Given a graph $G=(V,E)$, a $β$-ruling set is a subset $S\subseteq V$ that is i) independent, and ii) every node $v\in V$ has a node of $S$ within distance $β$. In this paper we present almost optimal distributed algorithms for finding ruling sets in trees and high girth graphs in the classic LOCAL model. As our first contribution we present an $O(\log\log n)$-round randomized algorithm for computing $2$-ruling sets on trees, almost matching the $Ω(\log\log n/\log\log\log n)$ lower bound given by Balliu et al. [FOCS'20]. Second, we show that $2$-ruling sets can be solved in $\widetilde{O}(\log^{5/3}\log n)$ rounds in high-girth graphs. Lastly, we show that $O(\log\log\log n)$-ruling sets can be computed in $\widetilde{O}(\log\log n)$ rounds in high-girth graphs matching the lower bound up to triple-log factors. All of these results either improve polynomially or exponentially on the previously best algorithms and use a smaller domination distance $β$.

翻译：给定图 $G=(V,E)$，$\beta$-控制集是一个子集 $S\subseteq V$，满足：i) 独立集，且 ii) 每个节点 $v\in V$ 在距离 $\beta$ 内均有 $S$ 中的节点。本文在经典 LOCAL 模型中提出了针对树和高围图寻找控制集的几乎最优分布式算法。首先，我们给出一个 $O(\log\log n)$ 轮随机算法来计算树上的 $2$-控制集，几乎匹配了 Balliu 等人 [FOCS'20] 给出的 $\Omega(\log\log n/\log\log\log n)$ 下界。其次，我们证明在高围图中 $2$-控制集可在 $\widetilde{O}(\log^{5/3}\log n)$ 轮内求解。最后，我们证明在高围图中 $O(\log\log\log n)$-控制集可在 $\widetilde{O}(\log\log n)$ 轮内计算，与下界仅相差三层对数因子。所有结果要么将此前最优算法实现多项式级指数级改进，要么使用了更小的支配距离 $\beta$。