We use network coding to improve the speed of distributed computation in the dynamic network model of Kuhn, Lynch and Oshman [STOC '10]. In this model an adversary adaptively chooses a new network topology in every round, making even basic distributed computations challenging. show that n nodes, each starting with a d-bit token, can broadcast them to all nodes in time O(n^2) using b-bit messages, where b > d + log n. Their algorithms take the natural approach of token forwarding : in every round each node broadcasts some particular token it knows. They prove matching Omega(n^2) lower bounds for a natural class of token forwarding algorithms and an Omega(n log n) lower bound that applies to all token-forwarding algorithms. We use network coding, transmitting random linear combinations of tokens, to break both lower bounds. Our algorithm's performance is quadratic in the message size b, broadcasting the n tokens in roughly d b^2 * n^2 rounds. For b = d = O(log n) our algorithms use O(n^2 log n) rounds, breaking the first lower bound, while for larger message sizes we obtain linear-time algorithms. We also consider networks that change only every T rounds, and achieve an additional factor T^2 speedup. This contrasts with related lower and upper bounds of implying that for natural token-forwarding algorithms a speedup of T, but not more, can be obtained. Lastly, we give a general way to derandomize random linear network coding, that also leads to new deterministic information dissemination algorithms.
Next to @cite_7 the line of research most relevant to this work is network coding for gossip problems @cite_13 @cite_0 @cite_8 @cite_12 and most specifically work by Haeupler @cite_1 . Since its introduction @cite_6 @cite_4 network coding has revolutionized the understanding of information flow in networks and found many practical applications (see, e.g., the books @cite_5 @cite_9 ).
@cite_4 @cite_7 @cite_8 @cite_9 @cite_1 @cite_6 @cite_0 @cite_5 @cite_13 @cite_12
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In this classic book, first published in 1962, L. R. Ford, Jr., and D. R. Fulkerson set the foundation for the study of network flow problems. The models and algorithms introduced in Flows in Networks are used widely today in the fields of transportation systems, manufacturing, inventory planning, image processing, and Internet traffic. The techniques presented by Ford and Fulkerson spurred the development of powerful computational tools for solving and analyzing network flow models, and also furthered the understanding of linear programming. In addition, the book helped illuminate and unify results in combinatorial mathematics while emphasizing proofs based on computationally efficient construction. Flows in Networks is rich with insights that remain relevant to current research in engineering, management, and other sciences. This landmark work belongs on the bookshelf of every researcher working with networks. We introduce a new class of problems called network information flow which is inspired by computer network applications. Consider a point-to-point communication network on which a number of information sources are to be multicast to certain sets of destinations. We assume that the information sources are mutually independent. The problem is to characterize the admissible coding rate region. This model subsumes all previously studied models along the same line. We study the problem with one information source, and we have obtained a simple characterization of the admissible coding rate region. Our result can be regarded as the max-flow min-cut theorem for network information flow. Contrary to one's intuition, our work reveals that it is in general not optimal to regard the information to be multicast as a "fluid" which can simply be routed or replicated. Rather, by employing coding at the nodes, which we refer to as network coding, bandwidth can in general be saved. This finding may have significant impact on future design of switching systems. We study the stopping times of gossip algorithms for network coding. We analyze algebraic gossip (i.e., random linear coding) and consider three gossip algorithms for information spreading Pull, Push, and Exchange. The stopping time of algebraic gossip is known to be linear for the complete graph, but the question of determining a tight upper bound or lower bounds for general graphs is still open. We take a major step in solving this question, and prove that algebraic gossip on any graph of size n is O(Δn) where Δ is the maximum degree of the graph. This leads to a tight bound of Θ(n) for bounded degree graphs and an upper bound of O(n2) for general graphs. We show that the latter bound is tight by providing an example of a graph with a stopping time of Ω(n2). Our proofs use a novel method that relies on Jackson's queuing theorem to analyze the stopping time of network coding; this technique is likely to become useful for future research. Store-and-forward had been the predominant technique for transmitting information through a network until its optimality was refuted by network coding theory. Network coding offers a new paradigm for network communications and has generated abundant research interest in information and coding theory, networking, switching, wireless communications, cryptography, computer science, operations research, and matrix theory. To comprehend the multipartite organization of large-scale biological and social systems, we introduce an information theoretic approach that reveals community structure in weighted and directed networks. We use the probability flow of random walks on a network as a proxy for information flows in the real system and decompose the network into modules by compressing a description of the probability flow. The result is a map that both simplifies and highlights the regularities in the structure and their relationships. We illustrate the method by making a map of scientific communication as captured in the citation patterns of >6,000 journals. We discover a multicentric organization with fields that vary dramatically in size and degree of integration into the network of science. Along the backbone of the network—including physics, chemistry, molecular biology, and medicine—information flows bidirectionally, but the map reveals a directional pattern of citation from the applied fields to the basic sciences. Gupta and Kumar (see IEEE Transactions an Information Theory, vol.46, no.2, p.388-404, 2000) determined the capacity of wireless networks under certain assumptions, among them point-to-point coding, which excludes for example multi-access and broadcast codes. We consider essentially the same physical model of a wireless network under a different traffic pattern, namely the relay traffic pattern, but we allow for arbitrarily complex network coding. In our model, there is only one active source destination pair, while all other nodes assist this transmission. We show code constructions leading to achievable rates and derive upper bounds from the max-flow min-cut theorem. It is shown that lower and upper bounds meet asymptotically as the number of nodes in the network goes to infinity, thus proving that the capacity of the wireless network with n nodes under the relay traffic pattern behaves like log n bits per second. This demonstrates also that network coding is essential: under the point-to-point coding assumption considered by , the achievable rate is constant, independent of the number of nodes. Moreover, the result of this paper has implications' and extensions to fading channels and to sensor networks. This book contains a thorough discussion of the classical topics in information theory together with the first comprehensive treatment of network coding, a subject first emerged under information theory in the mid 1990's that has now diffused into coding theory, computer networks, wireless communications, complexity theory, cryptography, graph theory, etc. With a large number of examples, illustrations, and original problems, this book is excellent as a textbook or reference book for a senior or graduate level course on the subject, as well as a reference for researchers in related fields. Network coding has shown the promise of significant throughput improvement. In this paper, we study the network throughput using network coding and explore how the maximum throughput can be achieved in a two-way relay wireless network. Unlike previous studies, we consider a more general network with arbitrary structure of overhearing status between receivers and transmitters. To efficiently utilize the coding opportunities, we invent the concept of network coding cliques (NCCs), upon which a formal analysis on the network throughput using network coding is elaborated. In particular, we derive the closed-form expression of the network throughput under certain traffic load in a slotted ALOHA network with basic medium access control. Furthermore, the maximum throughput as well as optimal medium access probability at each node is studied under various network settings. Our theoretical findings have been validated by simulation as well. Network coding promises to significantly impact the way communications networks are designed, operated, and understood. The first book to present a unified and intuitive overview of the theory, applications, challenges, and future directions of this emerging field, this is a must-have resource for those working in wireline or wireless networking. *Uses an engineering approach - explains the ideas and practical techniques *Covers mathematical underpinnings, practical algorithms, code selection, security, and network management *Discusses key topics of inter-session (non-multicast) network coding, lossy networks, lossless networks, and subgraph-selection algorithms Starting with basic concepts, models, and theory, then covering a core subset of results with full proofs, Ho and Lun provide an authoritative introduction to network coding that supplies both the background to support research and the practical considerations for designing coded networks. This is an essential resource for graduate students and researchers in electronic and computer engineering and for practitioners in the communications industry. This comprehensive treatment of network information theory and its applications provides the first unified coverage of both classical and recent results. With an approach that balances the introduction of new models and new coding techniques, readers are guided through Shannon's point-to-point information theory, single-hop networks, multihop networks, and extensions to distributed computing, secrecy, wireless communication, and networking. Elementary mathematical tools and techniques are used throughout, requiring only basic knowledge of probability, whilst unified proofs of coding theorems are based on a few simple lemmas, making the text accessible to newcomers. Key topics covered include successive cancellation and superposition coding, MIMO wireless communication, network coding, and cooperative relaying. Also covered are feedback and interactive communication, capacity approximations and scaling laws, and asynchronous and random access channels. This book is ideal for use in the classroom, for self-study, and as a reference for researchers and engineers in industry and academia.