Analyze the following recurrences and show their time complexity functions using (1) iteration method and (2) Master Theorem.
Analyze the following recurrences and show their time complexity functions using (1) iteration method and (2) Master Theorem.
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
Section: Chapter Questions
Problem 1PE
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Analyze the following recurrences and show their time complexity functions using (1) iteration method and (2) Master Theorem.
![The following are examples of recurrence relations, which are often used to analyze the computational complexity of recursive algorithms. These equations express the running time of an algorithm in terms of the running time of smaller instances of the same problem.
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**A4. Recurrence Relation:**
\[ T(n) = 2T(n - 1) + 1 \]
This relation suggests that the time complexity for an input size \( n \) is twice the time complexity for an input size \( n-1 \), plus a constant time operation.
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**A5. Recurrence Relation:**
\[ T(n) = 4T\left(\frac{n}{2}\right) + n \log n \]
Here, \( T(n) \) is expressed in terms of four instances of the problem with half the size (\( \frac{n}{2} \)), plus an additional term \( n \log n \), which indicates a logarithmic time complexity proportional to \( n \).
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**A6. Recurrence Relation:**
\[ T(n) = 3T\left(\frac{n}{5}\right) + n \log n \]
In this case, \( T(n) \) is determined by three smaller subproblems of size \( \frac{n}{5} \), and an additional term \( n \log n \), showing a logarithmic complexity as a function of \( n \).
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**A7. Recurrence Relation:**
\[ T(n) = 2T\left(\frac{n}{3}\right) + n^2 \]
This relation shows that the time complexity for input size \( n \) depends on two subproblems of one-third the size (\( \frac{n}{3} \)), with an additional quadratic term \( n^2 \).
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These recurrence relations are vital in understanding the theoretical performance and efficiency of recursive algorithms in computer science and mathematical analysis. They allow us to predict how an algorithm will behave as the size of the input grows, assisting in decision-making about algorithm selection and optimization.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba28fbe0-fc04-46bc-bd01-9a9731588fb8%2Fb2dd72dc-e05b-459f-aece-d3389e5ca5fa%2Fmmaobx5_processed.png&w=3840&q=75)
Transcribed Image Text:The following are examples of recurrence relations, which are often used to analyze the computational complexity of recursive algorithms. These equations express the running time of an algorithm in terms of the running time of smaller instances of the same problem.
---
**A4. Recurrence Relation:**
\[ T(n) = 2T(n - 1) + 1 \]
This relation suggests that the time complexity for an input size \( n \) is twice the time complexity for an input size \( n-1 \), plus a constant time operation.
---
**A5. Recurrence Relation:**
\[ T(n) = 4T\left(\frac{n}{2}\right) + n \log n \]
Here, \( T(n) \) is expressed in terms of four instances of the problem with half the size (\( \frac{n}{2} \)), plus an additional term \( n \log n \), which indicates a logarithmic time complexity proportional to \( n \).
---
**A6. Recurrence Relation:**
\[ T(n) = 3T\left(\frac{n}{5}\right) + n \log n \]
In this case, \( T(n) \) is determined by three smaller subproblems of size \( \frac{n}{5} \), and an additional term \( n \log n \), showing a logarithmic complexity as a function of \( n \).
---
**A7. Recurrence Relation:**
\[ T(n) = 2T\left(\frac{n}{3}\right) + n^2 \]
This relation shows that the time complexity for input size \( n \) depends on two subproblems of one-third the size (\( \frac{n}{3} \)), with an additional quadratic term \( n^2 \).
---
These recurrence relations are vital in understanding the theoretical performance and efficiency of recursive algorithms in computer science and mathematical analysis. They allow us to predict how an algorithm will behave as the size of the input grows, assisting in decision-making about algorithm selection and optimization.
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