Population Protocols for Exact Plurality Consensus -- How a small chance of failure helps to eliminate insignificant opinions

TL;DR

The paper tackles exact plurality consensus in population protocols by permitting a vanishingly small failure probability to achieve tight space usage. It introduces three protocols that progressively reduce the number of opinions involved in tournaments: first leverages a fixed order among opinions, second removes the ordering via leader election, and third prunes insignificant opinions to minimize tournament count. Each protocol achieves favorable time-space trade-offs, with runtimes scaling as $O(k\log n)$, $O(k\log n+\log^2 n)$, and $O\bigl(\frac{n}{x_{\max}}\log n+\log^2 n\bigr)$ respectively, while maintaining high-probability correctness. The approach relies on phase clocks, leader election, and pruning to accelerate convergence and reduce state complexity, bringing near-linear improvements over prior quadratic lower bounds on space. These results have implications for scalable consensus in large populations of simple agents, where exact plurality must be identified efficiently with minimal memory.

Abstract

We consider the \emph{exact plurality consensus} problem for \emph{population protocols}. Here, $n$ anonymous agents start each with one of $k$ opinions. Their goal is to agree on the initially most frequent opinion (the \emph{plurality opinion}) via random, pairwise interactions. The case of $k = 2$ opinions is known as the \emph{majority problem}. Recent breakthroughs led to an always correct, exact majority population protocol that is both time- and space-optimal, needing $O(\log n)$ states per agent and, with high probability, $O(\log n)$ time~[Doty, Eftekhari, Gasieniec, Severson, Stachowiak, and Uznanski; 2021]. We know that any always correct protocol requires $Ω(k^2)$ states, while the currently best protocol needs $O(k^{11})$ states~[Natale and Ramezani; 2019]. For ordered opinions, this can be improved to $O(k^6)$~[Gasieniec, Hamilton, Martin, Spirakis, and Stachowiak; 2016]. We design protocols for plurality consensus that beat the quadratic lower bound by allowing a negligible failure probability. While our protocols might fail, they identify the plurality opinion with high probability even if the bias is $1$. Our first protocol achieves this via $k-1$ tournaments in time $O(k \cdot \log n)$ using $O(k + \log n)$ states. While it assumes an ordering on the opinions, we remove this restriction in our second protocol, at the cost of a slightly increased time $O(k \cdot \log n + \log^2 n)$. By efficiently pruning insignificant opinions, our final protocol reduces the number of tournaments at the cost of a slightly increased state complexity $O(k \cdot \log\log n + \log n)$. This improves the time to $O(n / x_{\max} \cdot \log n + \log^2 n)$, where $x_{\max}$ is the initial size of the plurality. Note that $n/x_{\max}$ is at most $k$ and can be much smaller (e.g., in case of a large bias or if there are many small opinions).

Figures (1)

• Figure 1: State Space $\mathcal{S}$. Note that $[i] = \set{1, \dots, i}$ and $[-i;j] = \set{-i, \dots, j}$.

Theorems & Definitions (3)

• theorem 1
• theorem 2
• lemma 3