From Mathematics to Generic Programming

In their new book From Mathematics to Generic Programming, C++ STL creator Alexander Stepanov and Information Retrieval researcher Daniel Rose 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 surpris...

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In their new book From Mathematics to Generic Programming, C++ STL creator Alexander Stepanov and Information Retrieval researcher Daniel Rose 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.

Aimed at a wider audience than the classic Elements of Programming by Stepanov and McJones, From Mathematics to Generic Programming takes a less formal approach, interleaving discussions of programming, mathematical results, and stories of the people who made these discoveries throughout history. The authors trace the development of common algorithms from ancient times to the present, and show how to generalize them to new applications — even to the secure protocols of modern Internet commerce.

From Mathematics to Generic Programming is a great introduction to the core principles of generic programming for the professional 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.

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 L...

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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.

Table of Contents
Acknowledgments
About the Authors
Authors’ Note
Chapter 1: What This Book Is About
1.1 Programming and Mathematics
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Table of Contents
Acknowledgments
About the Authors
Authors’ Note
Chapter 1: What This Book Is About
1.1 Programming and Mathematics
1.2 A Historical Perspective
1.3 Prerequisites
1.4 Roadmap
Chapter 2: The First Algorithm
2.1 Egyptian Multiplication
2.2 Improving the Algorithm
2.3 Thoughts on the Chapter
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
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
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
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
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
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
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
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
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
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
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
Chapter 14: Conclusions
Further Reading
Appendix A: Notation
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
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