The Art of Computer Programming

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The Art of Computer Programming
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The Art of Computer Programming (sometimes known by its initials TAOCP) is a comprehensive monograph written by Donald Knuth that covers many kinds of programming algorithms and their analysis.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, to be followed by Volume 4, Fascicle 5 ("Mathematical Preliminaries Redux; Backtracking; Dancing Links") in October 2018. Fascicles 5 and 6 are expected to comprise the first two thirds of Volume 4B.


(File:KnuthAtOpenContentAlliance.jpg|right|thumb|200px|Donald Knuth in 2005)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.WEB,weblink 11299/107413, An Interview with Donald E. Knuth, Philip L., Frana, 2001-11-08, 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 {{val|3000}} pages long: he had assumed that about five hand-written pages would translate into one printed page, but his publisher said instead that about 1½ hand-written pages translated to one printed page. This meant the book would be approximately {{val|2000}} 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 Olga 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.BOOK, Donald Knuth, Donald J., Albers, Mathematical People: Profiles and Interviews, A K Peters, 2, 2008, 1-56881-340-6, Donald J., Albers, Gerald L., Alexanderson, Gerald L. Alexanderson, 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 TEX, which is currently used for all volumes.The offer of a so-called Knuth reward check worth "one hexadecimal dollar" (100HEX 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 dedicatedto the Type 650 computer once installed atCase Institute of Technology,with whom I have spent many pleasant evenings.The dedication was worded slightly differently in the first edition.

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."WEB,weblink Donald E. Knuth – A. M. Turing Award Winner, AM Turing, 2017-01-25, American Scientist has included this work among "100 or so Books that shaped a Century of Science", referring to the twentieth century,JOURNAL, Philip, Morrison, Philip Morrison, Phylis, Morrison,weblink yes,weblink" title="">weblink 2008-08-20, 100 or so Books that shaped a Century of Science, 87, 6, American Scientist, November–December 1999, Sigma Xi, The Scientific Research Society, 2008-01-11, 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."WEB,weblink Bill Gates once said 'definitely send me a résumé' if you finish this fiendishly difficult book, Matt, Weinberger, Business Insider, 2016-06-13, The New York Times referred to it as "the profession's defining treatise".NEWS, Lohr, Steve, Frances E. Holberton, 84, Early Computer Programmer, The New York Times, 2001-12-17,weblink 2010-05-17,



  • Volume 1 â€“ Fundamental Algorithms

*Chapter 1 â€“ Basic concepts *Chapter 2 â€“ Information structures
  • Volume 2 â€“ Seminumerical Algorithms

*Chapter 3 â€“ Random numbers *Chapter 4 â€“ Arithmetic

*Chapter 5 â€“ Sorting *Chapter 6 â€“ Searching

*Chapter 7 â€“ Combinatorial searching (part 1)


  • Volume 4B... â€“ Combinatorial Algorithms (chapters 7 & 8 released in several subvolumes)

*Chapter 7 â€“ Combinatorial searching (continued) *Chapter 8 â€“ Recursion
  • Volume 5 â€“ Syntactic Algorithms ({{as of|2017|lc=on}}, estimated for release in 2025)

*Chapter 9 â€“ Lexical scanning (also includes string search and data compression) *Chapter 10 â€“ Parsing techniques

Chapter outlines


Volume 1 â€“ Fundamental Algorithms

    • Chapter 1 â€“ Basic concepts
      • 1.1. Algorithms
      • 1.2. Mathematical Preliminaries
1.2.1. Mathematical Induction1.2.2. Numbers, Powers, and Logarithms1.2.3. Sums and Products1.2.4. Integer Functions and Elementary Number Theory1.2.5. Permutations and Factorials1.2.6. Binomial Coefficients1.2.7. Harmonic Numbers1.2.8. Fibonacci Numbers1.2.9. Generating Functions1.2.10. Analysis of an Algorithm1.2.11. Asymptotic Representations* The O-notation* Euler's summation formula* Some asymptotic calculations
      • 1.3 MMIX (MIX in the hardback copy but updated by fascicle 1)
1.3.1. Description of MMIX1.3.2. The MMIX Assembly Language1.3.3. Applications to Permutations
      • 1.4. Some Fundamental Programming Techniques
1.4.1. Subroutines1.4.2. Coroutines1.4.3. Interpretive Routines* A MIX simulator* Trace routines1.4.4. Input and Output1.4.5. History and Bibliography
    • Chapter 2 â€“ Information Structures
      • 2.1. Introduction
      • 2.2. Linear Lists
2.2.1. Stacks, Queues, and Deques2.2.2. Sequential Allocation2.2.3. Linked Allocation2.2.4. Circular Lists2.2.5. Doubly Linked Lists2.2.6. Arrays and Orthogonal Lists 2.3.1. Traversing Binary Trees2.3.2. Binary Tree Representation of Trees2.3.3. Other Representations of Trees2.3.4. Basic Mathematical Properties of Trees* Free trees* Oriented trees* The "infinity lemma"* Enumeration of trees* Path length* History and bibliography2.3.5. Lists and Garbage Collection

Volume 2 â€“ Seminumerical Algorithms

3.2.1. The Linear Congruential Method* Choice of modulus* Choice of multiplier* Potency3.2.2. Other Methods
      • 3.3. Statistical Tests
3.3.1. General Test Procedures for Studying Random Data3.3.2. Empirical Tests3.3.3. Theoretical Tests3.3.4. The Spectral Test 3.4.1. Numerical Distributions3.4.2. Random Sampling and Shuffling 4.2.1. Single-Precision Calculations4.2.2. Accuracy of Floating Point Arithmetic4.2.3. Double-Precision Calculations4.2.4. Distribution of Floating Point Numbers 4.3.1. The Classical Algorithms4.3.2. Modular Arithmetic4.3.3. How Fast Can We Multiply? 4.5.1. Fractions4.5.2. The Greatest Common Divisor4.5.3. Analysis of Euclid's Algorithm4.5.4. Factoring into Primes 4.6.1. Division of Polynomials4.6.2. Factorization of Polynomials4.6.3. Evaluation of Powers4.6.4. Evaluation of Polynomials

Volume 3 â€“ Sorting and Searching

5.1.1. Inversions5.1.2. Permutations of a Multiset5.1.3. Runs5.1.4. Tableaux and Involutions 5.2.1. Sorting by Insertion5.2.2. Sorting by Exchanging5.2.3. Sorting by Selection5.2.4. Sorting by Merging5.2.5. Sorting by Distribution
      • 5.3. Optimum Sorting
5.3.1. Minimum-Comparison Sorting5.3.2. Minimum-Comparison Merging5.3.3. Minimum-Comparison Selection5.3.4. Networks for Sorting 5.4.1. Multiway Merging and Replacement Selection5.4.2. The Polyphase Merge5.4.3. The Cascade Merge5.4.4. Reading Tape Backwards5.4.5. The Oscillating Sort5.4.6. Practical Considerations for Tape Merging5.4.7. External Radix Sorting5.4.8. Two-Tape Sorting5.4.9. Disks and Drums
      • 5.5. Summary, History, and Bibliography
    • Chapter 6 â€“ Searching
      • 6.1. Sequential Searching
      • 6.2. Searching by Comparison of Keys
6.2.1. Searching an Ordered Table6.2.2. Binary Tree Searching6.2.3. Balanced Trees6.2.4. Multiway Trees
      • 6.3. Digital Searching
      • 6.4. Hashing
      • 6.5. Retrieval on Secondary Keys

Volume 4A â€“ Combinatorial Algorithms, Part 1

7.1.1. Boolean Basics7.1.2. Boolean Evaluation7.1.3. Bitwise Tricks and Techniques7.1.4. Binary Decision Diagrams
      • 7.2. Generating All Possibilities
7.2.1. Generating Basic Combinatorial Patterns* Generating all n-tuples* Generating all permutations* Generating all combinations* Generating all partitions* Generating all set partitions* Generating all trees* History and further references


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

    • Chapter 7 â€“ Combinatorial Searching (continued)
      • 7.2. Generating all possibilities (continued)
7.2.2. Backtrack programming* Dancing links* Satisfiability* Constraint satisfaction* Hamiltonian paths* Cliques* Covers* Squares* A potpourri of puzzles* Estimating backtrack costs7.2.3. Generating inequivalent patterns 7.4.1. Components and traversal7.4.2. Special classes of graphs7.4.3. Expander graphs7.4.4. Random graphs 7.5.1. Distinct representatives7.5.2. The assignment problem7.5.3. Network flows7.5.4. Optimum subtrees7.5.5. Optimum matching7.5.6. Optimum orderings
      • 7.6. Independence theory
7.6.1. Independence structures7.6.2. Efficient matroid algorithms

Volume 5 â€“ Syntactic Algorithms

  • {{as of|2017|lc=on}}, estimated for release in 2025
    • Chapter 9 â€“ Lexical scanning (includes also string search and data compression)
    • Chapter 10 â€“ Parsing techniques
weblink titleTAOCP – Future plans, ">

Volume 6 â€“ The Theory of Context-free LanguagesWEB, urweblink titleTAOCP – Future plans,

Volume 7 â€“ 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: weblink (2011-01-08), weblink (2017-09-18, 27th printing). Addenda: weblink (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: weblink (2011-01-08), weblink (2017-09-18, 26th printing). Addenda: weblink (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: weblink (2011-01-08), weblink (2017-09-18, 27th printing). Addenda: weblink (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: weblink (2017-09-18, ? printing).
  • Volume 1, Fascicle 1: MMIX â€“ 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: weblink (2016-08-02).
  • Volume 4, Fascicle 5: Mathematical Preliminaries Redux; Backtracking; Dancing Links. (Addison-Wesley, 2018-10-18) 320pp, {{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: weblink (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}}.JOURNAL, Wells, Mark B., Review: The Art of Computer Programming, Volume 1. Fundamental Algorithms and Volume 2. Seminumerical Algorithms by Donald E. Knuth, Bulletin of the American Mathematical Society, 1973, 79, 3, 501–509,weblink 10.1090/s0002-9904-1973-13173-8,
  • Volume 2: Seminumerical Algorithms. First edition, 1969, xi+624pp, {{ISBN|0-201-03802-1}}.
  • Volume 3: Sorting and Searching''. First edition, 1973, xi+723pp+foldout, {{ISBN|0-201-03803-X}}. Errata: weblink.
  • Volume 1: Fundamental Algorithms''. Second edition, 1973, xxi+634pp, {{ISBN|0-201-03809-9}}. Errata: weblink.
  • Volume 2: Seminumerical Algorithms''. Second edition, 1981, xiii+ 688pp, {{ISBN|0-201-03822-6}}. Errata: weblink.
  • 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}}


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: weblink (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: weblink (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: weblink (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: weblink (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: weblink (2011-01-01).


Volume 4{{'s}} pre-fascicles 5A–5C and 6A were revised and published as fascicles 5 and 6.

See also


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

External links

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