In population protocols, the underlying distributed network consists of $n$ nodes (or agents), denoted by $V$, and a scheduler that continuously selects uniformly random pairs of nodes to interact. When two nodes interact, their states are updated by applying a state transition function that depends only on the states of the two nodes prior to the interaction. The efficiency of a population protocol is measured in terms of both time (which is the number of interactions until the nodes collectively have a valid output) and the number of possible states of nodes used by the protocol. By convention, we consider the parallel time cost, which is the time divided by $n$. In this paper we consider the majority problem, where each node receives as input a color that is either black or white, and the goal is to have all of the nodes output the color that is the majority of the input colors. We design a population protocol that solves the majority problem in $O(\log^{3/2}n)$ parallel time, both with high probability and in expectation, while using $O(\log n)$ states. Our protocol improves on a recent protocol of Berenbrink et al. that runs in $O(\log^{5/3}n)$ parallel time, both with high probability and in expectation, using $O(\log n)$ states.

中文翻译：

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具有$ O（\ log n）$个状态的多数人的$ O（\ log ^ {3/2} n）$并行时间人口协议

在总体协议中，底层的分布式网络由$ n $个节点（或代理）（由$ V $表示）和一个调度程序组成，该调度程序连续选择均匀随机的节点对进行交互。当两个节点交互时，通过应用仅依赖于交互之前两个节点的状态的状态转换函数来更新其状态。总体协议的效率通过时间（即直到节点共同具有有效输出之前的交互次数）和协议使用的节点可能状态的数量来衡量。按照惯例，我们考虑并行时间成本，即时间除以$ n $。在本文中，我们考虑了多数问题，其中每个节点接收的输入为黑色或白色，目标是让所有节点输出大多数输入颜色的颜色。我们设计了一个人口协议，以$ O（\ log ^ {3/2} n）$个并行状态同时解决了$ O（\ log ^ {3/2} n）$个并行时间中的多数问题，既有很高的概率，也有期望。我们的协议对Berenbrink等人的最新协议进行了改进。使用$ O（\ log n）$状态在$ O（\ log ^ {5/3} n）$并行时间内运行，既有很高的概率，也有期望。