Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 1: Q31E (page 50)

Consider the problem of computing N!=1·2·3···N.

(a) If Nis an role="math" localid="1658397956489" n-bit number, how many bits long is N!, approximately ( inΘ(·)form)?

(b) Give an algorithm to compute N!and analyze its running time.

Short Answer

Expert verified

The running time complexity of the given algorithm can be analyzed as it runs for n iteration and that is multiplied and the final time complexity O(NlogN).

Step by step solution

01

Introduction

The factorial can be defined as the product of all the numbers between 1 and the given number including the given number. It can be represented as an integer with an exclamation mark.

02

Calculation of bits long with N!

We have,

N!=1·2·3···N.

It is known that,

Log2(N!)=Θ(Nlog2N)

So, N-factorial can be written as,

N!=Θ(Nlog2N)

It can be written as,Θ(n2n)

Given that, N is a n-bit number, then

N!isΘ(n2n)bitslong.

03

Algorithm to compute N!

Normal algorithm flow is shown below,

Factorial(N)If(N==0)Return1ElseReturnNfactorial(N-1)

The running time complexity of the given algorithm can be analyzing as it runs for n iteration and that are multiplied and the final time complexity O(NlogN).

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

Let[m]denote the set{0,1,…,m−1}. For each of the following families of hash functions, say whether or not it is universal, and determine how many random bits are needed to choose a function from the family.

(a) H={ha1,a2:a1,a2∈[m]}, wheremis a fixed prime and

ha1·ha1,a2(x1,x2)=a1x1+a2x2modm

Notice that each of these functions has signatureha1,a2:[m]2→[m]that is, it maps a pair of integers in[m]to a single integer in[m].

(b) His as before, except that nowm=2kis some fixed power of.2

(c) His the set of all functionsf:[m]→[m−1].

Find the inverse of:.20mod79,3mod62,21mod91,5mod23

Unlike a decreasing geometric series, the sum of the1,12,13,14,15,..... diverges; that is,∑i=1n1i=∞

It turns out that, for large n , the sum of the first n terms of this series can be well approximated as

∑i=1n1i≈Inn+y

where is natural logarithm (log base e=2.718...) and y is a particular constant 0.57721...... Show that

∑i=1n1i=θ(logn)

(Hint: To show an upper bound, decrease each denominator to the next power of two. For a lower bound, increase each denominator to the next power of 2 .)

Suppose you want to compute the nth Fibonacci number Fn , modulo an integer p. Can you find an efficient way to do this?

Quadratic residues. Fix a positive integer N. We say that a is a quadratic residue modulo N ifthere exists a such that a≡x2modN.
(a) Let N be an odd prime and be a non-zero quadratic residue modulo N. Show that there are exactly two values in{0,1,....,N-1} satisfying x2≡amodN.
(b) Show that if N is an odd prime, there are exactly(N+1)2 quadratic residues in {0,1,...,N-1}.
(c) Give an example of positive integers a and N such thatx2≡amodNhas more than two solutions in {0,1,...,N-1}.
See all solutions

Recommended explanations on Computer Science Textbooks

Data Structures

Read Explanation

Big Data

Read Explanation

Issues in Computer Science

Read Explanation

Computer Network

Read Explanation

Algorithms in Computer Science

Read Explanation

Functional Programming

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