We present a loosely-stabilizing phase clock for population protocols. In the population model we are given a system of $n$ identical agents which interact in a sequence of randomly chosen pairs. Our phase clock is leaderless and it requires $O(\log n)$ states. It runs forever and is, at any point of time, in a synchronous state w.h.p. When started in an arbitrary configuration, it recovers rapidly and enters a synchronous configuration within $O(n\log n)$ interactions w.h.p. Once the clock is synchronized, it stays in a synchronous configuration for at least poly $n$ parallel time w.h.p. We use our clock to design a loosely-stabilizing protocol that solves the comparison problem introduced by Alistarh et al., 2021. In this problem, a subset of agents has at any time either $A$ or $B$ as input. The goal is to keep track which of the two opinions is (momentarily) the majority. We show that if the majority has a support of at least $\Omega(\log n)$ agents and a sufficiently large bias is present, then the protocol converges to a correct output within $O(n\log n)$ interactions and stays in a correct configuration for poly $n$ interactions, w.h.p.
翻译:我们为人口协议提供了一个松散的稳定时钟。 在人口模型中, 我们得到了一个由一对随机选择的一对序列互动的一美元相同的代理器系统。 我们的一对时钟没有领导力, 它需要$O(\log n) 美元状态。 它会永远运行, 在任何时刻, 在一个任意的配置中开始同步状态 w.h.p。 它会迅速恢复, 并进入一个同步配置 $(n\log n) 的一美元互动 n.h.p.。 一旦时钟同步, 它会保持一个同步的组合, 至少有一美元( 美元) 的一对一对一。 我们用时钟设计一个松散的稳定协议, 解决Alistarah 等人( 2021) 引入的比较问题。 在这个问题中, 一组代理器随时都有美元或美元作为投入。 目标是追踪两种意见中的哪一种( momental n.h) 多数。 我们显示, 如果大多数人支持至少是$( w\\ om\\\\\ $) com com commexactual exactivactal practivactal practactactactal press practivacty.