Algorithms in C++ Part 5: Graph Algorithms / Edition 3

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Once again, Robert Sedgewick provides a current and comprehensive introduction to important algorithms. The focus this time is on graph algorithms, which are increasingly critical for a wide range of applications, such as network connectivity, circuit design, scheduling, transaction processing, and resource allocation. In this book, Sedgewick offers the same successful blend of theory and practice that has made his work popular with programmers for many years. Christopher van Wyk and Sedgewick have developed concise new C++ implementations that both express the methods in a natural and direct manner and also can be used in real applications.

Algorithms in C++, Third Edition, Part 5: Graph Algorithms is the second book in Sedgewick's thoroughly revised and rewritten series. The first book, Parts 1-4, addresses fundamental algorithms, data structures, sorting, and searching. A forthcoming third book will focus on strings, geometry, and a range of advanced algorithms. Each book's expanded coverage features new algorithms and implementations, enhanced descriptions and diagrams, and a wealth of new exercises for polishing skills. A focus on abstract data types makes the programs more broadly useful and relevant for the modern object-oriented programming environment.

Coverage includes:

  • A complete overview of graph properties and types
  • Diagraphs and DAGs
  • Minimum spanning trees
  • Shortest paths
  • Network flows
  • Diagrams, sample C++ code, and detailed algorithm descriptions

The Web site for this book ( provides additional source code for programmers along with a wide range of academic support materials for educators.

A landmark revision, Algorithms in C++, Third Edition, Part 5 provides a complete tool set for programmers to implement, debug, and use graph algorithms across a wide range of computer applications.


The algorithms included here cover a broad range of fundamental and more advanced methods: sorting, searching, string-processing, geometric, graph, and mathematical algorithms. Readers will see how key algorithms can be implemented, run, debugged, and used in real applications--plus why some algorithms are to be preferred over others.

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Product Details

  • ISBN-13: 9780201361186
  • Publisher: Addison-Wesley
  • Publication date: 12/27/2001
  • Edition description: Third Edition
  • Edition number: 3
  • Pages: 496
  • Sales rank: 1,433,207
  • Product dimensions: 7.70 (w) x 9.20 (h) x 1.20 (d)

Meet the Author

Robert Sedgewick is the William O. Baker Professor of Computer Science at Princeton University. He is a Director of Adobe Systems and has served on the research staffs at Xerox PARC, IDA, and INRIA. He earned his Ph.D from Stanford University under Donald E. Knuth.


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Read an Excerpt


This book is intended to survey the most important computeralgorithms in use today and to teach fundamental techniques tothe growing number of people in need of knowing them. It can beused as a textbook for a second, third, or fourth course incomputer science, after students have acquired some programmingskills and familiarity with computer systems, but before theyhave taken specialized courses in advanced areas of computerscience or computer applications.

Additionally, the book may be useful for self-study or as areference for those engaged in the development of computersystems or applications programs, since it contains a number ofimplementations of useful algorithms and detailed information ontheir performance characteristics. The broad perspective takenin the book makes it an appropriate introduction to the field.

The algorithms are expressed in the C++ programming language(versions of the book in Pascal and C are also available). Nospecific knowledge about the language is assumed—the treatmenthere is self-contained (though fast-paced). Readers who havesome familiarity with C++ will find the language a useful vehiclefor learning a variety of methods of practical interest. Readerswho have some familiarity with basic algorithms will find thetreatment here a useful vehicle for learning a variety offeatures of the C++ language, while at the same time learningsome new algorithms.


The book contains forty-five chapters grouped into eight majorparts: fundamentals, sorting, searching, string processing,geometric algorithms, graph algorithms, mathematical algorithmsand advancedtopics. A major goal in developing this book hasbeen to bring together the fundamental methods from these diverseareas, in order to provide access to the best methods known forsolving problems by computer. Some of the chapters giveintroductory treatments of advanced material. It is hoped thatthe descriptions here can give readers some understanding of thebasic properties of fundamental algorithms ranging from priorityqueues and hashing to simplex and the fast Fourier transform.

One or two previous courses in computer science or equivalentprogramming experience are recommended for a reader to be able toappreciate the material in this book: one course in programmingin high-level languages such as C++, C or Pascal, and perhapsanother course which teaches fundamental concepts of programmingsystems. This book is thus intended for anyone conversant with amodern programming language and with the basic features of moderncomputer systems. References that might help fill in gaps inone's background are suggested in the text.

Most of the mathematical material supporting the analytic resultsis self-contained (or labeled as "beyond the scope" of thisbook), so little specific preparation in mathematics is requiredfor the bulk of the book, though a certain amount deal withalgorithms related to more advanced mathematical material—theseare intended to place the algorithms in context with othermethods throughout the book, not to teach the mathematicalmaterial. Thus the discussion of advanced mathematical conceptsis brief, general, and descriptive.


There is a great deal of flexibility in how the material here canbe taught. To a large extent, the individual chapters in thebook can be read independently of the others, though in somecases algorithms in one chapter make use of methods from aprevious chapter. The material can be adapted for use forvarious courses by selecting perhaps 25 or 30 of the 45 chapters,according to the taste of the instructor and the preparation ofthe students.

The book begins with an introductory section on data structuresand the design and analysis of algorithms. This sets the tonefor the rest of the book and provides a framework within whichmore advanced algorithms are treated. Some readers may skip orskim this section; others may learn the basics there.

An elementary course on "data structures and algorithms" mightomit some of the mathematical algorithms and some of the advancedtopics, then emphasize how various data structures are used inthe implementations. An intermediate course on "design andanalysis of algorithms" might omit some of the more practicallyoriented sections, then emphasize the identification and study ofthe ways in which algorithms achieve good asymptotic performance. A course on "software tools" might omit the mathematical andadvanced algorithmic material, then emphasize how to integratethe implementations given here into large programs or systems. Acourse on "algorithms" might take a survey approach and introduceconcepts from all these areas.

Some instructors may wish to add supplementary material to thecourses described above to reflect their particular orientation. For "data structures and algorithms," more mathematical analysiscould be added; and for "software tools", software engineeringtechniques could be covered in more depth. In this book,attention is paid to all these areas, but the emphasis is on thealgorithms themselves.

Earlier versions of this book have been used in recent years atscores of colleges and universities around the country as a textfor the second or third course in computer science and assupplemental reading for other courses. At Princeton, ourexperience has been that the breadth of coverage of material inthis book provides our majors with an introduction to computerscience that can be expanded upon in later courses on analysis ofalgorithms, systems programming and theoretical computer science,while at the same time providing all the students with a largeset of techniques that they can immediately put to good use.

There are 450 exercises ten following each chapter, thatgenerally divide into one of two types. Most are intended totest students' understanding of material in the next and askstudents to work through an example or apply concepts describedin the text. A few of them, however, involve implementing andputting together some of the algorithms, perhaps runningempirical studies to compare algorithms and to learn theirproperties.


The orientation of the book is toward algorithms likely to be ofpractical use. The emphasis is on teaching students the tools oftheir trade to the point that they can confidently implement, runand debug useful algorithms. Full implementations of the methodsdiscussed are included in the text, along with descriptions ofthe operations of these programs on a consistent set of examples. Indeed, as discussed in the epilog, hundreds of figures areincluded in the book that have been created by the algorithmsthemselves. Many algorithms are brought to light on an intuitivelevel through the visual dimension provided by these figures.

Characteristics of the algorithms and situations in which theymight be useful are discussed in detail. Though not emphasized,connections to the analysis of algorithms and theoreticalcomputer science are not ignored. When appropriate, empiricaland analytic results are discussed to illustrate why certainalgorithms are preferred. When interesting, the relationship ofthe practical algorithms being discussed to purely theoreticalresults is described. Specific information performancecharacteristics of algorithms is encapsulated throughout in"properties," important facts about the algorithms that deservefurther study.

Some algorithms are used in relatively small programs to solvespecific problems. Others play an integral part in relativelylarge systems. Many fundamental algorithms find application inboth domains. We indicate as appropriate how algorithms might bespecialized for use as problem-solving tools or generalized forintegration into much bigger programs. Such considerations areof particular interest for algorithms expressed in anobject-oriented language such as C++. In this book we providerelevant information that can be used to make intelligenttradeoffs between utility and performance in implementing widelyapplicable algorithms.

While there is little direct treatment of specific uses of thealgorithms in science and engineering applications, the potentialfor such use is mentioned when appropriate. Our experience hasbeen that when students learn good algorithms in a computerscience context early in their education, they are able to applythem to solve problems when warranted later on.


The programming language used throughout the book is C++ (Pascaland C versions of the book are also available). Any particularlanguage has advantages and disadvantages—our intention here isto provide access to the fundamental algorithms that have beendeveloped over the years to the growing number of people who aremoving to C++ as a primary language and using it forapplications. The programs can easily be translated to othermodern programming languages, since they are written in astylized form that are relatively language-independent. Indeed,many of the programs have been translated from Pascal, C, andother languages, though we try to use standard C++ idioms whenappropriate. On the other hand, C++ is particularly well-suitedto our task, because its basic support for data abstraction andmodular programming allows us to clearly express relationshipsamong algorithms and data structures.

Some of the programs can simplified by using more advancedlanguage feature, but this is true less often than one might think. Although language features are discussed when appropriate, thisbook is not intended to be reference work on C++ or onobject-oriented programming. While we use C++ classes heavily,we do not use templates, inheritance or virtual functions. Butthe algorithms are coded so as to ease the process of embeddingthem in large systems where such features can be used toadvantage in an object-oriented programming approach. Whenforced to make a choice, we concentrate on the algorithms, notlanguage feature or implementations details.

A goal of this book is to present the algorithms in as simple anddirect a form as possible. The programs are intended to be readnot by themselves, but as part of the surrounding text. Thisstyle was chosen as an alternative, for example, to having inlinecomments. The style is consistent whenever possible, so thatprograms that are similar look similar.


Many people gave me helpful feedback on earlier versions of thisbook. In particular, students at Princeton and Brown sufferedthrough preliminary versions ofthe material in this book in the1980's. Special thanks are due to Trina Avery, Tom Freeman andJanet Incerpi for their help in producing the first edition. Iwould particularly like to thank Janet for converting the bookinto Tex format, adding the thousands of changes I made after the"last draft" of the first edition, guiding the files throughvarious systems to produce printed pages and even writing thescan-conversion routine for Tex used to produce draftmanuscripts, among many other things. Only after performing manyof these tasks myself for later versions do I truly appreciateJanet's contribution. I would also like to thank the manyreaders who provided me with detailed comments about the secondedition, including Guy Almes, Jay Gischer, Kennedy Lemke, UdiManber, Dana Richards, John Reif, M. Rosenfield, Stephen Seidman,and Michael Quinn.

Many of the designs in the figures are based on joint work withMarc Brown in the "electronic classroom" project at BrownUniversity in 1983. Marc's support and assistance in creatingthe designs (not tomention the system with which we worked) aregratefully acknowledged. I would like to acknowledge SarantosKapidakis' help in producing the endpapers.

This C++ version owes its existence to the persistence of KeithWollman who convinced me to proceed, and the patience of PeterGordon, who was confident that I would get around to doing so. Dave Hanson's willingness to answer questions about C and C++ wasinvaluable. I also would like to thank Darcy Cotten and SkipPlank for their help in producing the book.

Much of what I've written here I've learned from the teachingsand writings of Don Knuth, my advisor at Stanford. Though Donhad no direct influence on this work, his presence may be felt inthe book, for it was he who put the study of algorithms on ascientific footing that makes a work such as this possible.

I am very thankful for the support of Brown University and INRIAwhere I did most of the work on the book, and the Institute forDefense Analyses and the Xerox Palo Alto Research Center, where Idid some work on the book while visiting. Many parts of the bookare dependent on research that has been generously supported bythe National Science Foundation and the Office of Naval Research.Finally, I would like to thank Bill Bowen, Aaron Lemonick, andNeil Rudenstine at Princeton University for their support inbuilding an academic environment in which I was able to preparethis book, despite numerous other responsibilities.

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Table of Contents

17. Graph Properties and Types.


Graph ADT.

Adjacency-Matrix Representation.

Adjacency-Lists Representation.

Variations, Extensions, and Costs.

Graph Generators.

Simple, Euler, and Hamilton Paths.

Graph-Processing Problems.

18. Graph Search.

Exploring a Maze.

Depth-First Search.

Graph-Search ADT Functions.

Properties of DFS Forests.

DFS Algorithms.

Separability and Biconnectivity.

Breadth-First Search.

Generalized Graph Search.

Analysis of Graph Algorithms.

19. Digraphs and DAGs.

Glossary and Rules of the Game.

Anatomy of DFS in Digraphs.

Reachability and Transitive Closure.

Equivalence Relations and Partial Orders.


Topological Sorting.

Reachability in DAGs.

Strong Components in Digraphs.

Transitive Closure Revisited.


20. Minimum Spanning Trees.


Underlying Principles of MST Algorithms.

Prim's Algorithm and Priority-First Search.

Kruskal's Algorithm.

Boruvka's Algorithm.

Comparisons and Improvements.

Euclidean MST.

21. Shortest Paths.

Underlying Principles.

Dijkstra's algorithm.

All-Pairs Shortest Paths.

Shortest Paths in Acyclic Networks.

Euclidean Networks.


Negative Weights.


22. Network Flow.

Flow Networks.

Augmenting-Path Maxflow Algorithms.

Preflow-Push Maxflow Algorithms.

Maxflow Reductions.

Mincost Flows.

Network Simplex Algorithm.

Mincost-Flow Reductions.


References for Part Five.

Index. 0201361183T12172001

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Graphs and graph algorithms are pervasive in modern computing applications. This book describes the most important known methods for solving the graph-processing problems that arise in practice. Its primary aim is to make these methods and the basic principles behind them accessible to the growing number of people in need of knowing them. The material is developed from first principles, starting with basic information and working through classical methods up through modern techniques that are still under development. Carefully chosen examples, detailed figures, and complete implementations supplement thorough descriptions of algorithms and applications.


This book is the second of three volumes that are intended to survey the most important computer algorithms in use today. The first volume (Parts 1-4) covers fundamental concepts (Part 1), data structures (Part 2), sorting algorithms (Part 3), and searching algorithms (Part 4); this volume (Part 5) covers graphs and graph algorithms; and the (yet to be published) third volume (Parts 6-8) covers strings (Part 6), computational geometry (Part 7), and advanced algorithms and applications (Part 8).

The books are useful as texts early in the computer science curriculum, after students have acquired basic programming skills and familiarity with computer systems, but before they have taken specialized courses in advanced areas of computer science or computer applications. The books also are useful for self-study or as a reference for people engaged in the development of computer systems or applications programs because they contain implementations of useful algorithms and detailed information on these algorithms' performance characteristics. The broad perspective taken makes the series an appropriate introduction to the field.

Together the three volumes comprise the Third Edition of a book that has been widely used by students and programmers around the world for many years. I have completely rewritten the text for this edition, and I have added thousands of new exercises, hundreds of new figures, dozens of new programs, and detailed commentary on all the figures and programs. This new material provides both coverage of new topics and fuller explanations of many of the classic algorithms. A new emphasis on abstract data types throughout the books makes the programs more broadly useful and relevant in modern object-oriented programming environments. People who have read previous editions will find a wealth of new information throughout; all readers will find a wealth of pedagogical material that provides effective access to essential concepts.

These books are not just for programmers and computer science students. Everyone who uses a computer wants it to run faster or to solve larger problems. The algorithms that we consider represent a body of knowledge developed during the last 50 years that is the basis for the efficient use of the computer for a broad variety of applications. From N-body simulation problems in physics to genetic-sequencing problems in molecular biology, the basic methods described here have become essential in scientific research; and from database systems to Internet search engines, they have become essential parts of modern software systems. As the scope of computer applications becomes more widespread, so grows the impact of basic algorithms, particularly the fundamental graph algorithms covered in this volume. The goal of this book is to serve as a resource so that students and professionals can know and make intelligent use of graph algorithms as the need arises in whatever computer application they might undertake.


This book, Algorithms in C++, Third Edition, Part 5: Graph Algorithms, contains six chapters that cover graph properties and types, graph search, directed graphs, minimal spanning trees, shortest paths, and networks. The descriptions here are intended to give readers an understanding of the basic properties of as broad a range of fundamental graph algorithms as possible.

You will most appreciate the material here if you have had a course covering basic principles of algorithm design and analysis and programming experience in a high-level language such as C++, Java, or C. Algorithms in C++, Third Edition, Parts 1-4 is certainly adequate preparation. This volume assumes basic knowledge about arrays, linked lists, and ADT design, and makes use of priority-queue, symbol table, and union-find ADTs--all of which are described in detail in Parts 1-4 (and in many other introductory texts on algorithms and data structures).

Basic properties of graphs and graph algorithms are developed from first principles, but full understanding often can lead to deep and difficult mathematics. Although the discussion of advanced mathematical concepts is brief, general, and descriptive, you certainly need a higher level of mathematical maturity to appreciate graph algorithms than you do for the topics in Parts 1-4. Still, readers at various levels of mathematical maturity will be able to profit from this book. The topic dictates this approach: some elementary graph algorithms that should be understood and used by everyone differ only slightly from some advanced algorithms that are not understood by anyone. The primary intent here is to place important algorithms in context with other methods throughout the book, not to teach all of the mathematical material. But the rigorous treatment demanded by good mathematics often leads us to good programs, so I have tried to provide a balance between the formal treatment favored by theoreticians and the coverage needed by practitioners, without sacrificing rigor.

Use in the Curriculum

There is a great deal of flexibility in how the material here can be taught, depending on the taste of the instructor and the preparation of the students. The algorithms described have found widespread use for years, and represent an essential body of knowledge for both the practicing programmer and the computer science student. There is sufficient coverage of basic material for the book to be used in a course on data structures and algorithms, and there is sufficient detail and coverage of advanced material for the book to be used for a course on graph algorithms. Some instructors may wish to emphasize implementations and practical concerns; others may wish to emphasize analysis and theoretical concepts.

For a more comprehensive course, this book is also available in a special bundle with Parts 1-4; thereby instructors can cover fundamentals, data structures, sorting, searching, and graph algorithms in one consistent style. A set of slide masters for use in lectures, sample programming assignments, interactive exercises for students, and other course materials may be found by accessing the book's home page.

The exercises--nearly all of which are new to this edition--fall into several types. Some are intended to test understanding of material in the text, and simply ask readers to work through an example or to apply concepts described in the text. Others involve implementing and putting together the algorithms, or running empirical studies to compare variants of the algorithms and to learn their properties. Still other exercises are a repository for important information at a level of detail that is not appropriate for the text. Reading and thinking about the exercises will pay dividends for every reader.

Algorithms of Practical Use

Anyone wanting to use a computer more effectively can use this book for reference or for self-study. People with programming experience can find information on specific topics throughout the book. To a large extent, you can read the individual chapters in the book independently of the others, although, in some cases, algorithms in one chapter make use of methods from a previous chapter.

The orientation of the book is to study algorithms likely to be of practical use. The book provides information about the tools of the trade to the point that readers can confidently implement, debug, and put to work algorithms to solve a problem or to provide functionality in an application. Full implementations of the methods discussed are included, as are descriptions of the operations of these programs on a consistent set of examples. Because we work with real code, rather than write pseudo-code, the programs can be put to practical use quickly. Program listings are available from the book's home page.

Indeed, one practical application of the algorithms has been to produce the hundreds of figures throughout the book. Many algorithms are brought to light on an intuitive level through the visual dimension provided by these figures.

Characteristics of the algorithms and of the situations in which they might be useful are discussed in detail. Connections to the analysis of algorithms and theoretical computer science are developed in context. When appropriate, empirical and analytic results are presented to illustrate why certain algorithms are preferred. When interesting, the relationship of the practical algorithms being discussed to purely theoretical results is described. Specific information on performance characteristics of algorithms and implementations is synthesized, encapsulated, and discussed throughout the book.

Programming Language

The programming language used for all of the implementations is C++. The programs use a wide range of standard C++ idioms, and the text includes concise descriptions of each construct.

Chris Van Wyk and I developed a style of C++ programming based on classes, templates, and overloaded operators that we feel is an effective way to present the algorithms and data structures as real programs. We have striven for elegant, compact, efficient, and portable implementations. The style is consistent whenever possible, so that programs that are similar look similar.

A goal of this book is to present the algorithms in as simple and direct a form as possible. For many of the algorithms, the similarities remain regardless of which language is used: Dijkstra's algorithm (to pick one prominent example) is Dijkstra's algorithm, whether expressed in Algol-60, Basic, Fortran, Smalltalk, Ada, Pascal, C, C++, Modula-3, PostScript, Java, or any of the countless other programming languages and environments in which it has proved to be an effective graph-processing method. On the one hand, our code is informed by experience with implementing algorithms in these and numerous other languages (a C version of this book is also available, and a Java version will appear soon); on the other hand, some of the properties of some of these languages are informed by their designers' experience with some of the algorithms and data structures that we consider in this book. In the end, we feel that the code presented in the book both precisely defines the algorithms and is useful in practice.
Robert Sedgewick

C++ Consultant's Preface

Bob Sedgewick and I wrote many versions of most of these programs in our quest to implement graph algorithms in clear and natural programs. Because there are so many kinds of graphs and so many different questions to ask about them, we agreed early on not to pursue a single class scheme that would work across the whole book. Remarkably, we ended up using only two schemes: a simple one in Chapters 17 through 19, where the edges of a graph are either present or absent; and an approach similar to STL containers in Chapters 20 through 22, where more information is associated with edges.

C++ classes offer many advantages for presenting graph algorithms. We use classes to collect useful generic functions on graphs (like input/output). In Chapter 18, we use classes to factor out the operations common to several different graph-search methods. Throughout the book, we use an iterator class on the edges emanating from a vertex so that the programs work no matter how the graph is stored. Most important, we package graph algorithms in classes whose constructor processes the graph and whose member functions give us access to the answers discovered. This organization allows graph algorithms to readily use other graph algorithms as subroutines--see, for example, Program 19.13 (transitive closure via strong components), Program 20.8 (Kruskal's algorithm for minimum spanning tree), Program 21.4 (all shortest paths via Dijkstra's algorithm), Program 21.6 (longest path in a directed acyclic graph). This trend culminates in Chapter 22, where most of the programs are built at a higher level of abstraction, using classes that are defined earlier in the book.

For consistency with Algorithms in C++, Third Edition, Parts 1-4, our programs rely on the stack and queue classes defined there, and we write explicit pointer operations on singly-linked lists in two low-level implementations. We have adopted two stylistic changes from Parts 1-4: Constructors use initialization rather than assignment and we use STL vectors instead of arrays. Here is a summary of the STL vector functions we use in our programs:

  • The default constructor creates an empty vector.
  • The constructor vec(n) creates a vector of n elements.
  • The constructor vec(n, x) creates a vector of n elements each initialized to the value x.
  • Member function vec.assign(n, x) makes vec a vector of n elements each initialized to the value x.
  • Member function vec.resize(n) grows or shrinks vec to have capacity n.
  • Member function vec.resize(n, x) grows or shrinks vec to have capacity n and initializes any new elements to the value x.

The STL also defines the assignment operator, copy constructor, and destructor needed to make vectors first-class objects.

Before I started working on these programs, I had read informal descriptions and pseudocode for many of the algorithms, but had only implemented a few of them. I have found it very instructive to work out the details needed to turn algorithms into working programs, and fun to watch them in action. I hope that reading and running the programs in this book will also help you to understand the algorithms better.
Christopher Van Wyk


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