a. What is the recurrence relation? b. Solve the recurrence relation from (a) in terms of n and show the O(
a. What is the recurrence relation? b. Solve the recurrence relation from (a) in terms of n and show the O(
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
Related questions
Question
![### Algorithm Analysis Exercise
**Problem Statement:**
1. Consider the following program:
```plaintext
Algorithm recursive (int n){
IF n <= 3 then
return (0);
ELSE
recursive(n/3);
recursive(n/3);
recursive(n/3);
FOR i = 1 to n
do something;
ENDFOR
ENDIF
}
```
**Questions:**
a. What is the recurrence relation?
b. Solve the recurrence relation from (a) in terms of *n* and show the O( ) notation.
**Explanation:**
The provided algorithm is a recursive function that executes different operations based on the value of *n*. If *n* is less than or equal to 3, the function terminates by returning 0. If *n* is greater than 3, the function calls itself three times with a reduced argument, `n/3`, and performs a loop that runs *n* times to "do something."
**Recurrence Relation Analysis:**
- **Base Case:** When *n* is less than or equal to 3, the operation completes in constant time *O(1)*.
- **Recursive Case:**
- Calls the recursive function thrice for inputs reduced to *n/3*.
- Executes an additional loop that operates in *O(n)* time.
The recurrence relation, therefore, can be expressed as:
\[ T(n) = 3T(n/3) + O(n) \]
**Solution Approach:**
To solve this recurrence relation, you might employ the Master Theorem, which provides a way to solve recurrences of the form:
\[ T(n) = aT(n/b) + f(n) \]
In this example, we identify:
- \( a = 3 \): Number of recursive calls
- \( b = 3 \): Factor by which the subproblem size is reduced
- \( f(n) = O(n) \): Additional cost outside the recursive calls
Using the Master Theorem's criteria:
- Compare \( f(n) \) = \( O(n) \) with \( n^{\log_b a} = n^{\log_3 3} = n^1 \)
Here, \( f(n) = \Theta(n^1) \), fitting the criteria for \( f(n) = \Theta(n^{\log_b a}) \). According to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4735b9fe-1f62-4c1c-b375-80e5700bde0c%2F04315f69-97f9-45bb-a2fe-c9fc0a056015%2Fhuzpmss_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Algorithm Analysis Exercise
**Problem Statement:**
1. Consider the following program:
```plaintext
Algorithm recursive (int n){
IF n <= 3 then
return (0);
ELSE
recursive(n/3);
recursive(n/3);
recursive(n/3);
FOR i = 1 to n
do something;
ENDFOR
ENDIF
}
```
**Questions:**
a. What is the recurrence relation?
b. Solve the recurrence relation from (a) in terms of *n* and show the O( ) notation.
**Explanation:**
The provided algorithm is a recursive function that executes different operations based on the value of *n*. If *n* is less than or equal to 3, the function terminates by returning 0. If *n* is greater than 3, the function calls itself three times with a reduced argument, `n/3`, and performs a loop that runs *n* times to "do something."
**Recurrence Relation Analysis:**
- **Base Case:** When *n* is less than or equal to 3, the operation completes in constant time *O(1)*.
- **Recursive Case:**
- Calls the recursive function thrice for inputs reduced to *n/3*.
- Executes an additional loop that operates in *O(n)* time.
The recurrence relation, therefore, can be expressed as:
\[ T(n) = 3T(n/3) + O(n) \]
**Solution Approach:**
To solve this recurrence relation, you might employ the Master Theorem, which provides a way to solve recurrences of the form:
\[ T(n) = aT(n/b) + f(n) \]
In this example, we identify:
- \( a = 3 \): Number of recursive calls
- \( b = 3 \): Factor by which the subproblem size is reduced
- \( f(n) = O(n) \): Additional cost outside the recursive calls
Using the Master Theorem's criteria:
- Compare \( f(n) \) = \( O(n) \) with \( n^{\log_b a} = n^{\log_3 3} = n^1 \)
Here, \( f(n) = \Theta(n^1) \), fitting the criteria for \( f(n) = \Theta(n^{\log_b a}) \). According to
Expert Solution
Step 1
1. Recurrence Relation can be defined as
"an equation that recursively defines a progression or multi-dimensional array of values, in which one or more primary terms are specified; each additional term of the sequence or array is distinct as a function of the previous terms."
Step by step
Solved in 2 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-science and related others by exploring similar questions and additional content below.Recommended textbooks for you
Database System Concepts
Computer Science
ISBN:
9780078022159
Author:
Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:
McGraw-Hill Education
Starting Out with Python (4th Edition)
Computer Science
ISBN:
9780134444321
Author:
Tony Gaddis
Publisher:
PEARSON
Digital Fundamentals (11th Edition)
Computer Science
ISBN:
9780132737968
Author:
Thomas L. Floyd
Publisher:
PEARSON
Database System Concepts
Computer Science
ISBN:
9780078022159
Author:
Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:
McGraw-Hill Education
Starting Out with Python (4th Edition)
Computer Science
ISBN:
9780134444321
Author:
Tony Gaddis
Publisher:
PEARSON
Digital Fundamentals (11th Edition)
Computer Science
ISBN:
9780132737968
Author:
Thomas L. Floyd
Publisher:
PEARSON
C How to Program (8th Edition)
Computer Science
ISBN:
9780133976892
Author:
Paul J. Deitel, Harvey Deitel
Publisher:
PEARSON
Database Systems: Design, Implementation, & Manag…
Computer Science
ISBN:
9781337627900
Author:
Carlos Coronel, Steven Morris
Publisher:
Cengage Learning
Programmable Logic Controllers
Computer Science
ISBN:
9780073373843
Author:
Frank D. Petruzella
Publisher:
McGraw-Hill Education