From Mathematics to Generic Programming

Series
Addison-Wesley
Author
Alexander A. Stepanov / Daniel E. Rose  
Publisher
Addison-Wesley
Cover
Softcover
Edition
1
Language
English
Total pages
320
Pub.-date
November 2014
ISBN13
9780321942043
ISBN
0321942043
Related Titles


Product detail

Product Price CHF Available  
9780321942043
From Mathematics to Generic Programming
44.50 approx. 7-9 days

Description

This book is a great introduction to the core principles of generic programming for the experienced programmer. The authors work through examples showing how to analyze the requirements of an algorithm and make it as general as possible. The book includes several programming “laws” of particular interest to those building software components. The authors show how programmers can become more effective by learning about the idea of abstraction and the math it relies on. In an engaging and accessible fashion, they describe how these mathematical results were first discovered and are surprisingly useful in programming.  

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

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

Author

Alexander A. Stepanov studied mathematics at Moscow State University from 1967 to 1972. He has been programming since 1972: first in the Soviet Union and, after emigrating in 1977, in the United States. He has programmed operating systems, programming tools, compilers, and libraries. His work on foundations of programming has been supported by GE, Polytechnic University, Bell Labs, HP, SGI, Adobe, and, since 2009, A9.com, Amazon’s search technology subsidiary. In 1995 he received the Dr. Dobb’s Journal Excellence in Programming Award for the design of the C++ Standard Template Library.

 

Daniel E. Rose is a research scientist who has held management positions at Apple, AltaVista, Xigo, Yahoo, and A9.com. His research focuses on all aspects of search technology, ranging from low-level algorithms for index compression to human–computer interaction issues in web search. Rose led the team at Apple that created desktop search for the Macintosh. He holds a Ph.D. in cognitive science and computer science from University of California, San Diego, and a B.A. in philosophy from Harvard University.