Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 1: Problem 17

Using the Properties of Order in Section \(1.4 .2,\) show that \\[ 5 n^{5}+4 n^{4}+6 n^{3}+2 n^{2}+n+7 \in \Theta\left(n^{5}\right) \\]

Short Answer

Expert verified
The function \(5n^{5} + 4n^{4} + 6n^{3} + 2n^{2} + n + 7\) is indeed in the complexity class Theta(n^5) as it was shown to be both \(O(n^{5})\) and \(Omega(n^{5})\).

Step by step solution

01

Prove Big O(n^5)

We start by proving the function is in Big O of \(n^{5}\), i.e., f(n) is less than or equal to \(c \cdot n^{5}\) for all \(n \geq n_0\) where c > 0 and \(n_0 \geq 1\) are constants. To show this, consider the function \(f(n) = 5n^{5} + 4n^{4} + 6n^{3} + 2n^{2} + n + 7\). One can note that: For \(n \geq 1\), this function is less than or equal to \(5n^{5} + 4n^{5} + 6n^{5} + 2n^{5} + n^{5} + 7n^{5}\), which equals to \(25n^{5}\). Therefore we can say that \(f(n) = O(n^{5})\)
02

Prove Big Omega(n^5)

We also need to show that the function is in Big Omega of \(n^{5}\), i.e. f(n) is more than or equal to \(c \cdot n^{5}\) for all \(n geq n_0\) where c > 0 and \(n_0 \geq 1\) are constants. Consider the function \(f(n) = 5n^{5} + 4n^{4} + 6n^{3} + 2n^{2} + n + 7\). One can note that: For \(n \geq 1\), this function is more than or equal to \(5n^{5}\) which is indeed bigger than or equal to \(c \cdot n^{5}\) when \(c = 5\). Therefore we can say \(f(n) = \Omega(n^{5})\)
03

Conclusion

Since the function \(f(n) = 5n^{5} + 4n^{4} + 6n^{3} + 2n^{2} + n + 7\) is both \(O(n^{5})\) and \(Omega(n^{5})\), we can conclude that \(f(n) \in Theta(n^{5})\), meaning that the function grows at the same rate as \(n^{5}\) in the limit as n approaches infinity.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Write a linear-time algorithm that sorts \(n\) distinct integers ranging from 1 to 500 , inclusive. (Hint: Use a 500 -element array.)

Algorithm 1.7 (nth Fibonacci Term, Iterative) is clearly linear in \(n,\) but is it a linear-time algorithm? In Section 1.3 .1 we defined the input size as the size of the input. In the case of the \(nth\) Fibonacci term, \(n\) is the input, and the number of bits it takes to encode \(n\) could be used as the input size. Using this measure the size of 64 is \(\lg 64=6,\) and the size of 1024 is \(\lg 1024=\) 10\. Show that Algorithm 1.7 is exponential-time in terms of its input size. Show further that any algorithm for computing the \(nth\) Fibonacci term must be an exponential-time algorithm because the size of the output is exponential in the input size. See Section 9.2 for a related discussion of the input size.

Group the following functions by complexity category. $$n \ln n \quad(\lg n)^{2} \quad 5 n^{2}+7 n \quad n^\frac{5}{2} $$ $$n ! \quad 2^{n !} \quad 4^{n} \quad n^{n} \quad n^{n}+\ln n$$ $$5^{\text {Ig n}} \quad \lg (n !) \quad(\lg n) ! \quad \sqrt{n} \quad e^{n} \quad 8 n+12 \quad 10^{n}+n^{20}$$

What is the time complexity \(T(n)\) of the nested loops below? For simplicity, you may assume that \(n\) is a power of \(2 .\) That is, \(n=2^{k}\) for some positive integer \(k\) \(:\) \\[ \begin{array}{} \text { for }(i=1 ; i<=n, i++)\\} \\ \ \begin{array}{} j=n \\ \text { while }(j>=1)\\{ \end{array} \end{array} \\] < body of the while loop> \(\quad\) I/ Needs \(\Theta(1)\) \\[ j=\lfloor j / 2\rfloor \\] } }

Write an Insertion Sort algorithm that uses Binary Search to find the position where the next insertion should take place.
See all solutions

Recommended explanations on Computer Science Textbooks

Data Representation in Computer Science

Read Explanation

Big Data

Read Explanation

Functional Programming

Read Explanation

Issues in Computer Science

Read Explanation

Problem Solving Techniques

Read Explanation

Theory of Computation

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free