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Contention resolution on a fading channel

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Abstract

In this paper, we study upper and lower bounds for contention resolution on a single hop fading channel; i.e., a channel where receive behavior is determined by a signal to interference and noise ratio equation. The best known previous solution solves the problem in this setting in \(O(\log ^2{n}/\log \log {n})\) rounds, with high probability in the system size n. We describe and analyze an algorithm that solves the problem in \(O(\log {n} + \log {R})\) rounds, where R is the ratio between the longest and shortest link, and is a value upper bounded by a polynomial in n for most feasible deployments. We complement this result with an \(\varOmega (\log {n})\) lower bound that proves the bound tight for reasonable R. We note that in the classical radio network model (which does not include signal fading), high probability contention resolution requires \(\varOmega (\log ^2{n})\) rounds. Our algorithm, therefore, affirms the conjecture that the spectrum reuse enabled by fading should allow distributed algorithms to achieve a significant improvement on this \(\log ^2{n}\) speed limit. In addition, we argue that the new techniques required to prove our upper and lower bounds are of general use for analyzing other distributed algorithms in this increasingly well-studied fading channel setting.

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Notes

  1. It is, of course, mathematically possible for R to be super-polynomial in n—say, exponential in n—which would cause the \(\log ^2{n}\) bound to dominate. In practice, however, once n grows beyond relatively small numbers, to maintain such a large gap between link distances becomes increasingly infeasible. Even a network size of only 20 nodes, for example, would require that the longest link be on the order of a million times longer than the shortest link.

  2. To establish that \(c>1\), note that \(2^{\epsilon } = 2^{\alpha /2-1}\). Because \(\alpha >2\), c is defined as 2 raised to some small value greater than 0, which implies \(c>1\).

  3. When we say a network has \(\ell \) link classes, we mean there are \(\ell \) link classes that contain at least one of the \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) \) possible links in the network. It is straightforward to show that the algorithm analyzed in this paper solves contention resolution in \(O(\log {n} + \ell )\) rounds.

  4. A technicality is that our lower bound network requires \(n\ge 6\). Therefore, for smaller values of k, our two-player algorithm \(\mathcal {B}\) will have to resort to some default strategy, like flipping coins and broadcasting if the coin comes up heads. For these small constant values of k, it is clear that this strategy will solve the problem with high probability in k (which is a constant) in \(\varOmega (\log {k}) = \varOmega (1)\) rounds. Another technicality is that we assume n is even. It is straightforward to adjust our network construction to accommodate an extra node in the case of odd n.

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Acknowledgements

This research was support in part by the following grants: NSF CCF 1314633, NSF CCF 1320279, NUS FRC T1 251RES1404 and ERC Grant No. 336495 (ACDC).

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Authors and Affiliations

  1. Georgetown University, Washington, DC, USA

    Jeremy T. Fineman & Calvin Newport

  2. National University of Singapore, Singapore, Singapore

    Seth Gilbert

  3. University of Freiburg, Freiburg, Germany

    Fabian Kuhn

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  1. Jeremy T. Fineman
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  2. Seth Gilbert
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  3. Fabian Kuhn
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  4. Calvin Newport
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Correspondence to Calvin Newport.

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Fineman, J.T., Gilbert, S., Kuhn, F. et al. Contention resolution on a fading channel. Distrib. Comput. 32, 517–533 (2019). https://doi.org/10.1007/s00446-018-0323-9

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