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1982 results in Distributed, Networked and Mobile Computing

Page 46 of 100

  • 19 - Gaussian Networks

  • from Part III - Multihop Networks
  • Abbas El Gamal, Young-Han Kim, University of California, San Diego
  • Book:
    Network Information Theory
    Published online:
    05 June 2012
    Print publication:
    08 December 2011, pp 484-504

  • Summary

    In this chapter, we discuss models for wireless multihop networks that generalize the Gaussian channel models we studied earlier. We extend the cutset bound and the noisy network coding inner bound on the capacity region of the multimessage DMN presented in Chapter 18 to Gaussian networks. We show through a Gaussian two-way relay channel example that noisy network coding can outperform decode–forward and amplify– forward, achieving rates within a constant gap of the cutset bound while the inner bounds achieved by these other schemes can have an arbitrarily large gap to the cutset bound. More generally, we show that noisy network coding for the Gaussian multimessage multicast network achieves rates within a constant gap of the capacity region independent of network topology and channel gains. For Gaussian networks with other messaging demands, e.g., general multiple-unicast networks, however, no such constant gap results exist in general. Can we still obtain some guarantees on the capacity of these networks?

    To address this question, we introduce the scaling-law approach to capacity, where we seek to find the order of capacity scaling as the number of nodes in the network becomes large. In addition to providing some guarantees on network capacity, the study of capacity scaling sheds light on the role of cooperation through relaying in combating interference and path loss in large wireless networks. We first illustrate the scaling-law approach via a simple unicast network example that shows how relaying can dramatically increase the capacity by reducing the effect of high path loss. We then present the Gupta–Kumar random network model in which the nodes are randomly distributed over a geographical area and the goal is to determine the capacity scaling law that holds for most such networks. We establish lower and upper bounds on the capacity scaling law for the multiple-unicast case. The lower bound is achieved via a cellular time-division scheme in which the messages are sent simultaneously using a simple multihop scheme with nodes in cells along the lines from each source to its destination acting as relays. We show that this scheme achieves much higher rates than direct transmission with time division, which demonstrates the role of relaying in mitigating interference in large networks.

  • 14 - Joint Source–Channel Coding

  • from Part II - Single-Hop Networks
  • Abbas El Gamal, Young-Han Kim, University of California, San Diego
  • Book:
    Network Information Theory
    Published online:
    05 June 2012
    Print publication:
    08 December 2011, pp 336-360

  • Summary

    In Chapters 4 through 9, we studied reliable communication of independent messages over noisy single-hop networks (channel coding), and in Chapters 10 through 13, we studied the dual setting of reliable communication of uncompressed sources over noiseless single-hop networks (source coding). These settings are special cases of the more general information flow problem of reliable communication of uncompressed sources over noisy single-hop networks. As we have seen in Section 3.9, separate source and channel coding is asymptotically sufficient for communicating a DMS over a DMC. Does such separation hold in general for communicating a k-DMS over a DM single-hop network?

    In this chapter, we show that such separation does not hold in general. Thus in some multiuser settings it is advantageous to perform joint source–channel coding. We demonstrate this breakdown in separation through examples of lossless communication of a 2-DMS over a DM-MAC and over a DM-BC.

    For the DM-MAC case, we show that joint source–channel coding can help communication by utilizing the correlation between the sources to induce statistical cooperation between the transmitters. We present a joint source–channel coding scheme that outperforms separate source and channel coding. We then show that this scheme can be improved when the sources have a common part, that is, a source that both senders can agree on with probability one.

    For the DM-BC case, we show that joint source–channel coding can help communication by utilizing the statistical compatibility between the sources and the channel. We first consider a separate source and channel coding scheme based on the Gray–Wyner source coding system and Marton's channel coding scheme. The optimal rate–region for the Gray–Wyner system naturally leads to several definitions of common information between correlated sources. We then describe a joint source–channel coding scheme that outperforms the separate Gray–Wyner and Marton coding scheme.

    Finally, we present a general single-hop network that includes as special cases many of themultiuser source and channel settings we discussed in previous chapters. We describe a hybrid source–channel coding scheme for this network.

  • 15 - Graphical Networks

  • from Part III - Multihop Networks
  • Abbas El Gamal, Young-Han Kim, University of California, San Diego
  • Book:
    Network Information Theory
    Published online:
    05 June 2012
    Print publication:
    08 December 2011, pp 363-381

  • Summary

    So far we have studied single-hop networks in which each node is either a sender or a receiver. In this chapter, we begin the discussion of multihop networks, where some nodes can act as both senders and receivers and hence communication can be performed over multiple rounds. We consider the limits on communication of independent messages over networks modeled by a weighted directed acyclic graph. This network model represents, for example, a wired network or a wireless mesh network operated in time or frequency division, where the nodes may be servers, handsets, sensors, base stations, or routers. The edges in the graph represent point-to-point communication links that use channel coding to achieve close to error-free communication at rates below their respective capacities. We assume that each node wishes to communicate a message to other nodes over this graphical network. The nodes can also act as relays to help other nodes communicate their messages. What is the capacity region of this network?

    Although communication over such a graphical network is not hampered by noise or interference, the conditions on optimal information flow are not known in general. The difficulty arises in determining the optimal relaying strategies when several messages are to be sent to different destination nodes.

    We first consider the graphical multicast network, where a source node wishes to communicate a message to a set of destination nodes. We establish the cutset upper bound on the capacity and show that it is achievable error-free via routing when there is only one destination, leading to the celebrated max-flow min-cut theorem. When there are multiple destinations, routing alone cannot achieve the capacity, however. We show that the cutset bound is still achievable, but using more sophisticated coding at the relays. The proof of this result involves linear network coding in which the relays perform simple linear operations over a finite field.

    We then consider graphical networks with multiple independent messages. We show that the cutset bound is tight when the messages are to be sent to the same set of destination nodes (multimessage multicast), and is achieved again error-free using linear network coding. When each message is to be sent to a different set of destination nodes, however, neither the cutset bound nor linear network coding is optimal in general.

  • 24 - Networking and InformationTheory

  • from Part IV - Extensions
  • Abbas El Gamal, Young-Han Kim, University of California, San Diego
  • Book:
    Network Information Theory
    Published online:
    05 June 2012
    Print publication:
    08 December 2011, pp 600-620

  • Summary

    The source and network models we discussed so far capture many essential ingredients of real-world communication networks, including

  • • noise,

  • • multiple access,

  • • broadcast,

  • • interference,

  • • time variation and uncertainty about channel statistics,

  • • distributed compression and computing,

  • • joint source–channel coding,

  • • multihop relaying,

  • • node cooperation,

  • • interaction and feedback, and

  • • secure communication.

    • Although a general theory for information flow under these models remains elusive, we have seen that there are several coding techniques—some of which are optimal or close to optimal—that promise significant performance improvements over today's practice. Still, the models we discussed do not capture other key aspects of real-world networks.

  • • We assumed that data is always available at the communication nodes. In real-world networks, data is bursty and the nodes have finite buffer sizes.

  • • We assumed that the network has a known and fixed number of users. In real-world networks, users can enter and leave the network at will.

  • • We assumed that the network operation is centralized and communication over the network is synchronous. Many real-world networks are decentralized and communication is asynchronous.

  • • We analyzed performance assuming arbitrarily long delays. In many networking applications, delay is a primary concern.

  • • We ignored the overhead (protocol) needed to set up the communication as well as the cost of feedback and channel state information.

    • While these key aspects of real-world networks have been at the heart of the field of computer networks, they have not been satisfactorily addressed by network information theory, either because of their incompatibility with the basic asymptotic approach of information theory or because the resulting models are messy and intractable. There have been several success stories at the intersection of networking and network information theory, however. In this chapter we discuss three representative examples.

    We first consider the channel coding problem for a DMC with random data arrival. We show that reliable communication is feasible provided that the data arrival rate is less than the channel capacity. Similar results can be established for multiuser channels and multiple data streams. A key new ingredient in this study is the notion of queue stability.

Advanced Topics in Bisimulation and Coinduction
  • Edited by Davide Sangiorgi, Jan Rutten
  • Published online:
    05 November 2011
    Print publication:
    13 October 2011
  • Coinduction is a method for specifying and reasoning about infinite data types and automata with infinite behaviour. In recent years, it has come to play an ever more important role in the theory of computing. It is studied in many disciplines, including process theory and concurrency, modal logic and automata theory. Typically, coinductive proofs demonstrate the equivalence of two objects by constructing a suitable bisimulation relation between them. This collection of surveys is aimed at both researchers and Master's students in computer science and mathematics and deals with various aspects of bisimulation and coinduction, with an emphasis on process theory. Seven chapters cover the following topics: history, algebra and coalgebra, algorithmics, logic, higher-order languages, enhancements of the bisimulation proof method, and probabilities. Exercises are also included to help the reader master new material.

Page 46 of 100