Contention resolution on a fading channel

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.