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 $$O(\log ^2{n}/\log \log {n})$$O(log2n/loglogn) rounds, with high probability in the system size n. We describe and analyze an algorithm that solves the problem in $$O(\log {n} + \log {R})$$O(logn+logR) 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 $$\varOmega (\log {n})$$Ω(logn) 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 $$\varOmega (\log ^2{n})$$Ω(log2n) 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 $$\log ^2{n}$$log2n 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.

中文翻译：

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衰落信道上的争用解决

在本文中，我们研究了单跳衰落信道上竞争解决的上限和下限；即，接收行为由信号干扰和噪声比方程确定的信道。之前最著名的解决方案在 $$O(\log ^2{n}/\log \log {n})$$O(log2n/loglogn) 轮中解决了这个设置中的问题，在系统大小 n 中的概率很高. 我们描述并分析了在 $$O(\log {n} + \log {R})$$O(logn+logR) 轮中解决问题的算法，其中 R 是最长和最短链接之间的比率，并且对于大多数可行的部署，是一个以 n 中的多项式为上限的值。我们用 $$\varOmega (\log {n})$$Ω(logn) 下界补充这个结果，证明合理 R 的边界很紧。我们注意到，在经典无线电网络模型中（不包括信号衰落), 高概率争用解决需要 $$\varOmega (\log ^2{n})$$Ω(log2n) 轮。因此，我们的算法肯定了这样一个猜想，即通过衰落实现的频谱重用应该允许分布式算法在此 $$\log ^2{n}$$log2n 速度限制上实现显着改进。此外，我们认为证明我们的上限和下限所需的新技术通常用于在这种越来越深入研究的衰落信道设置中分析其他分布式算法。