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Reviewer: Juraj Hromkovic

The communication complexity of two-party protocols was introduced by Abelson [1] and Yao [2] in 1978 and 1979. The initial goal was to develop a method for proving lower bounds on the complexity of distributed and parallel computations, with an emphasis on VLSI circuits. In the 20 years of its existence, communication complexity has established itself as an important area of complexity theory for three reasons: There is a well-developed mathematical machinery to approximately determine the communication complexity of concrete problems. Communication complexity has been successfully applied for proving lower bounds on fundamental complexity measures of various computing models. It has been more successful for this purpose than crossing sequence arguments or Kolmogorov complexity. It has contributed to the understanding of the nature and computational power of randomness and nondeterminism. This book is one of the first two monographs on communication complexity. Both consider all three subtopics mentioned above, but the other [3] is mainly devoted to the use of communication complexity for the study of the complexity measures of other fundamental computing models, while this book concentrates on randomness—one of the key topics in modern algorithmics and complexity theory. This monograph consists of three chapters, which are divided into sections. Chapter 1 starts by presenting the model of deterministic and nondeterministic two-party communication protocols and the mathematical models for proving lower bounds on communication complexity. The remaining two sections of this chapter are devoted to randomized protocols. Chapter 2 presents an overview of different versions of the two-party protocol model and its generalizations to multiparty protocol models. Chapter 3 is devoted to applications. First, applications for the study of fundamental computing models, such as VLSI and Boolean circuits, Turing machines, decision trees, and branching programs, are presented. In the rest of this chapter, the authors apply various communication complexity measures and results to the study of randomness in other areas and to cryptography. This excellent monograph surveys the recent research on communication complexity. Almost all of the important concepts, results, and methods are presented. I compliment the authors on the efficiency of their presentation. They have written a short message that presents the main ideas on this topic in a concise, easily readable form. They have thus proven themselves to be experts in efficient communication. I strongly recommend this book to everybody interested in this topic, and to researchers who primarily study the power of randomized computation or who try to prove lower bounds on the complexity of specific computing problems. Its readability and good organization also make it a suitable re source for preparing lectures for graduate students in theoretical computer science and mathematics.