Brief Announcement: Optimal Time and Space Leader Election in Population Protocols

Abstract

Population protocols are a model of distributed computing, where n agents with limited computational power and memory perform randomly scheduled pairwise interactions. Recently, a significant amount of work has been devoted to the study of the time and space complexity of leader election in this model. It is known that Ω (log log n) states per agent are needed to elect a leader in fewer than [EQUATION] expected interactions (Alistarh et al.; SODA'17) and that Ω (n log n) expected interactions are required regardless of the number of states (Sudo and Masuzawa; 2020). On the positive side, Gasieniec and Stachowiak (SODA'18) gave the first protocol that uses an optimal Θ(log log n) number or states and elects a leader in O(n log2 n) expected interactions. This running time was subsequently improved to O(n log n log log n) (Gasieniec et al.; SPAA'19). We provide the first leader election population protocol that is both time and space optimal, electing a leader in O(n log n) expected interactions and using Θ(log log n) states per agent. A novel component is a simple protocol that efficiently selects a small set of agents of polylog n size, given O(n∈) initially selected agents. Unlike existing approaches, which monotonically shrink this initially selected set, we first grow it in a controlled way to a specific size before shrinking it again.