An Almost Tight Lower Bound for Plurality Consensus with Undecided State Dynamics in the Population Protocol Model

TL;DR

This paper establishes an almost tight lower bound on the stabilization time for plurality consensus with undecided states in population protocols, revealing fundamental limits of the process in terms of interactions needed.

Contribution

It provides a nearly matching lower bound on stabilization time for plurality consensus with undecided states, extending understanding of the protocol's efficiency limits.

Findings

Lower bound of (kn \u2217 \u2212log n) interactions for certain initial configurations.

Bound is tight for k  n^{1/2 - \u03b5} with  O(k  log n) parallel time.

Shows fundamental limitations on stabilization time in population protocol models.

Abstract

We revisit the majority problem in the population protocol communication model, as first studied by Angluin et al. (Distributed Computing 2008). We consider a more general version of this problem known as plurality consensus, which has already been studied intensively in the literature. In this problem, each node in a system of $n$ nodes, has initially one of $k$ different opinions, and they need to agree on the (relative) majority opinion. In particular, we consider the important and intensively studied model of Undecided State Dynamics. Our main contribution is an almost tight lower bound on the stabilization time: we prove that there exists an initial configuration, even with bias $\Delta = \omega(\sqrt{n\log n})$, where stabilization requires $\Omega(kn\log \frac {\sqrt n} {k \log n})$ interactions, or equivalently, $\Omega(k\log \frac {\sqrt n} {k \log n})$ parallel time for any $k = o\left(\frac {\sqrt n}{\log n}\right)$. This bound is tight for any $ k \le n^{\frac 1 2 - \epsilon}$, where $\epsilon >0$ can be any small constant, as Amir et al.~(PODC'23) gave a $O(k\log n)$ parallel time upper bound for $k = O\left(\frac {\sqrt n} {\log ^2 n}\right)$.