Near-Optimal Distributed Ruling Sets for Trees and High-Girth Graphs
arXiv:2504.21777v2 Announce Type: replace Abstract: Given a graph $G=(V,E)$, a $\beta$-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 $\beta$. 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 $\Omega(\log\log n/\log\log\log n)$ 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\
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Abstract:Given a graph $G=(V,E)$, a $\beta$-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 $\beta$. 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 $\Omega(\log\log n/\log\log\log n)$ 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 $\beta$.
Subjects:
Data Structures and Algorithms (cs.DS); Distributed, Parallel, and Cluster Computing (cs.DC)
Cite as: arXiv:2504.21777 [cs.DS]
(or arXiv:2504.21777v2 [cs.DS] for this version)
https://doi.org/10.48550/arXiv.2504.21777
arXiv-issued DOI via DataCite
Journal reference: PODC '25: ACM Symposium on Principles of Distributed Computing, 2025, pp. 88 - 98
Related DOI:
https://doi.org/10.1145/3732772.3733547
DOI(s) linking to related resources
Submission history
From: Malte Baumecker [view email] [v1] Wed, 30 Apr 2025 16:29:14 UTC (53 KB) [v2] Thu, 2 Apr 2026 11:54:47 UTC (50 KB)
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