Algorithmics, 3rd ed.


Algorithmics: The Spirit of Computing, 3rd ed.     Table of contents


New to the Third Edition

Table of Contents

Part I. Preliminaries

  1. Introduction and historical review
    or, What's it all about?
  2. Algorithms and data
    or, Getting it done
  3. Programming languages and paradigms
    or, Getting it done by computer

Part II. Methods and Analysis

  1. Algorithmic methods
    or, Getting it done methodically
  2. The correctness of algorithms
    or, Getting it done right
  3. The efficiency of algorithms
    or, Getting it done cheaply

Part III. Limitations and Robustness

  1. Inefficiency and intractability
    or, You can't always get it done cheaply
  2. Noncomputability and undecidability
    or, Sometimes you can't get it done at all!
  3. Algorithmic universality and its robustness
    or, The simplest machines that get it done

Part IV. Relaxing the Rules

  1. Parallelism, concurrency and alternative models
    or, Getting lots of stuff done at once
  2. Probabilistic algorithms
    or, Getting it done by tossing coins
  3. Cryptography and reliable interaction
    or, Getting it done in secret

Part V. The Bigger Picture

  1. Software engineering
    or, Getting it done when it's big
  2. Reactive systems
    or, Getting it to behave properly over time
  3. Algorithmics and intelligence
    or, Are they better at it than us?


Read this, I pray thee
ISAIAH 29:12

This book tells a story. The story concerns the concepts, ideas, methods and results fundamental to computer science. It is not specifically about computer technology, nor is it about computer programming, though obviously it is heavily influenced by both.

The book is intended to fill a rather disturbing gap in the literature related to the computer revolution. Scores of excellent books can be found on computers themselves, with details of their structure, workings, and operation. There are also numerous books about the act of writing programs for the computers in any of a growing number of languages. These books come at a wide range of levels, some aimed at people with no computer-related background at all, and some aimed at the most computer-literate professionals. In addition, there are many books on subjects peripheral to the technology, such as the social and legal aspects of the revolution, as well as books describing the relevance of computers to a variety of application areas. All this comes as no surprise. People are curious about computers, and want to learn how to put them to use. They are typically interested in specific kinds of computers, and often for specific purposes, too.

Then there are textbooks. Indeed, computer science is a fast-growing academic discipline, with ever-larger numbers of potential students knocking at the doors of admission offices. Well-established academic disciplines have a habit of yielding excellent textbooks, and computer science is no exception. Over the years many comprehensive and clearly written textbooks have appeared, containing detailed technical accounts of the subjects deemed appropriate to students of computer science. However, despite the dizzying speed with which some of the technological innovations become obsolete and are replaced by new ones, the fundamentals of the science of computation, and hence many of the basic concepts that are considered important in a computer science curriculum, change slowly, if at all. Of course, new technologies and new languages require revisions in scientific emphasis, which are eventually reflected in the scientific literature. However, by and large, there is almost universal agreement on a core of fundamental topics that computer science students should be taught.

It would appear that anyone associated with computers ought to be aware of these topics, and not only those who have decided to spend three or four years getting a particular kind of academic diploma. Moreover, given that a revolution is indeed taking place before our very eyes, many of these topics, and the special ways of thinking that go with them, ought to be available to the enquiring person even if that person is not directly associated with a computer at all.

Books concerned primarily with computers or programming are intended to fulfill quite different needs. Computers are made of bits and bytes, and programming is carried out using languages with rigid rules of grammar and punctuation. Consequently, computer books often suffer from the "bit/byte syndrome" and programming books from the "semicolon syndrome". In other words, the reader becomes predominantly involved in the principles of a particular computer or the syntactic rules of a particular programming language (or both). It would seem that things cannot be explained without first describing, in detail, either a machine or a medium for communicating with one (or both).

Many advanced textbooks do treat the fundamentals, but by their very nature they concentrate on specific topics, and do so at an advanced technical level that is usually unsuitable for the general reader. Even professional programmers and systems analysts might lack the background or motivation required to get through books aimed at full-time computer science students.

Curiously, there appears to be very little written material devoted to the science of computing and aimed at the technically-oriented general reader as well as the computer professional. This fact is doubly curious in view of the abundance of precisely this kind of literature in most other scientific areas, such as physics, biology, chemistry, and mathematics, not to mention humanities and the arts. There appears to be an acute need for a technically-detailed, expository account of the fundamentals of computer science; one that suffers as little as possible from the bit/byte or semicolon syndromes an their derivatives, one that transcends the technological and linguistic whirlpool of specifics, and one that is useful both to a sophisticated layperson and to a computer expert. It seems that we have all been too busy with the revolution to be bothered with satisfying such a need.

This book is an attempt in this direction. Its objective is to present a readable account of some of the mot important and basic topics of computer science, stressing the fundamental and robust nature of the science in a form that is virtually independent of the details of specific computers, languages, and formalisms.

This book grew out of a series of lectures given by the author on "Galei Zahal", one of Israel's national radio channels, between October 1984 and January 1985. It is about what shall be called algorithmics in this book, that is, the study of algorithms. An algorithm is an abstract recipe, prescribing a process that might be carried out by a human, by a computer, or by other means. It thus represents a very general concept, with numerous applications. Its principal interest and use, however, is in those areas where the process is to be carried out by a computer.

The book could be used as the basis of one-semester introductory course in computer science or a general computer science literacy course in science and engineering schools. Moreover, it can be used as supplementary reading in many kinds of computer-related educational activities, from basic programming courses to advanced graduate or undergraduate degree programs in computer science. The material covered herein, while not directly aimed at producing better programmers or system analysts, can aid people who work with computers by providing an overall picture of some of the most fundamental issues relevant to their work.

The preliminary chapters discuss the concept of an algorithmic problem and the algorithm that solves it, followed by cursory discussions of the structure of algorithms, the data they manipulate, and the languages in which they are programmed. With the stage thus set, the first chapter of Part Two turns to some general methods and paradigms for algorithmic design. This is followed by two chapters on the analysis of algorithms, treating, respectively, their correctness and efficiency (mainly time efficiency), including techniques for establishing the former and estimating the latter. Part Three of the book is devoted to the inherent limitations of effectively executable algorithms, and hence of the computers that implement them. Certain precisely defined problems, including important and practical ones, are shown to be provably not solvable by any computers of reasonable size in any reasonable amount of time (say, the lifetime of a person), and never will be. Worse still, it is shown that some problems are provably not solvable by computers at all, even with unlimited time! In Part Four of the book the requirements are relaxed, for example, by employing concurrent activities or coin tossing, in order to overcome some of these difficulties. These chapters also discuss reactive and distributed systems, and cryptography. Finally, the relationship of computers to human intelligence is discussed, emphasizing the "soft" heuristic, or intuitive, nature of the latter, and the problems involved in relating it to the "hard" scientific subject of algorithmics.

The book is intended to be read or studied sequentially, not be used as a reference. It is organized so that each chapter depends on the previous ones, but with smooth readability in mind. Most of the material in the preliminary Part One should be familiar to people with a background in programming. Thus, Chapters 1 and 2 and parts of Chapter 3 can be browsed through by such readers.

Certain sections contain relatively technical material and can be skipped by the reader without too much loss of continuity. They are indented, set in smaller type and are prefixed by a small square. It is recommended, however, that even those sections be skimmed, at least to get a superficial idea of their contents.

Whenever appropriate, brief discussions of the research topics that are of current interest to computer scientists are included. The text is followed by Bibliographic Notes for each chapter, with "backward" pointers connecting the discussions in the text with the relevant literature.

It is hoped that his book will facilitate communication between the various groups of people who are actively involved in the computer revolution, and between that group, and those who, for the time being, are observers only.

David Harel; Pittsburgh, PA; February 1987

New to the Second Edition

See, this is new; but it has already been

The first edition of this book was intended to be read from beginning to end; it could also be used as a supplementary reading in a number of courses. Teaching a course based exclusively on it was possible, but would have required that the instructor prepare exercises and add examples and more detail in certain places. The present edition contains numerous exercises, as well as solutions to about a third of them. The solved exercises can thus be used to supplement the text.

Three chapters do not have exercises: Chapter 1 is an introduction, the bulk of Chapter 3 is really just a brief survey of several programming languages, and Chapter 12 is a nontechnical account of some topics in artificial intelligence. In a sense, these chapters are not integral parts of the topic of the book -- algorithmics -- and hence in teaching a course based on the book these should probably be assigned as homework reading.

Apart from the inclusion of exercises and solutions, which mark the most obvious change made in this edition, the text has been revised and updated. The reader may wonder why a more extensive revision of the text was not called for. Have computer scientists been idle during the five years since the first edition was published? Rather than taking this as a criticism of the field, I think that it shows that the topics selected for inclusion in the book are really of fundamental nature, so that no significant changes had to be made. The issues discussed herein are thus probably basic and lasting; maybe the term "classical" is most fitting.

David Harel; Rehovot, Israel; May, 1991

New to the Third Edition

they three were of one measure

This time around, a significant revision was carried out. There are several important changes in this edition of the book, compared to the first and second editions, including two brand new chapters, new sections, and more.

The first noticeable difference is that for this revision I needed real help..., and was fortunately joined by Yishai Feldman. He has taken part in all aspects of the revision, but most significantly took upon himself the thorough revision of the material on programming languages and the writing of the new chapter on software engineering .

The main changes are as follows:

The book now has five Parts, rather than four. In Part I (Preliminaries) Chapter 3 has been completely rewritten, and is now titled "Programming Languages and Paradigms." The list of languages discussed has been revised and is organized into paradigms, thus giving a far more informative and updated exposition of the media we use when we program computers. Discussions of some languages (e.g., APL and Snobol) have been dropped altogether and those of others (e.g., C, C++, and Java) have been added.

Part II (Methods and Analysis) and Part III (Limitations and Robustness), i.e., Chapters 4 through 9, required no sweeping changes. This can again be attributed to the "classical" nature of the topics chosen for these, as mentioned in the "New to the Second Edition" section above.

The first chapter of Part IV (Relaxing the Rules) was previously titled "Parallelism and Concurrency" and is now called "Parallelism, Concurrency, and Alternative Models." It incorporates new sections on quantum computing, including Shor's factoring algorithm, and a discussion of molecular computing. These topics may be considered to be additional forms of parallelism, albeit more radical ones. The remaining two chapters of Part IV were constructed by separating out the material on probabilistic algorithms (Chapter 11) from that on cryptography (now Chapter 12) -- presented together in a single chapter in the previous editions -- and extending both by discussions of some of the new developments in these fields.

Part V (The Bigger Picture) ends with the closing chapter of the previous editions, "Algorithms and Intelligence," which is now Chapter 15. However, this is now preceded by two new chapters. Chapter 13, "Software Engineering," and Chapter 14, "Reactive Systems." The first of these is an attempt to provide a general introduction to the issues and problems arising in the development of large software systems. The second new chapter zeros in on the particular difficulties arising in the special case of reactive systems, as a result of their complex behavior over time.

Besides these more noticeable changes, the entire text has been brought up to date in many less subtle and more subtle ways. There are discussions on abstract data types, on the non-approximability of certain NP-complete problems, on probabilistically checkable proofs, and, of course, on the brand new AKS polynomial-time algorithm for primality. The final chapter has been modified in many places too, e.g., with a discussion added on the Chinese room argument.

While we have left the exercises and solutions essentially as they were in the second edition, the bibliographic notes were a completely different story. Twelve years in Computer Science is almost an eternity... The format of the notes is the same as in the previous editions; i.e., a general section at the start of each chapter, which lists relevant books and periodicals, followed by detailed notes that progress with the text of the chapter itself and point back to its page numbers. In revising them, we had to prepare new notes for the large amount of newly added material, of course, but we also had to painstakingly reconsider and thoroughly revise the entire set of existing notes. Hopefully, the result of all of this will turn out to be a useful and up-to-date tool linking the text of this expository book with the accepted archival scientific literature.

Now that the revision is done, if hard-pressed to give my list of the most significant developments in pure, "classical" algorithmics (i.e., excluding software and systems engineering) in the last dozen or so years, it would probably contain three: the non-approximability results for NP-complete problems, Shor's QPTIME factoring algorithm, and the AKS PTIME primality test. And all I can say about these is this: wouldn't it be wonderful if the bulk of the work on the next edition of this book -- if and when, of course -- will be spent on results of similar calibre and importance.

David Harel; Rehovot, Israel; August, 2003

a threefold cord is not quickly broken

Write the vision, and make it plain upon tablets
that he who reads it may run