[Solved] Consider a bottom-up dynamic programming solution that uses a 1D array to count the number of ways to make change for an amount W using n... What is the overall time complexity of this approach? | Dynamic Programming: Knapsack

Dynamic Programming: Knapsack - Number of Ways to Make Change

Consider a bottom-up dynamic programming solution that uses a 1D array to count the number of ways to make change for an amount W using n coin denominations. What is the overall time complexity of this approach?

AO(W^n)

BO(n + W)

CO(n * log W)

DO(n * W)

Step-by-Step Solution

Solution:

Step 1: Understand the DP structure
The solution iterates over each coin and for each coin iterates over amounts from coin value up to W.
Step 2: Calculate iterations
For each of the n coins, it processes up to W amounts, resulting in n * W operations.
Final Answer:
Option D -> Option D
Quick Check:
Nested loops over coins and amounts [OK]

Quick Trick: Nested loops over coins and amounts give O(n*W) [OK]

Common Mistakes:

MISTAKES

Assuming time is O(n + W) by adding instead of multiplying
Confusing with exponential complexity due to recursion
Thinking sorting coins affects complexity

Trap Explanation:

PITFALL

Adding complexities instead of multiplying leads to underestimating time

Interviewer Note:

CONTEXT

Tests understanding of DP time complexity for coin change problem

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More Dynamic Programming: Knapsack Quizzes

What is the overall time complexity of this approach?

Step 1: Understand the DP structure

Step 2: Calculate iterations

Final Answer:

Quick Check:

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