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From Mathematics to Generic Programming

From Mathematics to Generic Programming

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Description

  • Copyright 2015
  • Dimensions: 7" x 9-1/8"
  • Pages: 320
  • Edition: 1st
  • eBook (Watermarked)
  • ISBN-10: 0-13-349180-3
  • ISBN-13: 978-0-13-349180-7

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

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

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