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 $β$.

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