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).

翻译：我们考虑群体协议中的精确多数共识问题。在此问题中，$n$个匿名智能体初始时各持有$k$种意见中的一种，目标是通过随机成对交互就初始出现频率最高的意见（即多数意见）达成一致。当$k=2$时，该问题被称为多数问题。近期突破性进展提出了首个始终正确且兼具时间和空间最优性的精确多数群体协议，每个智能体仅需$O(\log n)$个状态，且高概率在$O(\log n)$时间内完成[Doty, Eftekhari, Gasieniec, Severson, Stachowiak, and Uznanski; 2021]。已知任何始终正确的协议都需要$\Omega(k^2)$个状态，而当前最优协议需要$O(k^{11})$个状态[Natale and Ramezani; 2019]。对于有序意见，该复杂度可降至$O(k^6)$[Gasieniec, Hamilton, Martin, Spirakis, and Stachowiak; 2016]。我们设计了能突破二次下界且允许可忽略失败概率的多数共识协议。尽管我们的协议可能失效，但即使在偏差为1的情况下，它们仍能以高概率识别多数意见。首个协议通过$k-1$场锦标赛实现，时间复杂度为$O(k \cdot \log n)$，状态复杂度为$O(k + \log n)$。该协议假设意见具有有序性，我们在第二项协议中取消了此限制，但代价是时间复杂度略有增加至$O(k \cdot \log n + \log^2 n)$。通过高效剪枝不重要的意见，最终协议在状态复杂度小幅提升至$O(k \cdot \log\log n + \log n)$的条件下减少了锦标赛场次。这将时间复杂度优化至$O(n / x_{\max} \cdot \log n + \log^2 n)$，其中$x_{\max}$为多数意见的初始规模。注意$n/x_{\max}$最大为$k$，且在实际中可能更小（例如存在大偏差或大量小规模意见时）。