The model of population protocols refers to the growing in popularity theoretical framework suitable for studying pairwise interactions within a large collection of simple indistinguishable entities, frequently called agents . In this article, the emphasis is on the space complexity of fast leader election in population protocols governed by the random scheduler , which uniformly at random selects pairwise interactions between n agents. One of the main results of this article is the first fast space optimal leader election protocol , which works with high probability. The new protocol operates in parallel time O (log 2 n ) equivalent to O ( n log 2 n ) sequential pairwise interactions with each agent’s memory space limited to O (log log n ) states. This double logarithmic space utilisation matches asymptotically the lower bound ½log log n on the number of states utilised by agents in any leader election algorithm with the running time o ( n \polylog n ); see Reference [7]. Our new solution expands also on the classical concept of phase clocks used to synchronise and to coordinate computations in distributed algorithms. In particular, we formalise the concept and provide a rigorous analysis of phase clocks operating in nested modes. Our arguments are also valid for phase clocks propelled by multiple leaders. The combination of the two results in the first time-space efficient leader election algorithm. We also provide a complete formal argumentation, indicating that our solution is always correct, fast, and it works with high probability.

中文翻译：

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增强的相位时钟、种群协议和快速空间最优领导者选举

型号人口协议指日益流行的适合学习的理论框架成对交互在大量简单的无法区分的实体中，通常称为代理. 在这篇文章中，重点是快速的空间复杂度领袖选举在人口协议中由随机调度器，它均匀地随机选择成对的相互作用n代理。本文的主要成果之一是第一个快速空间最优领导人选举协议，它的工作概率很高。新协议运行于并行时间 ○（日志2 n） 相当于○(n日志2 n) 顺序成对交互每个代理的内存空间限制为○（日志日志n） 状态。这种双对数空间利用率渐近地匹配下限½log logn在运行时间的任何领导者选举算法中代理使用的状态数○(n\polylogn); 参见参考文献 [7]。我们的新解决方案还扩展了用于同步和协调分布式算法中的计算的相位时钟的经典概念。特别是，我们形式化了这个概念，并对在嵌套模式下运行的相位时钟进行了严格的分析。我们的论点也适用于由多个领导者推动的相位时钟。两者的结合产生了第一个时空高效的领导者选举算法。我们还提供了完整的形式论证，表明我们的解决方案始终是正确的、快速的，并且很有可能奏效。