What is the time complexity of the space-optimized bottom-up dynamic programming solution for the coin change minimum coins problem, given n coins and amount S?

medium🪤 Complexity Trap Q13 of Q15

Dynamic Programming: Knapsack - Coin Change (Minimum Coins)

AO(n * S) because for each amount up to S, all n coins are checked

BO(n^S) due to exploring all combinations recursively

CO(S^2) since the dp array is updated for each sub-amount multiple times

DO(n + S) as the algorithm iterates once over coins and once over amounts

Step-by-Step Solution

Solution:

Step 1: Identify loops in the bottom-up DP
Outer loop runs from 1 to S (amount), inner loop runs over n coins.
Step 2: Calculate total operations
Each dp[i] is updated by checking all n coins, so total operations = n * S.
Final Answer:
Option A -> Option A
Quick Check:
Nested loops over amount and coins -> O(n * S) [OK]

Quick Trick: Nested loops over amount and coins yield O(n*S) [OK]

Common Mistakes:

MISTAKES

Confusing recursion exponential time with DP
Assuming quadratic due to dp updates

Trap Explanation:

PITFALL

Candidates often confuse brute force exponential time with DP or misinterpret dp array updates as quadratic.

Interviewer Note:

CONTEXT

Tests understanding of DP time complexity versus naive recursion.

Master "Coin Change (Minimum Coins)" in Dynamic Programming: Knapsack

3 interactive learning modes - each teaches the same concept differently

Try It Solution Trace

Want More Practice?

15+ quiz questions · All difficulty levels · Free

Free Signup - Practice All Questions

More Dynamic Programming: Knapsack Quizzes

What is the time complexity of the space-optimized bottom-up dynamic programming solution for the coin change minimum coins problem, given n coins and amount S?

Step 1: Identify loops in the bottom-up DP

Step 2: Calculate total operations

Final Answer:

Quick Check:

Want More Practice?