Population protocols are a model of distributed computing, where n agents with limited computational power and memory perform randomly scheduled pairwise interactions. A fundamental problem in this setting is that of leader election, where all agents start from the same state, and they seek to reach and maintain a global state where exactly one agent is in a dedicated leader state.A significant amount of work has been devoted to the study of the time and space complexity of this problem. Alistarh et al. (SODA'17) have shown that Ω(log log n) states per agent are needed in order to elect a leader in fewer thanΘ(n 2 ) expected interactions. Moreover, Ω(n log n) expected interactions are required regardless of the number of states (Sudo and Masuzawa, 2019). On the upper bound side, Gasieniec and Stachowiak (SODA'18) have presented the first protocol that uses an optimal, Θ(log log n), number or states and elects a leader in O(n log 2 n) expected interactions. This running time was subsequently improved to O(n log n log log n) (Gasieniec et al., SPAA'19).In this paper we provide the first leader election population protocol that is both time and space optimal: it uses Θ(log log n) states per agent, and elects a leader in O(n log n) interactions in expectation. A key novel component of our approach is a simple protocol that efficiently selects a small set of agents, of poly(log n) size, given an initial set of s = O(n ϵ ) selected agents. Unlike existing approaches, which proceed by shrinking the initial set monotonically over time, our protocol first increases the set in a controlled way to a specific size (which is independent of s), before it shrinks the set to a poly(log n) size. CCS CONCEPTS• Mathematics of computing → Stochastic processes; • Theory of computation → Self-organization.

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Population protocols are a model for distributed computing that is focused on simplicity and robustness. A system of n identical agents (finite state machines) performs a global task like electing a unique leader or determining the majority opinion when each agent has one of two opinions. Agents communicate in pairwise interactions with randomly assigned communication partners. Quality is measured in two ways: the number of interactions to complete the task and the number of states per agent. We present protocols for the majority problem that allow for a trade-off between these two measures. Compared to the only other trade-off result (Alistarh et al. in Proceedings of the 2015 ACM symposium on principles of distributed computing, Donostia-San Sebastián, 2015), we improve the number of interactions by almost a linear factor. Furthermore, our protocols can be made uniform (working correctly without any information on the population size n), yielding the first uniform majority protocols that stabilize in a subquadratic number of interactions.

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