functions, this exercise helps you to better understand its definition and properties. (a) Suppose n is the input size, we have the following commonly seen functions in complexity analysis: f1(n) = 1, f2(n) = log n, f3(n) = n, f4(n) = n log n, f5(n) = n^2, f6(n) = 2^n, f7(n) = n!, f8(n) = n^n. Intuitively, the growth rate of the functions satisfy 1 < log n < n < n log n < n^2 < 2^n < n! < n^n. Prove this is true. [Hint: You are expected to prove the following asymptotics by using the definition of big-O notation: 1 = O(log n), log n = O(n), n = O(n log n), n log n = O(n^2), n^2 = O(2^n), 2^n = O(n!), n! = O(n^n). Note: Chap 3.2 of our textbook provides some math facts in case you need.]

Computer Networking: A Top-Down Approach (7th Edition)
7th Edition
ISBN:9780133594140
Author:James Kurose, Keith Ross
Publisher:James Kurose, Keith Ross
Chapter1: Computer Networks And The Internet
Section: Chapter Questions
Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
icon
Related questions
Question

1. Big-O notation. We have learnt big-O notation to compare the growth rates of functions, this exercise helps you to better understand its definition and properties.
(a) Suppose n is the input size, we have the following commonly seen
functions in complexity analysis: f1(n) = 1, f2(n) = log n, f3(n) = n, f4(n) = n log n, f5(n) = n^2, f6(n) = 2^n, f7(n) = n!, f8(n) = n^n. Intuitively, the growth rate of the functions satisfy 1 < log n < n < n log n < n^2 < 2^n < n! < n^n. Prove
this is true.
[Hint: You are expected to prove the following asymptotics by using the definition of big-O notation: 1 = O(log n), log n = O(n), n = O(n log n), n log n = O(n^2), n^2 = O(2^n), 2^n = O(n!), n! = O(n^n). Note: Chap 3.2 of our textbook provides some math facts in case you need.]

Expert Solution
Step 1

The question has been answered in step2 

steps

Step by step

Solved in 2 steps

Blurred answer
Knowledge Booster
Computational Systems
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, computer-engineering and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Computer Networking: A Top-Down Approach (7th Edi…
Computer Networking: A Top-Down Approach (7th Edi…
Computer Engineering
ISBN:
9780133594140
Author:
James Kurose, Keith Ross
Publisher:
PEARSON
Computer Organization and Design MIPS Edition, Fi…
Computer Organization and Design MIPS Edition, Fi…
Computer Engineering
ISBN:
9780124077263
Author:
David A. Patterson, John L. Hennessy
Publisher:
Elsevier Science
Network+ Guide to Networks (MindTap Course List)
Network+ Guide to Networks (MindTap Course List)
Computer Engineering
ISBN:
9781337569330
Author:
Jill West, Tamara Dean, Jean Andrews
Publisher:
Cengage Learning
Concepts of Database Management
Concepts of Database Management
Computer Engineering
ISBN:
9781337093422
Author:
Joy L. Starks, Philip J. Pratt, Mary Z. Last
Publisher:
Cengage Learning
Prelude to Programming
Prelude to Programming
Computer Engineering
ISBN:
9780133750423
Author:
VENIT, Stewart
Publisher:
Pearson Education
Sc Business Data Communications and Networking, T…
Sc Business Data Communications and Networking, T…
Computer Engineering
ISBN:
9781119368830
Author:
FITZGERALD
Publisher:
WILEY