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

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 \(\Omega(k^2)\) 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).