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What is the time complexity of the bottom-up dynamic programming solution for the minimum coin change problem with n coin denominations and target amount M?

medium📊 Complexity Q5 of Q15

Dynamic Programming: Knapsack - Coin Change (Minimum Coins)

What is the time complexity of the bottom-up dynamic programming solution for the minimum coin change problem with n coin denominations and target amount M?

AO(M^n)

BO(n + M)

CO(n * M)

DO(n^2 * M)

Step-by-Step Solution

Solution:

Step 1: Analyze loops
The solution uses a nested loop: outer loop over amounts (M), inner loop over coins (n).
Step 2: Calculate complexity
Each amount requires checking all n coins, so total operations are proportional to n * M.
Final Answer:
Option C -> Option C
Quick Check:
Nested loops over coins and amounts [OK]

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

Common Mistakes:

MISTAKES

Assuming linear or exponential complexity incorrectly
Confusing n and M roles
Ignoring nested loop structure

Trap Explanation:

PITFALL

Ignoring the nested loops leads to underestimating complexity

Interviewer Note:

CONTEXT

Tests understanding of algorithmic time complexity for DP solutions

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