We study the Undecided-State Dynamics (USD), a fundamental consensus process in which each vertex holds one of $k$ decided opinions or the undecided state. We consider both the gossip model and the population protocol model. Prior work established tight bounds on the consensus time of this process only for the regime $k = O(\sqrt{n}/(\log n)^2)$ (for the population protocol model) and $k = O((n/\log n)^{1/3})$ (for the gossip model), often under restrictive assumptions on the initial configuration. In this paper, we obtain the first consensus-time guarantees for USD that hold for \emph{arbitrary} $2\le k\le n$ and for \emph{arbitrary} initial configurations in both the gossip model and the population protocol model. In the gossip model, USD reaches consensus within $\widetilde O(\min\{k,\sqrt n\})$ synchronous rounds with probability $1-p_{\bot}-n^{-c}$, where $p_{\bot}$ is the gossip-specific probability of collapsing to the all-undecided state in the first round. In the population protocol model, USD reaches consensus within $\widetilde O(\min\{kn,n^{3/2}\})$ asynchronous interactions with high probability. We also present lower bounds that match the upper bounds up to polylogarithmic factors for a specific initial configuration and show that our upper bounds are essentially optimal.

翻译：本研究探讨未定状态动态（USD）这一基本共识过程，其中每个顶点持有$k$种已定意见之一或处于未定状态。我们同时考察闲谈模型与群体协议模型。先前研究仅针对$k = O(\sqrt{n}/(\log n)^2)$（群体协议模型）与$k = O((n/\log n)^{1/3})$（闲谈模型）的范围建立了该过程共识时间的紧确界，且通常对初始配置施加严格限制。本文首次获得适用于\emph{任意}$2\le k\le n$与\emph{任意}初始配置的USD共识时间保证，同时涵盖闲谈模型与群体协议模型。在闲谈模型中，USD以$1-p_{\bot}-n^{-c}$的概率在$\widetilde O(\min\{k,\sqrt n\})$个同步轮次内达成共识，其中$p_{\bot}$是首轮即坍缩至全未定状态的闲谈模型特有概率。在群体协议模型中，USD以高概率在$\widetilde O(\min\{kn,n^{3/2}\})$次异步交互内达成共识。我们还针对特定初始配置给出了与上界匹配（至多相差多对数因子）的下界，并证明所得上界本质最优。