The Art of Computer Programming, Volume 1: Fundamental Algorithms | |

Author | Donald Knuth |
---|---|

Country | United States |

Language | English |

Genre | Non-fiction Monograph |

Publisher | Addison-Wesley |

Publication date | 1968– (the book is still incomplete) |

Media type | Print (Hardcover) |

ISBN | 0-201-03801-3 |

519 | |

LC Class | QA76.75 |

* The Art of Computer Programming* (

*) is a comprehensive monograph written by Donald Knuth that covers many kinds of programming algorithms and their analysis.*

**TAOCP**Knuth began the project, originally conceived as a single book with twelve chapters, in 1962. The first three volumes of what was then expected to be a seven-volume set were published in 1968, 1969, and 1973. The first published installment of Volume 4 appeared in paperback as Fascicle 2 in 2005. The hardback Volume 4A, combining Volume 4, Fascicles 0–4, was published in 2011. Volume 4, Fascicle 6 ("Satisfiability") was released in December 2015; Volume 4, Fascicle 5 ("Mathematical Preliminaries Redux; Backtracking; Dancing Links") is planned for release in September 2019. Fascicles 5 and 6 are expected to comprise the first two-thirds of Volume 4B, which is scheduled to be published on May 17, 2019.

## Contents

## History

After winning a Westinghouse Talent Search scholarship, Knuth enrolled at the Case Institute of Technology (now Case Western Reserve University), where his performance was so outstanding that the faculty voted to award him a master of science upon his completion of the baccalaureate degree. During his summer vacations, Knuth was hired by the Burroughs Corporation to write compilers, earning more in his summer months than full professors did for an entire year.^{[1]} Such exploits made Knuth a topic of discussion among the mathematics department, which included Richard S. Varga.

Knuth started to write a book about compiler design in 1962, and soon realized that the scope of the book needed to be much larger. In June 1965, Knuth finished the first draft of what was originally planned to be a single volume of twelve chapters. His hand-written first-draft manuscript (completed in 1966) was pages long: he had assumed that about five hand-written pages would translate into one printed page, but his publisher said instead that about 30001 ^{1}⁄_{2} hand-written pages translated to one printed page. This meant the book would be approximately pages in length. The publisher was nervous about accepting such a project from a graduate student. At this point, Knuth received support from Richard S. Varga, who was the scientific adviser to the publisher. Varga was visiting 2000Olga Taussky-Todd and John Todd at Caltech. With Varga's enthusiastic endorsement, the publisher accepted Knuth's expanded plans. In its expanded version, the book would be published in seven volumes, each with just one or two chapters.^{[2]} Due to the growth in the material, the plan for Volume 4 has since expanded to include Volumes 4A, 4B, 4C, 4D, and possibly more.

In 1976, Knuth prepared a second edition of Volume 2, requiring it to be typeset again, but the style of type used in the first edition (called hot type) was no longer available. In 1977, he decided to spend some time creating something more suitable. Eight years later, he returned with T_{E}X, which is currently used for all volumes.

The offer of a so-called Knuth reward check worth "one hexadecimal dollar" (100_{HEX} base 16 cents, in decimal, is $2.56) for any errors found, and the correction of these errors in subsequent printings, has contributed to the highly polished and still-authoritative nature of the work, long after its first publication. Another characteristic of the volumes is the variation in the difficulty of the exercises. The level of difficulty ranges from "warm-up" exercises to unsolved research problems.

Knuth's dedication reads:

This series of books is affectionately dedicated

to the Type 650 computer once installed at

Case Institute of Technology,

with whom I have spent many pleasant evenings.^{[a]}

## Assembly language in the book

All examples in the books use a language called "MIX assembly language", which runs on the hypothetical MIX computer. Currently, the MIX computer is being replaced by the MMIX computer, which is a RISC version. Software such as GNU MDK exists to provide emulation of the MIX architecture. Knuth considers the use of assembly language necessary for the speed and memory usage of algorithms to be judged.

## Critical response

Knuth was awarded the 1974 Turing Award "for his major contributions to the analysis of algorithms […], and in particular for his contributions to the 'art of computer programming' through his well-known books in a continuous series by this title."^{[3]} *American Scientist* has included this work among "100 or so Books that shaped a Century of Science", referring to the twentieth century,^{[4]} and within the computer science community it is regarded as the first and still the best comprehensive treatment of its subject. Covers of the third edition of Volume 1 quote Bill Gates as saying, "If you think you're a really good programmer… read (Knuth's) *Art of Computer Programming*… You should definitely send me a résumé if you can read the whole thing."^{[5]} *The New York Times* referred to it as "the profession's defining treatise".^{[6]}

## Volumes

### Completed

- Volume 1 – Fundamental Algorithms

- Chapter 1 – Basic concepts
- Chapter 2 – Information structures

- Volume 2 ��� Seminumerical Algorithms

- Chapter 3 – Random numbers
- Chapter 4 – Arithmetic

- Volume 4A – Combinatorial Algorithms

- Chapter 7 – Combinatorial searching (part 1)

### Planned

- Volume 4B... – Combinatorial Algorithms (chapters 7 & 8 released in several subvolumes)

- Chapter 7 – Combinatorial searching (continued)
- Chapter 8 – Recursion

- Volume 5 – Syntactic Algorithms (as of 2017
^{[update]}, estimated for release in 2025)

- Chapter 9 – Lexical scanning (also includes string search and data compression)
- Chapter 10 – Parsing techniques

- Volume 6 – The Theory of Context-Free Languages
- Volume 7 ��� Compiler Techniques

## Chapter outlines

### Completed

#### Volume 1 – Fundamental Algorithms

- Chapter 1 – Basic concepts
- 1.1. Algorithms
- 1.2. Mathematical Preliminaries
- 1.2.1. Mathematical Induction
- 1.2.2. Numbers, Powers, and Logarithms
- 1.2.3. Sums and Products
- 1.2.4. Integer Functions and Elementary Number Theory
- 1.2.5. Permutations and Factorials
- 1.2.6. Binomial Coefficients
- 1.2.7. Harmonic Numbers
- 1.2.8. Fibonacci Numbers
- 1.2.9. Generating Functions
- 1.2.10. Analysis of an Algorithm
- 1.2.11. Asymptotic Representations
- 1.2.11.1. The O-notation
- 1.2.11.2. Euler's summation formula
- 1.2.11.3. Some asymptotic calculations

- 1.3 MMIX (MIX in the hardback copy but updated by fascicle 1)
- 1.3.1. Description of MMIX
- 1.3.2. The MMIX Assembly Language
- 1.3.3. Applications to Permutations

- 1.4. Some Fundamental Programming Techniques
- 1.4.1. Subroutines
- 1.4.2. Coroutines
- 1.4.3. Interpretive Routines
- 1.4.3.1. A MIX simulator
- 1.4.3.2. Trace routines

- 1.4.4. Input and Output
- 1.4.5. History and Bibliography

- Chapter 2 – Information Structures
- 2.1. Introduction
- 2.2. Linear Lists
- 2.2.1. Stacks, Queues, and Deques
- 2.2.2. Sequential Allocation
- 2.2.3. Linked Allocation
- 2.2.4. Circular Lists
- 2.2.5. Doubly Linked Lists
- 2.2.6. Arrays and Orthogonal Lists

- 2.3. Trees
- 2.3.1. Traversing Binary Trees
- 2.3.2. Binary Tree Representation of Trees
- 2.3.3. Other Representations of Trees
- 2.3.4. Basic Mathematical Properties of Trees
- 2.3.4.1. Free trees
- 2.3.4.2. Oriented trees
- 2.3.4.3. The "infinity lemma"
- 2.3.4.4. Enumeration of trees
- 2.3.4.5. Path length
- 2.3.4.6. History and bibliography

- 2.3.5. Lists and Garbage Collection

- 2.4. Multilinked Structures
- 2.5. Dynamic Storage Allocation
- 2.6. History and Bibliography

- Chapter 1 – Basic concepts

#### Volume 2 – Seminumerical Algorithms

- Chapter 3 – Random Numbers
- 3.1. Introduction
- 3.2. Generating Uniform Random Numbers
- 3.2.1. The Linear Congruential Method
- 3.2.1.1. Choice of modulus
- 3.2.1.2. Choice of multiplier
- 3.2.1.3. Potency

- 3.2.2. Other Methods

- 3.2.1. The Linear Congruential Method
- 3.3. Statistical Tests
- 3.3.1. General Test Procedures for Studying Random Data
- 3.3.2. Empirical Tests
- 3.3.3. Theoretical Tests
- 3.3.4. The Spectral Test

- 3.4. Other Types of Random Quantities
- 3.4.1. Numerical Distributions
- 3.4.2. Random Sampling and Shuffling

- 3.5. What Is a Random Sequence?
- 3.6. Summary

- Chapter 4 – Arithmetic
- 4.1. Positional Number Systems
- 4.2. Floating Point Arithmetic
- 4.2.1. Single-Precision Calculations
- 4.2.2. Accuracy of Floating Point Arithmetic
- 4.2.3. Double-Precision Calculations
- 4.2.4. Distribution of Floating Point Numbers

- 4.3. Multiple Precision Arithmetic
- 4.3.1. The Classical Algorithms
- 4.3.2. Modular Arithmetic
- 4.3.3. How Fast Can We Multiply?

- 4.4. Radix Conversion
- 4.5. Rational Arithmetic
- 4.5.1. Fractions
- 4.5.2. The Greatest Common Divisor
- 4.5.3. Analysis of Euclid's Algorithm
- 4.5.4. Factoring into Primes

- 4.6. Polynomial Arithmetic
- 4.6.1. Division of Polynomials
- 4.6.2. Factorization of Polynomials
- 4.6.3. Evaluation of Powers
- 4.6.4. Evaluation of Polynomials

- 4.7. Manipulation of Power Series

- Chapter 3 – Random Numbers

#### Volume 3 – Sorting and Searching

- Chapter 5 – Sorting
- 5.1. Combinatorial Properties of Permutations
- 5.1.1. Inversions
- 5.1.2. Permutations of a Multiset
- 5.1.3. Runs
- 5.1.4. Tableaux and Involutions

- 5.2. Internal sorting
- 5.2.1. Sorting by Insertion
- 5.2.2. Sorting by Exchanging
- 5.2.3. Sorting by Selection
- 5.2.4. Sorting by Merging
- 5.2.5. Sorting by Distribution

- 5.3. Optimum Sorting
- 5.3.1. Minimum-Comparison Sorting
- 5.3.2. Minimum-Comparison Merging
- 5.3.3. Minimum-Comparison Selection
- 5.3.4. Networks for Sorting

- 5.4. External Sorting
- 5.4.1. Multiway Merging and Replacement Selection
- 5.4.2. The Polyphase Merge
- 5.4.3. The Cascade Merge
- 5.4.4. Reading Tape Backwards
- 5.4.5. The Oscillating Sort
- 5.4.6. Practical Considerations for Tape Merging
- 5.4.7. External Radix Sorting
- 5.4.8. Two-Tape Sorting
- 5.4.9. Disks and Drums

- 5.5. Summary, History, and Bibliography

- 5.1. Combinatorial Properties of Permutations
- Chapter 6 – Searching

- Chapter 5 – Sorting

#### Volume 4A – Combinatorial Algorithms, Part 1

- Chapter 7 – Combinatorial Searching
- 7.1. Zeros and Ones
- 7.1.1. Boolean Basics
- 7.1.2. Boolean Evaluation
- 7.1.3. Bitwise Tricks and Techniques
- 7.1.4. Binary Decision Diagrams

- 7.2. Generating All Possibilities
- 7.2.1. Generating Basic Combinatorial Patterns
- 7.2.1.1. Generating all n-tuples
- 7.2.1.2. Generating all permutations
- 7.2.1.3. Generating all combinations
- 7.2.1.4. Generating all partitions
- 7.2.1.5. Generating all set partitions
- 7.2.1.6. Generating all trees
- 7.2.1.7. History and further references

- 7.2.1. Generating Basic Combinatorial Patterns

- 7.1. Zeros and Ones

- Chapter 7 – Combinatorial Searching

### Planned

#### Volume 4B, 4C, 4D – Combinatorial Algorithms

- Chapter 7 – Combinatorial Searching (continued)
- 7.2. Generating all possibilities (continued)
- 7.2.2. Backtrack programming (online as pre-fascicle 5b)
- 7.2.2.1. Dancing links (online as pre-fascicle 5c)
- 7.2.2.2. Satisfiability (published in Fascicle 6)
- 7.2.2.3. Constraint satisfaction
- 7.2.2.4. Hamiltonian paths
- 7.2.2.5. Cliques
- 7.2.2.6. Covers (Vertex cover, Set cover problem, Exact cover, Clique cover)
- 7.2.2.7. Squares
- 7.2.2.8. A potpourri of puzzles
- 7.2.2.9. Estimating backtrack costs (chapter 6 of "Selected Papers on Analysis of Algorithms", and pre-fascicle 5b in Section 7.2.2 under the heading "Running time estimates")

- 7.2.3. Generating inequivalent patterns (includes discussion of Pólya enumeration theorem)

- 7.2.2. Backtrack programming (online as pre-fascicle 5b)
- 7.3. Shortest paths
- 7.4. Graph algorithms
- 7.4.1. Components and traversal
- 7.4.2. Special classes of graphs
- 7.4.3. Expander graphs
- 7.4.4. Random graphs

- 7.5. Network algorithms
- 7.5.1. Distinct representatives
- 7.5.2. The assignment problem
- 7.5.3. Network flows
- 7.5.4. Optimum subtrees
- 7.5.5. Optimum matching
- 7.5.6. Optimum orderings

- 7.6. Independence theory
- 7.6.1. Independence structures
- 7.6.2. Efficient matroid algorithms

- 7.7. Discrete dynamic programming (see also Transfer-matrix method)
- 7.8. Branch-and-bound techniques
- 7.9. Herculean tasks (aka NP-hard problems)
- 7.10. Near-optimization

- 7.2. Generating all possibilities (continued)
- Chapter 8 – Recursion (chapter 22 of "Selected Papers on Analysis of Algorithms")

- Chapter 7 – Combinatorial Searching (continued)

#### Volume 5 – Syntactic Algorithms

*as of 2017*^{[update]}, estimated for release in 2025- Chapter 9 – Lexical scanning (includes also string search and data compression)
- Chapter 10 – Parsing techniques

#### Volume 6 – The Theory of Context-free Languages^{[7]}

#### Volume 7 – Compiler Techniques

## English editions

### Current editions

These are the current editions in order by volume number:

*The Art of Computer Programming, Volumes 1-4A Boxed Set*. Third Edition (Reading, Massachusetts: Addison-Wesley, 2011), 3168pp. ISBN 978-0-321-75104-1, 0-321-75104-3*Volume 1: Fundamental Algorithms*. Third Edition (Reading, Massachusetts: Addison-Wesley, 1997), xx+650pp. ISBN 978-0-201-89683-1, 0-201-89683-4. Errata: [1] (2011-01-08), [2] (2017-09-18, 27th printing). Addenda: [3] (2011).*Volume 2: Seminumerical Algorithms*. Third Edition (Reading, Massachusetts: Addison-Wesley, 1997), xiv+762pp. ISBN 978-0-201-89684-8, 0-201-89684-2. Errata: [4] (2011-01-08), [5] (2017-09-18, 26th printing). Addenda: [6] (2011).*Volume 3: Sorting and Searching*. Second Edition (Reading, Massachusetts: Addison-Wesley, 1998), xiv+780pp.+foldout. ISBN 978-0-201-89685-5, 0-201-89685-0. Errata: [7] (2011-01-08), [8] (2017-09-18, 27th printing). Addenda: [9] (2011).*Volume 4A: Combinatorial Algorithms, Part 1*. First Edition (Reading, Massachusetts: Addison-Wesley, 2011), xv+883pp. ISBN 978-0-201-03804-0, 0-201-03804-8. Errata: [10] (2017-09-18, ? printing).

*Volume 1, Fascicle 1: MMIX – A RISC Computer for the New Millennium*. (Addison-Wesley, 2005-02-14) ISBN 0-201-85392-2 (will be in the fourth edition of volume 1). Errata: [11] (2016-08-02).*Volume 4, Fascicle 5: Mathematical Preliminaries Redux; Backtracking; Dancing Links*. (Addison-Wesley, 2019-09-12) 350pp, ISBN 978-0-13-467179-6 (will become part of volume 4B)*Volume 4, Fascicle 6: Satisfiability*. (Addison-Wesley, 2015-12-08) xiii+310pp, ISBN 978-0-13-439760-3. Errata: [12] (2017-06-01) (will become part of volume 4B)

### Previous editions

#### Complete volumes

These volumes were superseded by newer editions and are in order by date.

*Volume 1: Fundamental Algorithms*. First edition, 1968, xxi+634pp, ISBN 0-201-03801-3.^{[8]}*Volume 2: Seminumerical Algorithms*. First edition, 1969, xi+624pp, ISBN 0-201-03802-1.^{[8]}*Volume 3: Sorting and Searching*. First edition, 1973, xi+723pp+foldout, ISBN 0-201-03803-X. Errata: [13].*Volume 1: Fundamental Algorithms*. Second edition, 1973, xxi+634pp, ISBN 0-201-03809-9. Errata: [14].*Volume 2: Seminumerical Algorithms*. Second edition, 1981, xiii+ 688pp, ISBN 0-201-03822-6. Errata: [15].*The Art of Computer Programming, Volumes 1-3 Boxed Set*. Second Edition (Reading, Massachusetts: Addison-Wesley, 1998), pp. ISBN 978-0-201-48541-7, 0-201-48541-9

#### Fascicles

Volume 4's fascicles 0–4 were revised and published as Volume 4A.

*Volume 4, Fascicle 0: Introduction to Combinatorial Algorithms and Boolean Functions*. (Addison-Wesley Professional, 2008-04-28) vi+240pp, ISBN 0-321-53496-4. Errata: [16] (2011-01-01).*Volume 4, Fascicle 1: Bitwise Tricks & Techniques; Binary Decision Diagrams*. (Addison-Wesley Professional, 2009-03-27) viii+260pp, ISBN 0-321-58050-8. Errata: [17] (2011-01-01).*Volume 4, Fascicle 2: Generating All Tuples and Permutations*. (Addison-Wesley, 2005-02-14) v+127pp, ISBN 0-201-85393-0. Errata: [18] (2011-01-01).*Volume 4, Fascicle 3: Generating All Combinations and Partitions*. (Addison-Wesley, 2005-07-26) vi+150pp, ISBN 0-201-85394-9. Errata: [19] (2011-01-01).*Volume 4, Fascicle 4: Generating All Trees; History of Combinatorial Generation*. (Addison-Wesley, 2006-02-06) vi+120pp, ISBN 0-321-33570-8. Errata: [20] (2011-01-01).*Volume 4, Fascicle 6: Satisfiability*. (Addison-Wesley, 2015-12-18) xiii+310pp, ISBN 978-0-13-439760-3. Errata: [21] (2017-06-01)

#### Pre-fascicles

Volume 4's pre-fascicle 6A was revised and published as fascicle 6.

*Volume 4B, Pre-fascicle 5A: Mathematical Preliminaries Redux (available for download)**Volume 4B, Pre-fascicle 5B: Introduction to Backtracking (available for download)**Volume 4B, Pre-fascicle 5C: Dancing Links (available for download)*

## See also

## References

**Notes**

**^**The dedication was worded slightly differently in the first edition.

**Citations**

**^**Frana, Philip L. (2001-11-08). "An Interview with Donald E. Knuth". hdl:11299/107413.**^**Albers, Donald J. (2008). "Donald Knuth". In Albers, Donald J.; Alexanderson, Gerald L. (eds.).*Mathematical People: Profiles and Interviews*(2 ed.). A K Peters. ISBN 1-56881-340-6.**^**"Donald E. Knuth – A. M. Turing Award Winner".*AM Turing*. Retrieved 2017-01-25.**^**Morrison, Philip; Morrison, Phylis (November–December 1999). "100 or so Books that shaped a Century of Science".*American Scientist*. Sigma Xi, The Scientific Research Society.**87**(6). Archived from the original on 2008-08-20. Retrieved 2008-01-11.**^**Weinberger, Matt. "Bill Gates once said 'definitely send me a résumé' if you finish this fiendishly difficult book".*Business Insider*. Retrieved 2016-06-13.**^**Lohr, Steve (2001-12-17). "Frances E. Holberton, 84, Early Computer Programmer".*The New York Times*. Retrieved 2010-05-17.**^**"TAOCP – Future plans".- ^
^{a}^{b}Wells, Mark B. (1973). "Review:*The Art of Computer Programming, Volume 1. Fundamental Algorithms*and*Volume 2. Seminumerical Algorithms*by Donald E. Knuth" (PDF).*Bulletin of the American Mathematical Society*.**79**(3): 501–509. doi:10.1090/s0002-9904-1973-13173-8.

**Sources**

- Slater, Robert (1987).
*Portraits in Silicon*. MIT Press. ISBN 0-262-19262-4. - Shasha, Dennis; Lazere, Cathy (1995).
*Out of Their Minds: The Lives and Discoveries of 15 Great Computer Scientists*. Copernicus. ISBN 0-387-97992-1.

## External links

- Overview of topics (Knuth's personal homepage)
- Oral history interview with Donald E. Knuth at Charles Babbage Institute, University of Minnesota, Minneapolis. Knuth discusses software patenting, structured programming, collaboration and his development of TeX. The oral history discusses the writing of
*The Art of Computer Programming*. - "Robert W Floyd, In Memoriam", by Donald E. Knuth - (on the influence of Bob Floyd)
*TAoCP*and its Influence of Computer Science (Softpanorama)