What is the time complexity of the space-optimized bottom-up dynamic programming solution for counting the number of ways to make change with n coins and amount W?

medium🪤 Complexity Trap Q13 of Q15

Dynamic Programming: Knapsack - Number of Ways to Make Change

AO(n * W) because for each coin we iterate over all amounts up to W

BO(n + W) because we iterate over coins and amounts separately

CO(W^n) due to nested loops over coins and amounts

DO(n * W) plus O(W) auxiliary space for the dp array

Step-by-Step Solution

Solution:

Step 1: Identify loops and their ranges
Outer loop runs n times (coins), inner loop runs up to W (amount), so time is O(n * W).
Step 2: Consider space usage
DP array size is W+1, so auxiliary space is O(W). Total complexity includes both time and space, but time complexity is the main focus here.
Final Answer:
Option A -> Option A
Quick Check:
Time O(n*W) matches DP approach [OK]

Quick Trick: Time is product of coins and amount; space is dp array size [OK]

Common Mistakes:

MISTAKES

Confusing time with space complexity
Forgetting auxiliary space for dp array
Mistaking exponential complexity for DP

Trap Explanation:

PITFALL

Option D includes space but question asks for time complexity; option B correctly states time complexity.

Interviewer Note:

CONTEXT

Checks understanding of DP time complexity.

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More Dynamic Programming: Knapsack Quizzes

What is the time complexity of the space-optimized bottom-up dynamic programming solution for counting the number of ways to make change with n coins and amount W?

Step 1: Identify loops and their ranges

Step 2: Consider space usage

Final Answer:

Quick Check:

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