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ExcerptGraphs and graph algorithms are pervasive in modern computing applications. This book describes the most important known methods for solving the graphprocessing 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. AlgorithmsThis book is the second of three volumes that are intended to survey the most important computer algorithms in use today. The first volume (Parts 14) 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 68) 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 selfstudy 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 objectoriented 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 Nbody simulation problems in physics to geneticsequencing 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. ScopeThis 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 highlevel language such as C++, Java, or C. Algorithms in C++, Third Edition, Parts 14 is certainly adequate preparation. This volume assumes basic knowledge about arrays, linked lists, and ADT design, and makes use of priorityqueue, symbol table, and unionfind ADTsall of which are described in detail in Parts 14 (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 14. 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 CurriculumThere 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 14; 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 exercisesnearly all of which are new to this editionfall 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 UseAnyone wanting to use a computer more effectively can use this book for reference or for selfstudy. 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 pseudocode, 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 LanguageThe 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 Algol60, Basic, Fortran, Smalltalk, Ada, Pascal, C, C++, Modula3, PostScript, Java, or any of the countless other programming languages and environments in which it has proved to be an effective graphprocessing 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 PrefaceBob 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 graphsearch 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 subroutinessee, 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 14, our programs rely on the stack and queue classes defined there, and we write explicit pointer operations on singlylinked lists in two lowlevel implementations. We have adopted two stylistic changes from Parts 14: 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 STL also defines the assignment operator, copy constructor, and destructor needed to make vectors firstclass 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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