A stable majority population protocol using logarithmic time and states

TL;DR

This paper presents a population protocol for the majority problem that achieves logarithmic time and state complexity, is stable with probability 1, and is optimal within certain natural constraints.

Contribution

The authors introduce a stable, population protocol for majority consensus that uses O(log n) states and expected O(log n) time, improving upon prior protocols and establishing optimality under specific conditions.

Findings

Achieves stable majority consensus in expected O(log n) time.

Uses O(log n) states, proven to be optimal for output dominant, monotone protocols.

Can be adapted to a uniform protocol with slightly higher state complexity.

Abstract

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 $A$, $B$, or $T$, 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 stable protocols that are "output dominant" and "monotone". These are two natural constraints satisfied by our protocol, making it state-optimal for that class. We use, and develop novel analysis of, a key technique called a "fixed resolution clock" due to Gasieniec, Stachowiak, and Uznanski, who showed a majority protocol using $O(\log n)$ time and states that has a positive probability of error. 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 $\Theta(\log n \log \log n)$.