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 Many aspects of the internal and external workings of computers can be viewed as a series of communication processes. Communication complexity is the mathematical theory of such communication processes. It is also often used as an abstract model of other aspects of computation. This book surveys this mathematical theory, concentrating on the question of how much communication is necessary for any particular process. The first part of the book is devoted to the simple two-party model introduced by Yao in 1979, which is still the most widely studied model. The second part treats newer models developed to deal with more complicated communication processes. Finally, applications of these models, including computer networks, VLSI circuits, and data structures, are treated in the third part of the book. This is an essential resource for graduate students and researchers in theoretical computer science, circuits, networks and information theory.

 The communication complexity of a function f(x, y) measures the number of bits that two players, one who knows x and the other who knows y, must exchange to determine the value f(x, y). Communication complexity is a fundamental measure of complexity of functions. Lower bounds on this measure lead to lower bounds on many other measures of computational complexity. This monograph surveys lower bounds in the field of communication complexity. Our focus is on lower bounds that work by first representing the communication complexity measure in Euclidean space. That is to say, the first step in these lower bound techniques is to find a geometric complexity measure, such as rank or trace norm, that serves as a lower bound to the underlying communication complexity measure. Lower bounds on this geometric complexity measure are then found using algebraic and geometric tools.

 Communication Complexity describes a new intuitive model for studying circuit networks that captures the essence of circuit depth. Although the complexity of boolean functions has been studied for almost 4 decades, the main problems the inability to show a separation of any two classes, or to obtain nontrivial lower bounds remain unsolved. The communication complexity approach provides clues as to where to took for the heart of complexity and also sheds light on how to get around the difficulty of proving lower bounds.Karchmer's approach looks at a computation device as one that separates the words of a language from the non-words. It views computation in a top down fashion, making explicit the idea that flow of information is a crucial term for understanding computation. Within this new setting, Communication Complexity gives simpler proofs to old results and demonstrates the usefulness of the approach by presenting a depth lower bound for st-connectivity. Karchmer concludes by proposing open problems which point toward proving a general depth lower bound.Mauricio Karchmer received his doctorate from Hebrew University and is currently a Postdoctoral Fellow at the University of Toronto. Communication Complexity received the 1988 ACM Doctoral Dissertation Award.

 This text collects the lecture notes from the author's course "Communication Complexity (for Algorithm Designers)," taught at Stanford in the winter quarter of 2015. The two primary goals of the text are: (1) Learn several canonical problems in communication complexity that are useful for proving lower bounds for algorithms (disjointness, index, gap-hamming, etc.). (2) Learn how to reduce lower bounds for fundamental algorithmic problems to communication complexity lower bounds. Along the way, readers will also: (3) Get exposure to lots of cool computational models and some famous results about them -- data streams and linear sketches, compressive sensing, space-query time trade-offs in data structures, sublinear-time algorithms, and the extension complexity of linear programs. (4) Scratch the surface of techniques for proving communication complexity lower bounds (fooling sets, corruption arguments, etc.).

 This book constitutes the thoroughly refereed post-conference proceedings of the 16th International Colloquium on Structural Information and Communication Complexity, SIROCCO 2009, held in Piran, Slovenia, in May 2009. The 23 revised full papers presented were carefully reviewed and selected from 53 submissions. The volume also contains two invited papers. SIROCCO addresses topics such as distributed computing, parallel computing, game theory, social networks, networking, mobile computing, peer to peer systems, communication complexity, combinatorial optimization; special focus is put to compact data structures, information dissemination, informative labeling schemes, distributed scheduling, wireless networks and scheduling of transmissions, routing, broadcasting, and localization.

 The communication complexity of two-party protocols is an only 15 years old complexity measure, but it is already considered to be one of the fundamen tal complexity measures of recent complexity theory. Similarly to Kolmogorov complexity in the theory of sequential computations, communication complex ity is used as a method for the study of the complexity of concrete computing problems in parallel information processing. Especially, it is applied to prove lower bounds that say what computer resources (time, hardware, memory size) are necessary to compute the given task. Besides the estimation of the compu tational difficulty of computing problems the proved lower bounds are useful for proving the optimality of algorithms that are already designed. In some cases the knowledge about the communication complexity of a given problem may be even helpful in searching for efficient algorithms to this problem. The study of communication complexity becomes a well-defined indepen dent area of complexity theory. In addition to a strong relation to several funda mental complexity measures (and so to several fundamental problems of com plexity theory) communication complexity has contributed to the study and to the understanding of the nature of determinism, nondeterminism, and random ness in algorithmics. There already exists a non-trivial mathematical machinery to handle the communication complexity of concrete computing problems, which gives a hope that the approach based on communication complexity will be in strumental in the study of several central open problems of recent complexity theory.

 This book constitutes the refereed proceedings of the 17th International Colloquium on Structural Information and Communication Complexity, SIROCCO 2010, held in Sirince, Turkey, in June 2010. The 19 revised full papers presented were carefully reviewed and selected from 37 submissions. The volume also contains the abstract of one invited talk. The papers are organized in topical section on game theory, network algorithms, motion planning, asynchrony, network algorithms, motion planning, topology algorithms, and graph algorithms.

 This book constitutes the refereed proceedings of the 18th International Colloquium on Structural Information and Communication Complexity, SIROCCO 2011, held in Gdańsk, Poland, in June 2011. The 24 revised full papers presented together with 1 survey lecture and 2 invited talks were carefully reviewed and selected from 57 submissions. The papers are organized in topical section on fault tolerance, routing, mobile agents, mobile robots, probabilistic methods, distributed algorithms on graphs, and ad-hoc networks.

 The communication complexity of two-party protocols is an only 15 years old complexity measure, but it is already considered to be one of the fundamen tal complexity measures of recent complexity theory. Similarly to Kolmogorov complexity in the theory of sequential computations, communication complex ity is used as a method for the study of the complexity of concrete computing problems in parallel information processing. Especially, it is applied to prove lower bounds that say what computer resources (time, hardware, memory size) are necessary to compute the given task. Besides the estimation of the compu tational difficulty of computing problems the proved lower bounds are useful for proving the optimality of algorithms that are already designed. In some cases the knowledge about the communication complexity of a given problem may be even helpful in searching for efficient algorithms to this problem. The study of communication complexity becomes a well-defined indepen dent area of complexity theory. In addition to a strong relation to several funda mental complexity measures (and so to several fundamental problems of com plexity theory) communication complexity has contributed to the study and to the understanding of the nature of determinism, nondeterminism, and random ness in algorithmics. There already exists a non-trivial mathematical machinery to handle the communication complexity of concrete computing problems, which gives a hope that the approach based on communication complexity will be in strumental in the study of several central open problems of recent complexity theory.

 The ultimate goal of research in Distributed Computing is to understand the nature, properties and limits of computing in a system of autonomous communicating agents. To this end, it is crucial to identify those factors which are significant for the computability and the communication complexity of problems. A crucial role is played by those factors which can be termed Structural Information: its identification, characterization, analysis, and its impact on communication complexity is an important theoretical task which has immediate practical importance. The purpose of the Colloquia on Structural Information and Communication Complexity (SIROCCO) is to focus explicitly on the interaction between structural information and communication complexity. The Colloquia comprise position papers, presentations of current research, and group discussions. Series 1 contains papers presented at the 1st Colloquium on Structural Information and Communication Complexity, held in Ottawa, Canada. Series 2 contains papers presented at the 2nd Colloquium held in Olympia, Greece.

 This book constitutes the refereed proceedings of the 13th International Colloquium on Structural Information and Communication Complexity, SIROCCO 2006, held in Chester, UK, July 2006. The book presents 24 revised full papers together with three invited talks, on topics in distributed and parallel computing, information dissemination, communication complexity, interconnection networks, high speed networks, wireless and sensor networks, mobile computing, optical computing, autonomous robots, and related areas.

 The ultimate goal of research in Distributed Computing is to understand the nature, properties and limits of computing in a system of autonomous communicating agents. To this end, it is crucial to identify those factors which are significant for the computability and the communication complexity of problems. A crucial role is played by those factors which can be termed Structural Information: its identification, characterization, analysis, and its impact on communication complexity is an important theoretical task which has immediate practical importance. The purpose of the Colloquia on Structural Information and Communication Complexity (SIROCCO) is to focus explicitly on the interaction between structural information and communication complexity. The Colloquia comprise position papers, presentations of current research, and group discussions. Series 1 contains papers presented at the 1st Colloquium on Structural Information and Communication Complexity, held in Ottawa, Canada. Series 2 contains papers presented at the 2nd Colloquium held in Olympia, Greece.

 SIROCCO 2005 was the twelfth in this series, held in Mont Saint-Michel, France, May 24 26, 2005.

 This book constitutes the thoroughly refereed post-conference proceedings of the 20th International Colloquium on Structural Information and Communication Complexity, SIROCCO 2013, held in Ischia, Italy, in July 2013. The 28 revised full papers presented were carefully reviewed and selected from 67 submissions. SIROCCO is devoted to the study of communication and knowledge in distributed systems. Special emphasis is given to innovative approaches and fundamental understanding, in addition to efforts to optimize current designs. The typical areas include distributed computing, communication networks, game theory, parallel computing, social networks, mobile computing (including autonomous robots), peer to peer systems, communication complexity, fault tolerant graph theories and randomized/probabilistic issues in networks.