Beeping models are models for networks of weak devices, such as sensor networks or biological networks. In these networks, nodes are allowed to communicate only via emitting beeps: unary pulses of energy. Listening nodes only the capability of {\it carrier sensing}: they can only distinguish between the presence or absence of a beep, but receive no other information. The noisy beeping model further assumes listening nodes may be disrupted by random noise. Despite this extremely restrictive communication model, it transpires that complex distributed tasks can still be performed by such networks. In this paper we provide an optimal procedure for simulating general message passing in the beeping and noisy beeping models. We show that a round of \textsf{Broadcast CONGEST} can be simulated in $O(\Delta\log n)$ round of the noisy (or noiseless) beeping model, and a round of \textsf{CONGEST} can be simulated in $O(\Delta^2\log n)$ rounds (where $\Delta$ is the maximum degree of the network). We also prove lower bounds demonstrating that no simulation can use asymptotically fewer rounds. This allows a host of graph algorithms to be efficiently implemented in beeping models. As an example, we present an $O(\log n)$-round \textsf{Broadcast CONGEST} algorithm for maximal matching, which, when simulated using our method, immediately implies a near-optimal $O(\Delta \log^2 n)$-round maximal matching algorithm in the noisy beeping model.

翻译：蜂鸣模型是针对弱设备网络（如传感器网络或生物网络）的模型。在这些网络中，节点仅允许通过发射蜂鸣（即能量的单脉冲信号）进行通信。监听节点仅具备载波侦听能力：它们只能区分蜂鸣的存在与否，但无法接收其他信息。嘈杂蜂鸣模型进一步假设监听节点可能受到随机噪声的干扰。尽管这种通信模型极为受限，但事实证明此类网络仍能执行复杂的分布式任务。本文提出了一种在蜂鸣模型与嘈杂蜂鸣模型中模拟通用消息传递的最优方案。我们证明，在嘈杂（或无噪声）蜂鸣模型中，一轮\textsf{Broadcast CONGEST}可在$O(\Delta\log n)$轮内完成模拟，而一轮\textsf{CONGEST}可在$O(\Delta^2\log n)$轮内完成模拟（其中$\Delta$为网络的最大度数）。同时，我们给出下界，证明任何模拟方法都无法使用渐近更少的轮数。这使得大量图算法可在蜂鸣模型中高效实现。例如，我们提出了一种用于最大匹配的$O(\log n)$轮\textsf{Broadcast CONGEST}算法，通过我们的方法模拟后，立即得到嘈杂蜂鸣模型中近乎最优的$O(\Delta \log^2 n)$轮最大匹配算法。