Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/non-Hamiltonian.In this paper a computational complexity theory of the “knowledge” contained in a proof is developed. Zero-knowledge proofs are defined as those proofs that convey no additional knowledge other than the correctness of the proposition in question. Examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and 'quadratic nonresiduosity. These are the first examples of zero-knowledge proofs for languages not known to be efficiently recognizable.

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The knowledge complexity of interactive proof-systems

The theory of directed graphs has developed enormously over recent decades, yet this book (first published in 2000) remains the only book to cover more than a small fraction of the results. New research in the field has made a second edition a necessity. Substantially revised, reorganised and updated, the book now comprises eighteen chapters, carefully arranged in a straightforward and logical manner, with many new results and open problems. As well as covering the theoretical aspects of the subject, with detailed proofs of many important results, the authors present a number of algorithms, and whole chapters are devoted to topics such as branchings, feedback arc and vertex sets, connectivity augmentations, sparse subdigraphs with prescribed connectivity, and also packing, covering and decompositions of digraphs. Throughout the book, there is a strong focus on applications which include quantum mechanics, bioinformatics, embedded computing, and the travelling salesman problem. Detailed indices and topic-oriented chapters ease navigation, and more than 650 exercises, 170 figures and 150 open problems are included to help immerse the reader in all aspects of the subject. Digraphs is an essential, comprehensive reference for undergraduate and graduate students, and researchers in mathematics, operations research and computer science. It will also prove invaluable to specialists in related areas, such as meteorology, physics and computational biology.

Digraphs Theory Algorithms And Applications

Investigates the class of epidemic algorithms that are commonly used for the lazy transmission of updates to distributed copies of a database. These algorithms use a simple randomized communication mechanism to ensure robustness. Suppose n players communicate in parallel rounds in each of which every player calls a randomly selected communication partner. In every round, players can generate rumors (updates) that are to be distributed among all players. Whenever communication is established between two players, each one must decide which of the rumors to transmit. The major problem is that players might not know which rumors their partners have already received. For example, a standard algorithm forwarding each rumor form the calling to the called players for /spl Theta/(ln n) rounds needs to transmit the rumor /spl Theta/(n ln n) times in order to ensure that every player finally receives the rumor with high probability. We investigate whether such a large communication overhead is inherent to epidemic algorithms. On the positive side, we show that the communication overhead can be reduced significantly. We give an algorithm using only O(n ln ln n) transmissions and O(ln n) rounds. In addition, we prove the robustness of this algorithm. On the negative side, we show that any address-oblivious algorithm needs to send /spl Omega/(n ln ln n) messages for each rumor, regardless of the number of rounds. Furthermore, we give a general lower bound showing that time and communication optimality cannot be achieved simultaneously using random phone calls, i.e. every algorithm that distributes a rumor in O(ln n) rounds needs /spl omega/(n) transmissions.

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Randomized rumor spreading

Randomized Rumor Spreading

We propose a new measure of the communicability of a complex network, which is a broad generalization of the concept of the shortest path. According to the new measure, most of the real-world networks display the largest communicability between the most connected (popular) nodes of the network (assortative communicability). There are also several networks with the disassortative communicability, where the most "popular" nodes communicate very poorly to each other. Using this information we classify a diverse set of real-world complex systems into a small number of universality classes based on their structure-dynamic correlation. In addition, the new communicability measure is able to distinguish finer structures of networks, such as communities into which a network is divided. A community is unambiguously defined here as a set of nodes displaying larger communicability among them than to the rest of the nodes in the network.

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Communicability in complex networks

Considerable attention in recent theoretical computer science is devoted to parallel computing. Here, we would like to present a special part of this large topic, namely, the part devoted to an abstract study of the dissemination of information in interconnection networks. The importance of this research area lies in the fact that the ability of a network to effectively disseminate information is an important qualitative measure for the suitabilty of the network for parallel computing. This follows simply from the observation that the communication among processes working in parallel is one of the main parts of the whole parallel computation. So, the effectivity of information exchange among processors essentially influences the effectivity of the whole computation process.

Dissemination of Information in Interconnection Networks (Broadcasting & Gossiping)

This book is devoted to the investigation of a special topic in theoretical computer science - communication complexity as an abstract measure of the complexity of computing problems. Its main aim is to show how the theoretical study of communication complexity can be useful in the process of designing effective parallel algorithms. The author shows how to get important information about the parallel complexity (parallel time, the number of processors, the descriptional complexity of the topology of the parallel architecture) of specific computing problems from knowledge of their communication complexity. The book is written as a textbook for undergraduate and graduate students and provides a careful explanation of the subject as well as motivation for further research.

Communication Complexity and Parallel Computing

There are several approaches to attack hard problems. All have their merits, but also their limitations, and need a large body of theory as their basis. A number of books for each one exist: books on complexity theory, others on approximation algorithms, heuristic approaches, parametrized complexity, and yet others on randomized algorithms. This book discusses thoroughly all of the above approaches. And, amazingly, at the same time, does this in a style that makes the book accessible not only to theoreticians, but also to the non-specialist, to the student or teacher, and to the programmer. Do you think that mathematical rigor and accessibility contradict? Look at this book to find out that they do not, due to the admirable talent of the author to present his material in a clear and concise way, with the idea behind the approach spelled out explicitly, often with a revealing example. Reading this book is a beautiful experience and I can highly recommend it to anyone interested in learning how to solve hard problems. It is not just a condensed union of material from other books. Because it discusses the different approaches in depth, it has the chance to compare them in detail, and, most importantly, to highlight under what circumstances which approach might be worth exploring. No book on a single type of solution can do that, but this book does it in an absolutely fascinating way that can serve as a pattern for theory textbooks with a high level of generality. (Peter Widmayer) The second edition extends the part on the method of relaxation to linear programming with an emphasis on rounding, LP-duality, and primal-dual schema, and provides a self-contained and transparent presentation of the design of randomized algorithms for primality testing.

Algorithmics for Hard Problems: Introduction to Combinatorial Optimization, Randomization, Approximation, and Heuristics

The Telegraph and Telephone Modes.- Fundamentals.- Broadcasting.- Gossiping.- Systolic Communication.- Fault-Tolerance.- Distributed Networks.- Broadcast on Distributed Networks.- Leader Election in Asynchronous Distributed Networks.- Fault-Tolerant Broadcast in Distributed Networks.

Juraj Hromkovič

Papers

Dissemination of Information in Interconnection Networks (Broadcasting & Gossiping)

Communication Complexity and Parallel Computing

Algorithmics for Hard Problems: Introduction to Combinatorial Optimization, Randomization, Approximation, and Heuristics

Dissemination of information in communication networks: Broadcasting, gossiping, leader election, and fault-tolerance

Algorithmics for Hard Problems