Home > Store

From Mathematics to Generic Programming

Register your product to gain access to bonus material or receive a coupon.

From Mathematics to Generic Programming

Best Value Purchase

Book + eBook Bundle

  • Your Price: $41.51
  • List Price: $67.18
  • We're temporarily out of stock, but order now and we'll send it to you later.
  • Includes EPUB and PDF
  • About eBook Formats
  • This eBook includes the following formats, accessible from your Account page after purchase:

    ePub EPUB The open industry format known for its reflowable content and usability on supported mobile devices.

    Adobe Reader PDF The popular standard, used most often with the free Acrobat® Reader® software.

    This eBook requires no passwords or activation to read. We customize your eBook by discreetly watermarking it with your name, making it uniquely yours.

More Purchase Options

Book

  • Your Price: $31.99
  • List Price: $39.99
  • We're temporarily out of stock, but order now and we'll send it to you later.

eBook (Watermarked)

  • Your Price: $21.75
  • List Price: $27.19
  • Includes EPUB and PDF
  • About eBook Formats
  • This eBook includes the following formats, accessible from your Account page after purchase:

    ePub EPUB The open industry format known for its reflowable content and usability on supported mobile devices.

    Adobe Reader PDF The popular standard, used most often with the free Acrobat® Reader® software.

    This eBook requires no passwords or activation to read. We customize your eBook by discreetly watermarking it with your name, making it uniquely yours.

About

Features

  • Practical insights for understanding key algorithms and efficiently implementing them in the broadest possible set of applications
  • Journeys through three key algorithms, revealing how they have played a profound role in the development of mathematics, and how they remain crucial to the work of modern programmers
  • By Alexander Stepanov and Daniel Rose: the ideal complement to Stepanov's classic Elements of Programming

Description

  • Copyright 2015
  • Dimensions: 6" x 9"
  • Pages: 320
  • Edition: 1st
  • Book
  • ISBN-10: 0-321-94204-3
  • ISBN-13: 978-0-321-94204-3

In this substantive yet accessible book, pioneering software designer Alexander Stepanov and his colleague Daniel Rose illuminate the principles of generic programming and the mathematical concept of abstraction on which it is based, helping you write code that is both simpler and more powerful.

If you’re a reasonably proficient programmer who can think logically, you have all the background you’ll need. Stepanov and Rose introduce the relevant abstract algebra and number theory with exceptional clarity. They carefully explain the problems mathematicians first needed to solve, and then show how these mathematical solutions translate to generic programming and the creation of more effective and elegant code. To demonstrate the crucial role these mathematical principles play in many modern applications, the authors show how to use these results and generalized algorithms to implement a real-world public-key cryptosystem.

As you read this book, you’ll master the thought processes necessary for effective programming and learn how to generalize narrowly conceived algorithms to widen their usefulness without losing efficiency. You’ll also gain deep insight into the value of mathematics to programming—insight that will prove invaluable no matter what programming languages and paradigms you use.

You will learn about

  • How to generalize a four thousand-year-old algorithm, demonstrating indispensable lessons about clarity and efficiency
  • Ancient paradoxes, beautiful theorems, and the productive tension between continuous and discrete
  • A simple algorithm for finding greatest common divisor (GCD) and modern abstractions that build on it
  • Powerful mathematical approaches to abstraction
  • How abstract algebra provides the idea at the heart of generic programming
  • Axioms, proofs, theories, and models: using mathematical techniques to organize knowledge about your algorithms and data structures
  • Surprising subtleties of simple programming tasks and what you can learn from them
  • How practical implementations can exploit theoretical knowledge

Sample Content

Online Sample Chapter

From Mathematics to Generic Programming: The First Algorithm

Sample Pages

Download the sample pages (includes Chapter 3 and Index)

Table of Contents

Acknowledgments ix

About the Authors xi

Authors’ Note xiii

Chapter 1: What This Book Is About 1

1.1 Programming and Mathematics 2

1.2 A Historical Perspective 2

1.3 Prerequisites 3

1.4 Roadmap 4

Chapter 2: The First Algorithm 7

2.1 Egyptian Multiplication 8

2.2 Improving the Algorithm 11

2.3 Thoughts on the Chapter 15

Chapter 3: Ancient Greek Number Theory 17

3.1 Geometric Properties of Integers 17

3.2 Sifting Primes 20

3.3 Implementing and Optimizing the Code 23

3.4 Perfect Numbers 28

3.5 The Pythagorean Program 32

3.6 A Fatal Flaw in the Program 34

3.7 Thoughts on the Chapter 38

Chapter 4: Euclid’s Algorithm 41

4.1 Athens and Alexandria 41

4.2 Euclid’s Greatest Common Measure Algorithm 45

4.3 A Millennium without Mathematics 50

4.4 The Strange History of Zero 51

4.5 Remainder and Quotient Algorithms 53

4.6 Sharing the Code 57

4.7 Validating the Algorithm 59

4.8 Thoughts on the Chapter 61

Chapter 5: The Emergence of Modern Number Theory 63

5.1 Mersenne Primes and Fermat Primes 63

5.2 Fermat’s Little Theorem 69

5.3 Cancellation 72

5.4 Proving Fermat’s Little Theorem 77

5.5 Euler’s Theorem 79

5.6 Applying Modular Arithmetic 83

5.7 Thoughts on the Chapter 84

Chapter 6: Abstraction in Mathematics 85

6.1 Groups 85

6.2 Monoids and Semigroups 89

6.3 Some Theorems about Groups 92

6.4 Subgroups and Cyclic Groups 95

6.5 Lagrange’s Theorem 97

6.6 Theories and Models 102

6.7 Examples of Categorical and Non-categorical Theories 104

6.8 Thoughts on the Chapter 107

Chapter 7: Deriving a Generic Algorithm 111

7.1 Untangling Algorithm Requirements 111

7.2 Requirements on A 113

7.3 Requirements on N 116

7.4 New Requirements 118

7.5 Turning Multiply into Power 119

7.6 Generalizing the Operation 121

7.7 Computing Fibonacci Numbers 124

7.8 Thoughts on the Chapter 127

Chapter 8: More Algebraic Structures 129

8.1 Stevin, Polynomials, and GCD 129

8.2 Göttingen and German Mathematics 135

8.3 Noether and the Birth of Abstract Algebra 140

8.4 Rings 142

8.5 Matrix Multiplication and Semirings 145

8.6 Application: Social Networks and Shortest Paths 147

8.7 Euclidean Domains 150

8.8 Fields and Other Algebraic Structures 151

8.9 Thoughts on the Chapter 152

Chapter 9: Organizing Mathematical Knowledge 155

9.1 Proofs 155

9.2 The First Theorem 159

9.3 Euclid and the Axiomatic Method 161

9.4 Alternatives to Euclidean Geometry 164

9.5 Hilbert’s Formalist Approach 167

9.6 Peano and His Axioms 169

9.7 Building Arithmetic 173

9.8 Thoughts on the Chapter 176

Chapter 10: Fundamental Programming Concepts 177

10.1 Aristotle and Abstraction 177

10.2 Values and Types 180

10.3 Concepts 181

10.4 Iterators 184

10.5 Iterator Categories, Operations, and Traits 185

10.6 Ranges 188

10.7 Linear Search 190

10.8 Binary Search 191

10.9 Thoughts on the Chapter 196

Chapter 11: Permutation Algorithms 197

11.1 Permutations and Transpositions 197

11.2 Swapping Ranges 201

11.3 Rotation 204

11.4 Using Cycles 207

11.5 Reverse 212

11.6 Space Complexity 215

11.7 Memory-Adaptive Algorithms 216

11.8 Thoughts on the Chapter 217

Chapter 12: Extensions of GCD 219

12.1 Hardware Constraints and a More Efficient Algorithm 219

12.2 Generalizing Stein’s Algorithm 222

12.3 Bézout’s Identity 225

12.4 Extended GCD 229

12.5 Applications of GCD 234

12.6 Thoughts on the Chapter 234

Chapter 13: A Real-World Application 237

13.1 Cryptology 237

13.2 Primality Testing 240

13.3 The Miller-Rabin Test 243

13.4 The RSA Algorithm: How and Why It Works 245

13.5 Thoughts on the Chapter 248

Chapter 14: Conclusions 249

Further Reading 251

Appendix A: Notation 257

Appendix B: Common Proof Techniques 261

B.1 Proof by Contradiction 261

B.2 Proof by Induction 262

B.3 The Pigeonhole Principle 263

Appendix C: C++ for Non-C++ Programmers 265

C.1 Template Functions 265

C.2 Concepts 266

C.3 Declaration Syntax and Typed Constants 267

C.4 Function Objects 268

C.5 Preconditions, Postconditions, and Assertions 269

C.6 STL Algorithms and Data Structures 269

C.7 Iterators and Ranges 270

C.8 Type Aliases and Type Functions with using in C++11 272

C.9 Initializer Lists in C++11 272

C.10 Lambda Functions in C++11 272

C.11 A Note about inline 273

Bibliography 275

Index 281

Updates

Submit Errata

More Information

InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.

Overview


Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

Collection and Use of Information


To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.

Surveys

Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.

Newsletters

If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

Other Collection and Use of Information


Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

Cookies and Related Technologies

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

Do Not Track

This site currently does not respond to Do Not Track signals.

Security


Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.

Children


This site is not directed to children under the age of 13.

Marketing


Pearson may send or direct marketing communications to users, provided that

  • Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
  • Such marketing is consistent with applicable law and Pearson's legal obligations.
  • Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
  • Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

Correcting/Updating Personal Information


If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.

Choice/Opt-out


Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

Sale of Personal Information


Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

Supplemental Privacy Statement for California Residents


California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

Sharing and Disclosure


Pearson may disclose personal information, as follows:

  • As required by law.
  • With the consent of the individual (or their parent, if the individual is a minor)
  • In response to a subpoena, court order or legal process, to the extent permitted or required by law
  • To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
  • In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
  • To investigate or address actual or suspected fraud or other illegal activities
  • To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
  • To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
  • To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.

Links


This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.

Requests and Contact


Please contact us about this Privacy Notice or if you have any requests or questions relating to the privacy of your personal information.

Changes to this Privacy Notice


We may revise this Privacy Notice through an updated posting. We will identify the effective date of the revision in the posting. Often, updates are made to provide greater clarity or to comply with changes in regulatory requirements. If the updates involve material changes to the collection, protection, use or disclosure of Personal Information, Pearson will provide notice of the change through a conspicuous notice on this site or other appropriate way. Continued use of the site after the effective date of a posted revision evidences acceptance. Please contact us if you have questions or concerns about the Privacy Notice or any objection to any revisions.

Last Update: November 17, 2020