We study population protocols, a model of distributed computing appropriate for modeling well-mixed chemical reaction networks and other physical systems where agents exchange information in pairwise interactions, but have no control over their schedule of interaction partners. The well-studied *majority* problem is that of determining in an initial population of $n$ agents, each with one of two opinions $A$ or $B$, whether there are more $A$, more $B$, or a tie. A *stable* protocol solves this problem with probability 1 by eventually entering a configuration in which all agents agree on a correct consensus decision of $\mathsf{A}$, $\mathsf{B}$, or $\mathsf{T}$, from which the consensus cannot change. We describe a protocol that solves this problem using $O(\log n)$ states ($\log \log n + O(1)$ bits of memory) and optimal expected time $O(\log n)$. The number of states $O(\log n)$ is known to be optimal for the class of polylogarithmic time stable protocols that are "output dominant" and "monotone". These are two natural constraints satisfied by our protocol, making it simultaneously time- and state-optimal for that class. We introduce a key technique called a "fixed resolution clock" to achieve partial synchronization. Our protocol is *nonuniform*: the transition function has the value $\left \lceil {\log n} \right \rceil$ encoded in it. We show that the protocol can be modified to be uniform, while increasing the state complexity to $\Theta(\log n \log \log n)$.
翻译:我们研究的是人口协议,一种适合模型化工反应网络和其他物理系统的分布计算模型,其代理商在对称互动中交换信息,但对其互动伙伴的日程表没有控制权。经过仔细研究的* 多数* 问题是在初始人群中确定美元代理商,每个代理商持有两种意见中的美元或美元,是否存在更多的A美元、更多的B美元或一条线条。一个* 稳定* 协议解决了这个问题,概率1,最终进入一个配置,所有代理商都同意一个正确的共识决定 $\ mathsf{A}$、 $\mathsf{B} 美元或$\mathsfsf{T} 问题。我们描述一个协议,用$(log n) 或美元来解决这个问题, $(log n) + O(1) 记忆百元) 和最佳的预期时间 $O(log n) 。所有代理商在配置中同意正确的 $(log n) $(log n) $(n), $ (n) 的金额) 的金额是已知的一致值, 或正态的过渡功能。