Time and Space Optimal Exact Majority Population Protocols

TL;DR

This paper introduces a new population protocol for exact majority that is optimal in both time and space, operating in logarithmic parallel time with each agent using logarithmic states, enabled by a novel phase clock mechanism.

Contribution

It presents the first time and space optimal exact majority population protocol that works with high probability, utilizing a new fixed-resolution phase clock for efficient counting.

Findings

Operates in O(log n) parallel time, optimal for population protocols.

Uses O(log n) states per agent, matching space optimality.

Introduces a novel fixed-resolution phase clock for counting in population protocols.

Abstract

In this paper we study population protocols governed by the {\em random scheduler}, which uniformly at random selects pairwise interactions between $n$ agents. The main result of this paper is the first time and space optimal {\em exact majority population protocol} which also works with high probability. The new protocol operates in the optimal {\em parallel time} $O(\log n),$ which is equivalent to $O(n\log n)$ sequential {\em pairwise interactions}, where each agent utilises the optimal number of $O(\log n)$ states. The time optimality of the new majority protocol is possible thanks to the novel concept of fixed-resolution phase clocks introduced and analysed in this paper. The new phase clock allows to count approximately constant parallel time in population protocols.