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from Mathematics to Generic Programming.pdf

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    About This eBook
    Title Page
    Copyright Page
    Contents
    Acknowledgments
    About the Authors
    Authors’ Note
    1. What This Book Is About
    1.1 Programming and Mathematics
    1.2 A Historical Perspective
    1.3 Prerequisites
    1.4 Roadmap
    2. The First Algorithm
    2.1 Egyptian Multiplication
    2.2 Improving the Algorithm
    2.3 Thoughts on the Chapter
    3. Ancient Greek Number Theory
    3.1 Geometric Properties of Integers
    3.2 Sifting Primes
    3.3 Implementing and Optimizing the Code
    3.4 Perfect Numbers
    3.5 The Pythagorean Program
    3.6 A Fatal Flaw in the Program
    3.7 Thoughts on the Chapter
    4. Euclid’s Algorithm
    4.1 Athens and Alexandria
    4.2 Euclid’s Greatest Common Measure Algorithm
    4.3 A Millennium without Mathematics
    4.4 The Strange History of Zero
    4.5 Remainder and Quotient Algorithms
    4.6 Sharing the Code
    4.7 Validating the Algorithm
    4.8 Thoughts on the Chapter
    5. The Emergence of Modern Number Theory
    5.1 Mersenne Primes and Fermat Primes
    5.2 Fermat’s Little Theorem
    5.3 Cancellation
    5.4 Proving Fermat’s Little Theorem
    5.5 Euler’s Theorem
    5.6 Applying Modular Arithmetic
    5.7 Thoughts on the Chapter
    6. Abstraction in Mathematics
    6.1 Groups
    6.2 Monoids and Semigroups
    6.3 Some Theorems about Groups
    6.4 Subgroups and Cyclic Groups
    6.5 Lagrange’s Theorem
    6.6 Theories and Models
    6.7 Examples of Categorical and Non-categorical Theories
    6.8 Thoughts on the Chapter
    7. Deriving a Generic Algorithm
    7.1 Untangling Algorithm Requirements
    7.2 Requirements on A
    7.3 Requirements on N
    7.4 New Requirements
    7.5 Turning Multiply into Power
    7.6 Generalizing the Operation
    7.7 Computing Fibonacci Numbers
    7.8 Thoughts on the Chapter
    8. More Algebraic Structures
    8.1 Stevin, Polynomials, and GCD
    8.2 Göttingen and German Mathematics
    8.3 Noether and the Birth of Abstract Algebra
    8.4 Rings
    8.5 Matrix Multiplication and Semirings
    8.6 Application: Social Networks and Shortest Paths
    8.7 Euclidean Domains
    8.8 Fields and Other Algebraic Structures
    8.9 Thoughts on the Chapter
    9. Organizing Mathematical Knowledge
    9.1 Proofs
    9.2 The First Theorem
    9.3 Euclid and the Axiomatic Method
    9.4 Alternatives to Euclidean Geometry
    9.5 Hilbert’s Formalist Approach
    9.6 Peano and His Axioms
    9.7 Building Arithmetic
    9.8 Thoughts on the Chapter
    10. Fundamental Programming Concepts
    10.1 Aristotle and Abstraction
    10.2 Values and Types
    10.3 Concepts
    10.4 Iterators
    10.5 Iterator Categories, Operations, and Traits
    10.6 Ranges
    10.7 Linear Search
    10.8 Binary Search
    10.9 Thoughts on the Chapter
    11. Permutation Algorithms
    11.1 Permutations and Transpositions
    11.2 Swapping Ranges
    11.3 Rotation
    11.4 Using Cycles
    11.5 Reverse
    11.6 Space Complexity
    11.7 Memory-Adaptive Algorithms
    11.8 Thoughts on the Chapter
    12. Extensions of GCD
    12.1 Hardware Constraints and a More Efficient Algorithm
    12.2 Generalizing Stein’s Algorithm
    12.3 Bézout’s Identity
    12.4 Extended GCD
    12.5 Applications of GCD
    12.6 Thoughts on the Chapter
    13. A Real-World Application
    13.1 Cryptology
    13.2 Primality Testing
    13.3 The Miller-Rabin Test
    13.4 The RSA Algorithm: How and Why It Works
    13.5 Thoughts on the Chapter
    14. Conclusions
    Further Reading
    Chapter 1
    Chapter 2
    Chapter 3
    Chapter 4
    Chapter 5
    Chapter 6
    Chapter 7
    Chapter 8
    Chapter 9
    Chapter 10
    Chapter 11
    Chapter 12
    Chapter 13
    Appendix A. Notation
    Examples
    Implication and the Contrapositive
    Appendix B. Common Proof Techniques
    B.1 Proof by Contradiction
    B.2 Proof by Induction
    B.3 The Pigeonhole Principle
    Appendix C. C++ for Non-C++ Programmers
    C.1 Template Functions
    C.2 Concepts
    C.3 Declaration Syntax and Typed Constants
    C.4 Function Objects
    C.5 Preconditions, Postconditions, and Assertions
    C.6 STL Algorithms and Data Structures
    C.7 Iterators and Ranges
    C.8 Type Aliases and Type Functions with using in C++11
    C.9 Initializer Lists in C++11
    C.10 Lambda Functions in C++11
    C.11 A Note about inline
    Bibliography
    Index
    Photo Credits
    Code Snippets
    About This eBook ePUB is an open, industry-standard format for eBooks. However, support of ePUB and its many features varies across reading devices and applications. Use your device or app settings to customize the presentation to your liking. Settings that you can customize often include font, font size, single or double column, landscape or portrait mode, and figures that you can click or tap to enlarge. For additional information about the settings and features on your reading device or app, visit the device manufacturer’s Web site. Many titles include programming code or configuration examples. To optimize the presentation of these elements, view the eBook in single- column, landscape mode and adjust the font size to the smallest setting. In addition to presenting code and configurations in the reflowable text format, we have included images of the code that mimic the presentation found in the print book; therefore, where the reflowable format may compromise the presentation of the code listing, you will see a “Click here to view code image” link. Click the link to view the print-fidelity code image. To return to the previous page viewed, click the Back button on your device or app.
    From Mathematics to Generic Programming Alexander A. Stepanov Daniel E. Rose Upper Saddle River, NJ • Boston • Indianapolis • San Francisco New York • Toronto • Montreal • London • Munich • Paris • Madrid Capetown • Sydney • Tokyo • Singapore • Mexico City
    Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book, and the publisher was aware of a trademark claim, the designations have been printed with initial capital letters or in all capitals. The authors and publisher have taken care in the preparation of this book, but make no expressed or implied warranty of any kind and assume no responsibility for errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of the use of the information or programs contained herein. For information about buying this title in bulk quantities, or for special sales opportunities (which may include electronic versions; custom cover designs; and content particular to your business, training goals, marketing focus, or branding interests), please contact our corporate sales department at corpsales@pearsoned.com or (800) 382-3419. For government sales inquiries, please contact governmentsales@pearsoned.com. For questions about sales outside the United States, please contact international@pearsoned.com. Visit us on the Web: informit.com/aw Library of Congress Cataloging-in-Publication Data Stepanov, Alexander A. From mathematics to generic programming / Alexander A. Stepanov, Daniel E. Rose. pages cm Includes bibliographical references and index. ISBN 978-0-321-94204-3 (pbk. : alk. paper) 1. Generic programming (Computer science)—Mathematics. 2. Computer algorithms. I. Rose, Daniel E. II. Title. QA76.6245.S74 2015 005.1'1—dc23 2014034539 Copyright © 2015 Pearson Education, Inc.
    All rights reserved. Printed in the United States of America. This publication is protected by copyright, and permission must be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. To obtain permission to use material from this work, please submit a written request to Pearson Education, Inc., Permissions Department, One Lake Street, Upper Saddle River, New Jersey 07458, or you may fax your request to (201) 236-3290. Photo credits are listed on page 293. ISBN-13: 978-0-321-94204-3 ISBN-10: 0-321-94204-3 Text printed in the United States on recycled paper at RR Donnelley in Crawfordsville, Indiana. First printing, November 2014
    Contents Acknowledgments About the Authors Authors’ Note 1 What This Book Is About 1.1 Programming and Mathematics 1.2 A Historical Perspective 1.3 Prerequisites 1.4 Roadmap 2 The First Algorithm 2.1 Egyptian Multiplication 2.2 Improving the Algorithm 2.3 Thoughts on the Chapter 3 Ancient Greek Number Theory 3.1 Geometric Properties of Integers 3.2 Sifting Primes 3.3 Implementing and Optimizing the Code 3.4 Perfect Numbers 3.5 The Pythagorean Program 3.6 A Fatal Flaw in the Program 3.7 Thoughts on the Chapter 4 Euclid’s Algorithm 4.1 Athens and Alexandria 4.2 Euclid’s Greatest Common Measure Algorithm 4.3 A Millennium without Mathematics 4.4 The Strange History of Zero
    4.5 Remainder and Quotient Algorithms 4.6 Sharing the Code 4.7 Validating the Algorithm 4.8 Thoughts on the Chapter 5 The Emergence of Modern Number Theory 5.1 Mersenne Primes and Fermat Primes 5.2 Fermat’s Little Theorem 5.3 Cancellation 5.4 Proving Fermat’s Little Theorem 5.5 Euler’s Theorem 5.6 Applying Modular Arithmetic 5.7 Thoughts on the Chapter 6 Abstraction in Mathematics 6.1 Groups 6.2 Monoids and Semigroups 6.3 Some Theorems about Groups 6.4 Subgroups and Cyclic Groups 6.5 Lagrange’s Theorem 6.6 Theories and Models 6.7 Examples of Categorical and Non-categorical Theories 6.8 Thoughts on the Chapter 7 Deriving a Generic Algorithm 7.1 Untangling Algorithm Requirements 7.2 Requirements on A 7.3 Requirements on N 7.4 New Requirements 7.5 Turning Multiply into Power 7.6 Generalizing the Operation
    7.7 Computing Fibonacci Numbers 7.8 Thoughts on the Chapter 8 More Algebraic Structures 8.1 Stevin, Polynomials, and GCD 8.2 Göttingen and German Mathematics 8.3 Noether and the Birth of Abstract Algebra 8.4 Rings 8.5 Matrix Multiplication and Semirings 8.6 Application: Social Networks and Shortest Paths 8.7 Euclidean Domains 8.8 Fields and Other Algebraic Structures 8.9 Thoughts on the Chapter 9 Organizing Mathematical Knowledge 9.1 Proofs 9.2 The First Theorem 9.3 Euclid and the Axiomatic Method 9.4 Alternatives to Euclidean Geometry 9.5 Hilbert’s Formalist Approach 9.6 Peano and His Axioms 9.7 Building Arithmetic 9.8 Thoughts on the Chapter 10 Fundamental Programming Concepts 10.1 Aristotle and Abstraction 10.2 Values and Types 10.3 Concepts 10.4 Iterators 10.5 Iterator Categories, Operations, and Traits 10.6 Ranges
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