How can I identify which case to use? Such that after knowing which case to use, how to use that case to find the complexity running time? What is the complexity of T(n) = 4T(n/2) + n (choose one from the options below):1. Θ(n³)2. Θ(n²)3. Θ(n log n)4. None of the above**Master Theorem**T(n) = aT(n/b) + f(n)- **Case 1:** Θ(n^logₐb), g(a)=logₐb, if f(n) = O(n^(logₐb)⁻ε), ε < 1- **Case 2:** Θ(n^logₐb log n), g(a)=logₐb, if f(n) = Θ(n^logₐb)- **Case 3:** Θ(f(n)), if f(n) = Ω(n^(logₐb)+ε), g(a)=logₐb, ε > 0, and there exists a constant c < 1 such that a*f(n/b) < c*f(n)The correct answer is option 4.**Answer Options:**- O 2- O 3- ● 4- O 1

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Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

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How can I identify which case to use? Such that after knowing which case to use, how to use that case to find the complexity running time?

What is the complexity of T(n) = 4T(n/2) + n (choose one from the options below):

1. Θ(n³)
2. Θ(n²)
3. Θ(n log n)
4. None of the above

**Master Theorem**

T(n) = aT(n/b) + f(n)

- **Case 1:** Θ(n^logₐb), g(a)=logₐb, if f(n) = O(n^(logₐb)⁻ε), ε < 1
- **Case 2:** Θ(n^logₐb log n), g(a)=logₐb, if f(n) = Θ(n^logₐb)
- **Case 3:** Θ(f(n)), if f(n) = Ω(n^(logₐb)+ε), g(a)=logₐb, ε > 0, and there exists a constant c < 1 such that a*f(n/b) < c*f(n)

The correct answer is option 4.

**Answer Options:**

- O 2
- O 3
- ● 4
- O 1

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Time Complexity:-

The volume of operations required by an algorithm to complete a job determines its temporal complexity (considering that each operation takes the same amount of time). According to the time complexity, the algorithm that completes the task with the fewest operations is regarded as the most effective one. We are excluding your operating system and hardware from this discussion because they have an impact on the space and temporal complexity as well, but they are relevant nonetheless.

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