iterate through all amounts up to target, resulting in O(n * amount) operations.
Final Answer:
Option C -> Option C
Quick Check:
Check loops in code match complexity [OK]

Quick Trick: DP loops over coins and amounts multiply for complexity [OK]

Common Mistakes:

MISTAKES

Assuming exponential complexity due to recursion
Ignoring the linear iteration over amount for each coin
Confusing addition with multiplication in complexity

Trap Explanation:

PITFALL

Exponential options look plausible but DP avoids recomputation.

Interviewer Note:

CONTEXT

Tests knowledge of DP time complexity analysis.

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