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Reviewer: John B. Slater

Some books take longer to review than others. When two eminent practitioners with a lot of experience in writing for various audiences produce a book of this length on the analysis of algorithms, it is unlikely that the critical reader is going to appreciate everything in one read. Even after a couple of months of delving, one continues to find new material, especially in the plentiful exercises, which are packed with examples that can be used in a course in this area. Anyone conducting such a course should have this book on the shelf. The authors claim that the work has a variety of uses—as a textbook in a third- or fourth-year course on mathematical analysis of algorithms, in a discrete math course for computer science students, and as backup to a course introducing mathematicians to data structures and algorithms. It might also be used as the basis of an introductory graduate course or as a reference work for researchers. There is no doubt whatsoever about the book's considerable advantages for these purposes, though. The book is divided into eight chapters, of which chapters 2 through 4 give the underlying mathematical theory with practical examples. Chapters 5 through 8 concentrate more on applications in actual algorithms and combinatorial structures and have more of a computer science feel. Chapter 1 is relatively short; it endeavors to illustrate the interplay between the two approaches and to set the book in context. It does this through a combination of high-level discussion and concrete examples likely to be familiar in some form to all possible readers, such as Quicksort. It effectively demonstrates the techniques and thinking behind the book, but many undergraduates may be unable to appreciate it immediately. In this chapter, the dense presentation is most intrusive. Later in the book, there is much more white space, with tables and formulas spread out, but here the words close in on the reader. The remainder of the book is excellently laid out and accurate. Each chapter has its own set of references, which are helpful and extensive and overlap appropriately. Each chapter also contains a summary, which gives an overview of the material and places the chapter in context. Other meta-discussion helps readers find their way through the book. The notation is clear, standard, and built up nicely as needed. A helpful list of themes and an extensive index are also provided. The exercises provided (without solutions) are, however, the best feature of the work. Their choice and positioning show the considerable care taken, and their use will repay any reader. The second chapter is the easiest for a third-year student to follow. The relationship with interactive and recursive techniques is brought out, as is the general way to approach a difference equation; the authors clearly differentiate between exact and precise methods. The third chapter introduces generating functions initially as formal entities, and this algebraic approach pervades. This chapter contains a number of excellent reference tables. The notation and analysis are clear. The fourth chapter, completing the mathematical part, is on asymptotics, and it will be much harder for students to follow without substantial help. The important Ramanujan distributions are covered alongside the more traditional normal and Poisson distributions. A wealth of examples help readers understand the inevitably hard math problems of bounding tails and approximating sums with integrals. In the computer science chapters, the authors visit trees and their enumeration and applications (in chapter 5), permutations and enumeration problems (in chapter 6), strings and tries (in chapter 7), and words and maps (in chapter 8). They give clear leads into much current research, and examples that start out with simple applications of the earlier material are rapidly shown to link with the state of the art. This is only possible with the wealth of knowledge that comes from being at the forefront of research in this area. This book is for teachers, researchers, or graduate students. It is a book from which to get examples and to have available to support many types of courses. Its suitability as a course book for undergraduates is less clear.