An $O(\log^{3/2}n)$ Parallel Time Population Protocol for Majority with $O(\log n)$ States

Abstract

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.