This paper considers the basic $${\mathcal {PULL}}$$PULL model of communication, in which in each round, each agent extracts information from few randomly chosen agents. We seek to identify the smallest amount of information revealed in each interaction (message size) that nevertheless allows for efficient and robust computations of fundamental information dissemination tasks. We focus on the Majority Bit Dissemination problem that considers a population of n agents, with a designated subset of source agents. Each source agent holds an input bit and each agent holds an output bit. The goal is to let all agents converge their output bits on the most frequent input bit of the sources (the majority bit). Note that the particular case of a single source agent corresponds to the classical problem of Broadcast (also termed Rumor Spreading). We concentrate on the severe fault-tolerant context of self-stabilization, in which a correct configuration must be reached eventually, despite all agents starting the execution with arbitrary initial states. In particular, the specification of who is a source and what is its initial input bit may be set by an adversary. We first design a general compiler which can essentially transform any self-stabilizing algorithm with a certain property (called “the bitwise-independence property”) that uses $$\ell $$ℓ-bits messages to one that uses only $$\log \ell $$logℓ-bits messages, while paying only a small penalty in the running time. By applying this compiler recursively we then obtain a self-stabilizing Clock Synchronization protocol, in which agents synchronize their clocks modulo some given integer T, within $$\tilde{\mathcal {O}}(\log n\log T)$$O~(lognlogT) rounds w.h.p., and using messages that contain 3 bits only. We then employ the new Clock Synchronization tool to obtain a self-stabilizing Majority Bit Dissemination protocol which converges in $$\tilde{\mathcal {O}}(\log n)$$O~(logn) time, w.h.p., on every initial configuration, provided that the ratio of sources supporting the minority opinion is bounded away from half. Moreover, this protocol also uses only 3 bits per interaction.

中文翻译：

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最小化随机通信模式中的消息大小：3 位快速自稳定协议

本文考虑了基本的 $${\mathcal {PULL}}$$PULL 通信模型，其中在每一轮中，每个代理从几个随机选择的代理中提取信息。我们试图识别在每次交互中显示的最小信息量（消息大小），但仍允许对基本信息传播任务进行有效和稳健的计算。我们关注多数位传播问题，该问题考虑了 n 个代理的总体，以及指定的源代理子集。每个源代理持有一个输入位，每个代理持有一个输出位。目标是让所有代理将其输出位收敛到源中最频繁的输入位（多数位）。请注意，单个源代理的特殊情况对应于广播的经典问题（也称为谣言传播）。我们专注于自稳定的严重容错上下文，尽管所有代理以任意初始状态开始执行，但最终必须达到正确的配置。特别地，谁是源以及它的初始输入位是什么的规范可以由对手设置。我们首先设计了一个通用编译器，它本质上可以将任何使用 $$\ell $$ℓ 位消息的具有特定属性（称为“位独立属性”）的自稳定算法转换为仅使用 $$\log 的算法\ell $$logℓ-bits 的消息，而在运行时间上只付出很小的代价。通过递归地应用这个编译器，我们获得了一个自稳定时钟同步协议，其中代理以某个给定整数 T 为模同步他们的时钟，在 $$\tilde{\mathcal {O}}(\log n\log T)$$O~(lognlogT) 内舍入 whp，并且使用仅包含 3 位的消息。然后我们使用新的时钟同步工具来获得一个自稳定的多数位传播协议，该协议收敛于 $$\tilde{\mathcal {O}}(\log n)$$O~(logn) 时间，whp，在每个初始配置，前提是支持少数意见的来源比例不超过一半。此外，该协议每次交互也仅使用 3 位。前提是支持少数派意见的来源比例不超过一半。此外，该协议每次交互也仅使用 3 位。前提是支持少数派意见的来源比例不超过一半。此外，该协议每次交互也仅使用 3 位。