Data Structures and Algorithms in Java
Data Structures and Algorithms in Java
6th Edition
ISBN: 9781119278023
Author: Michael T. Goodrich; Roberto Tamassia; Michael H. Goldwasser
Publisher: Wiley Global Education US
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Chapter 4, Problem 1R

Explanation of Solution

Graph of functions using logarithmic scale:

8n:

Let the function be:

f(n)=8n

Take the logarithm on both the sides of the above function as follows:

log f(n) = log 8n

logf(n)=logn+log8        (1)

According to the given data, let us take log f(n)=Y , log n= X . Substitute the values in Equation (1):

Y=X+log8        (2)

Thus, from the above Equation (2), the value of Yinterval= log 8 , Xinterval =log 8

The following is the graph representation corresponding to the values Yinterval= log 8, Xinterval =log 8 on respective x-axis and y-axis respectively.

Data Structures and Algorithms in Java, Chapter 4, Problem 1R , additional homework tip  1

4n logn:

Let the function be:

f(n)=4nlogn

Take the logarithm on both the sides of the above function as follows:

log f(n) = log 4nlogn

logf(n)=log4+log(nlogn)

logf(n)=log4+logn+log(logn)        (3)

According to the given data, let us take log f(n)=Y , log n= X . Substitute the values in Equation (3)

Y=X+log(X)+log4        (4)

The following is the graph representation corresponding for the function f(n)=4nlogn on respective x-axis and y-axis respectively.

Data Structures and Algorithms in Java, Chapter 4, Problem 1R , additional homework tip  2

2n2:

Let the function be:

f(n)=2n2

Take the logarithm on both the sides of the above function as follows:

log f(n) = log 2n2

logf(n)=log2+logn2

logf(n)=log2+2logn        (5)

According to the given data, let us take log f(n)=Y , log n= X . Substitute the values in Equation (5)

Y=2X+log2        (6)

Thus, from the above Equation (6), the value of Yinterval= log 2 while substituting X =0, Xinterval =log 22 while substituting Y=0.

The following is the graph representation corresponding to the values Yinterval= log 2, Xinterval =log 22 on respective x-axis and y-axis respectively.

Data Structures and Algorithms in Java, Chapter 4, Problem 1R , additional homework tip  3

n3:

Let the function be:

f(n)=n3

Take the logarithm on both the sides of the above function as follows:

log f(n) = log n3

logf(n)=3logn        (7)

According to the given data, let us take log f(n)=Y , log n= X . Substitute the values in Equation (6)

Y=3X        (8)

The following is the graph representation corresponding to the value on respective x-axis and y-axis respectively.

Data Structures and Algorithms in Java, Chapter 4, Problem 1R , additional homework tip  4

2n:

Let the function be:

f(n)=2n

Take the logarithm on both the sides of the above function as follows:

log f(n) = log 2n

logf(n)=nlog2

logf(n)=elognlog2        (7)

According to the given data, let us take log f(n)=Y , log n= X . Substitute the values in Equation (6)

Y=exlog2        (8)

The following is the graph representation corresponding to the value on respective x-axis and y-axis respectively.

Data Structures and Algorithms in Java, Chapter 4, Problem 1R , additional homework tip  5

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