Dynamic Programming: Knapsack - Perfect Squares
What is the worst-case time complexity of the bottom-up dynamic programming solution to find the minimum number of perfect squares summing to
n?What is the worst-case time complexity of the bottom-up dynamic programming solution to find the minimum number of perfect squares summing to n?
medium📊 Complexity Theory Q5 of Q15
Step-by-Step Solution
Solution:
Step 1: Identify loops
The DP solution iterates from 1 tonand for eachi, checks all perfect squares ≤i.Step 2: Count perfect squares
Number of perfect squares ≤iis approximatelysqrt(i).Step 3: Calculate complexity
Total operations ≈ sum of sqrt(i) for i=1 to n, which is O(n * sqrt(n)).Final Answer:
Option A -> Option AQuick Check:
Nested loops: outer n, inner up to sqrt(n) [OK]
Quick Trick: DP loops over n and sqrt(n) perfect squares [OK]
Common Mistakes:
MISTAKES- Assuming O(n^2) without considering sqrt(n) inner loop
- Confusing with logarithmic complexities
- Ignoring the number of perfect squares
Trap Explanation:
PITFALL- Mistaking inner loop as linear instead of sqrt(n)
Interviewer Note:
CONTEXT- Tests understanding of DP time complexity analysis
Master "Perfect Squares" in Dynamic Programming: Knapsack
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