Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 1: Problem 30

Explain in English what functions are in following sets. a. \(n^{O(1)}\) b. \(O\left(n^{O(1)}\right)\) c. \(O\left(O\left(n^{O(1)}\right)\right)\)

Short Answer

Expert verified
In short, \(n^{O(1)}\), \(O\left(n^{O(1)}\right)\) and \(O\left(O\left(n^{O(1)}\right)\right)\) are all sets of functions with growth rates bounded by some function of \(n\) raised to a constant power, i.e., they represent polynomial time complexity.

Step by step solution

01

Describe the function in \(n^{O(1)}\)

The set \(n^{O(1)}\) is the set of all functions whose growth rate can be bounded by some function of \(n\$ raised to a constant power. In simple terms, it refers to functions that run in polynomial time. In Big O notation, \(O(1)\) represents a constant amount of work that does not vary with the number of data items. Therefore, any function in the set is one whose growth rate is under the limit of a polynomial.
02

Describe the function in \(O\left(n^{O(1)}\right)\)

The set \(O\left(n^{O(1)}\right)\) is essentially equivalent to \(n^{O(1)}\), referring to the set of functions whose growth rates can be bounded by a function that grows polynomially. It represents the functions whose rate of growth is polynomial.
03

Describe the function in \(O\left(O\left(n^{O(1)}\right)\right)\)

The nested \(O\left(O\left(n^{O(1)}\right)\right)\) notation is also equivalent to the others. In Big O notation, nesting does not make a difference, because it is a set notation - it just means the output of one function is the input of another. So it is equivalent to the set of functions whose growth rate is also bounded by a polynomial.

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

Under what circumstances, when a searching operation is needed, would sequential Search (Algorithm 1.1) not be appropriate?

Write an algorithm that determines whether or not an almost complete binary tree is a heap.

Group the following function by complexity category. $$\begin{aligned}&n \ln n \quad(\lg n)^{2} \quad 5 n^{2}+7 n \quad n^{5 / 2}\\\&n ! \quad 2^{n !} \quad 4^{n} \quad n^{n} \quad n^{n}+\ln n\\\&5^{\lg n} \lg (n !) \quad(\lg n) ! \quad \sqrt{n} \quad e^{n} \quad 8 n+12 \quad 10^{n}+n^{20}\end{aligned}$$

Write an algorithm that finds the \(m\) smallest numbers in a list of \(n\) numbers.

Write an Insertion Sort algorithm (Insertion Sort is discussed in Section 7.2 ) that uses Binary Search to find the position where the next insertion should take place.
See all solutions

Recommended explanations on Computer Science Textbooks

Computer Network

Read Explanation

Databases

Read Explanation

Data Structures

Read Explanation

Functional Programming

Read Explanation

Algorithms in Computer Science

Read Explanation

Computer Systems

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