## 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*.

## References

- 1.
Abramson, N.: The ALOHA system: another alternative for computer communications. In: Proceedings of the Fall Joint Computer Conference (1970)

- 2.
Bar-Yehuda, R., Goldreich, O., Itai, A.: On the time complexity of broadcast in radio networks: an exponential gap between determinism and randomization. In: Proceedings of the International Symposium on Principles of Distributed Computing (1987)

- 3.
Chlamtac, I., Kutten, S.: On broadcasting in radio networks: problem analysis and protocol design. In: Proceedings of the Conference on Computer Communication (1985)

- 4.
Daum, S., Gilbert, S., Kuhn, F., Newport, C.: Broadcast in the ad hoc SINR model. In: Proceedings of the International Symposium on Distributed Computing (2013)

- 5.
Daum, S., Kuhn, F., Newport, C.: Efficient symmetry breaking in multi-channel radio networks. In: Proceedings of the International Symposium on Distributed Computing (2012)

- 6.
Gallager, R.: A perspective on multiaccess channels. IEEE Trans. Inf. Theory

**31**(2), 124–142 (1985) - 7.
Gasieniec, L., Pelc, A., Peleg, D.: The wakeup problem in synchronous broadcast systems. SIAM J. Discrete Math.

**14**(2), 207–222 (2001) - 8.
Goussevskaia, O., Moscibroda, T., Wattenhofer, R.: Local broadcasting in the physical interference model. In: Proceedings of the International Workshop on the Foundations of Mobile Computing. ACM (2008)

- 9.
Greenberg, A., Winograd, S.: A lower bound on the time needed in the worst case to resolve conflicts deterministically in multiple access channels. J. ACM

**32**(3), 589–596 (1985) - 10.
Hajek, B., van Loon, T.: Decentralized dynamic control of a multiaccess broadcast channel. IEEE Trans. Autom. Control

**27**(3), 559–569 (1982) - 11.
Halldorsson, M.M., Mitra, P.: Distributed connectivity of wireless networks. In: Proceedings of the ACM Conference on Distributed Computing (2012)

- 12.
Halldorsson, M.M., Mitra, P.: Towards tight bounds for local broadcasting. In: Proceedings of the International Workshop on the Foundations of Mobile Computing. ACM (2012)

- 13.
Jurdzinski, T., Kowalski, D.: Distributed backbone structure for algorithms in the SINR model of wireless networks. In: Proceedings of the International Conference on Distributed Computing (2012)

- 14.
Jurdzinski, T., Kowalski, D.R., Rozanski, M., Stachowiak, G.: On the impact of geometry on ad hoc communication in wireless networks. In: Proceedings of the ACM Conference on Distributed Computing (2014)

- 15.
Jurdziński, T., Stachowiak, G.: Probabilistic algorithms for the wakeup problem in single-hop radio networks. In: Bose, P., Morin, P. (eds.) Algorithms and Computation. ISAAC 2002. Lecture Notes in Computer Science, vol 2518. Springer, Berlin

- 16.
Jurdzinski, T., Stachowiak, G.: The cost of synchronizing multiple-access channels. In: Proceedings of the ACM Conference on Distributed Computing (2015)

- 17.
Kaplan, M.: A sufficient condition for non-ergodicity of a Markov chain. IEEE Trans. Inf. Theory

**25**, 470–471 (1979) - 18.
Komlos, J., Greenberg, A.: An asymptotically nonadaptive algorithm for conflict resolution in multiple-access channels. IEEE Trans. Inf. Theory

**31**(2), 302–306 (1985) - 19.
Moscibroda, T., Wattenhofer, R.: The complexity of connectivity in wireless networks. In: Proceedings of the IEEE International Conference on Computer Communications (2006)

- 20.
Newport, C.: Radio network lower bounds made easy. In: Proceedings of the International Symposium on Distributed Computing (2014)

- 21.
Roberts, L.G.: ALOHA packet system with and without slots and capture. ACM SIGCOMM Comput. Commun. Rev.

**5**(2), 28–42 (1975) - 22.
Scheideler, C., Richa, A., Santi, P.: An \(O(\log {n})\) dominating set protocol for wireless ad-hoc networks under the physical interference model. In: Proceedings of the International Symposium on Mobile Ad Hoc Networking and Computing (2008)

- 23.
Willard, D.E.: Log-logarithmic selection resolution protocols in a multiple access channel. SIAM J. Comput.

**15**(2), 468–477 (1986)

## 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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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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### Keywords

- Contention resolution
- Leader election
- Wireless channel
- Wireless algorithms
- SINR model