Given a graph $G=(V,E)$, a $β$-ruling set is a subset of nodes $S\subseteq V$ that is independent, and each node in $V$ is at distance at most $β$ from some node in $S$. In this paper, we present almost optimal distributed algorithms for finding $2$-ruling sets in the classical \LOCAL model. Our main contribution is a randomized algorithm that w.h.p.\ computes a $2$-ruling set on any $n$-node graph with bounded arboricity in $O(\log \log n)$ rounds. In fact, the algorithm works up to arboricity $O(\log\log n)$, improves exponentially over the prior state of the art that can be achieved by combining [Barenboim, Elkin, Pettie, Schneider; JACM'16], [Ghaffari; SODA'16], and [Bisht, Kothapalli and Pemmaraju; PODC'14], and nearly matches the lower bound of $Ω(\log \log n / \log \log \log n)$ [Balliu, Brandt, Kuhn, Olivetti; FOCS'20]. The domination parameter $β=2$ is optimal for algorithms with runtime $\log^{o(1)}n$: on graphs with arboricity $2$, there is a lower bound of $Ω(\sqrt{\log n})$ rounds for MIS (i.e., $β= 1$) [Khoury, Schild; FOCS'25]. Additionally, we obtain improved algorithms for larger arboricity. For general graphs with arboricity $α$, we present a randomized algorithm that computes a $2$-ruling set in $\widetilde{O}(\log^{5/8} α+\log^{5/3} \log n)$ rounds. This improves exponentially over the state of the art for a large range of non-constant arboricity. Our techniques extend beyond distributed computing. We present an $O(\log \log \log n)$-round algorithm in the low-space Massively Parallel Computation (\mpc) model that w.h.p.\ computes a $2$-ruling set on any graph with arboricity up to $2^{poly (\log \log n)}$, improving exponentially over the state of the art from [Kothapalli, Pai, Pemmaraju; FSTTCS'20] combined with [Fischer, Giliberti, Grunau; SPAA'23].

翻译：给定图 $G=(V,E)$，一个 $β$-统治集是节点子集 $S\subseteq V$，满足独立性质，且 $V$ 中每个节点到 $S$ 中某节点的距离至多为 $β$。本文提出在经典 \LOCAL 模型下计算 $2$-统治集的近乎最优分布式算法。主要贡献是一个随机算法，该算法以高概率在 $O(\log \log n)$ 轮内为任意 $n$ 节点有界树度图计算 $2$-统治集。实际上，该算法适用于树度高达 $O(\log\log n)$ 的图，相较于结合 [Barenboim, Elkin, Pettie, Schneider; JACM'16]、[Ghaffari; SODA'16] 和 [Bisht, Kothapalli, Pemmaraju; PODC'14] 可实现的先前最优成果，实现了指数级改进，并几乎匹配下界 $Ω(\log \log n / \log \log \log n)$ [Balliu, Brandt, Kuhn, Olivetti; FOCS'20]。支配参数 $β=2$ 对于运行时间为 $\log^{o(1)}n$ 的算法是最优的：在树度为 $2$ 的图上，MIS（即 $β=1$）存在下界 $Ω(\sqrt{\log n})$ 轮 [Khoury, Schild; FOCS'25]。此外，我们针对更大树度改进了算法。对于树度为 $α$ 的一般图，提出一个随机算法，在 $\widetilde{O}(\log^{5/8} α+\log^{5/3} \log n)$ 轮内计算 $2$-统治集。这在非恒定树度的大范围上相较于先前最优成果实现了指数级改进。我们的技术不仅限于分布式计算。在低空间大规模并行计算 (\mpc) 模型中，我们提出一个 $O(\log \log \log n)$ 轮算法，能够以高概率为树度高达 $2^{poly (\log \log n)}$ 的任意图计算 $2$-统治集，相较于 [Kothapalli, Pai, Pemmaraju; FSTTCS'20] 结合 [Fischer, Giliberti, Grunau; SPAA'23] 的先前最优成果实现了指数级改进。