Dynamic Programming: Knapsack - Perfect Squares
What is the time complexity of the bottom-up dynamic programming solution for the Perfect Squares problem using the unbounded knapsack pattern, where
n is the input number?What is the time complexity of the bottom-up dynamic programming solution for the Perfect Squares problem using the unbounded knapsack pattern, where n is the input number?
medium🪤 Complexity Trap Q13 of Q15
Step-by-Step Solution
Solution:
Step 1: Identify loops in the algorithm
The outer loop runs from 1 to n (O(n)). The inner loop iterates over all perfect squares less than or equal to i, which is approximately O(sqrt(n)) because the largest square root is about sqrt(n).Step 2: Combine complexities
Multiplying outer and inner loops gives O(n * sqrt(n)). Other options either overestimate (O(n^2)) or underestimate (O(n log n), O(sqrt(n) log n)) the complexity.Final Answer:
Option D -> Option DQuick Check:
Nested loops: n and sqrt(n) -> O(n * sqrt(n)) [OK]
Quick Trick: Inner loop runs up to sqrt(n), outer up to n [OK]
Common Mistakes:
MISTAKES- Assuming inner loop is O(n)
- Confusing recursion stack space with DP space
Trap Explanation:
PITFALL- Many think inner loop is O(n), leading to O(n^2), but it only iterates over squares ≤ i, about sqrt(n).
Interviewer Note:
CONTEXT- Checks if candidate understands loop bounds and complexity of nested loops in DP.
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