Symmetry Breaking in the Congest Model: Time- and Message-Efficient Algorithms for Ruling Sets

We study local symmetry breaking problems in the Congest model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. The time (round) complexity of MIS (and ruling sets) have attracted much attention in the Local model. Indeed, recent results (Barenboim et al., FOCS 2012, Ghaffari SODA 2016) for the MIS problem have tried to break the long-standing O(logn)-round “barrier” achieved by Luby’s algorithm, but these yield o(logn)-round complexity only when the maximum degree ∆ is somewhat small relative to n. More importantly, these results apply only in the Local model. In fact, the best known time bound in the Congest model is still O(logn) (via Luby’s algorithm) even for somewhat small ∆. Furthermore, message complexity has been largely ignored in the context of local symmetry breaking. Luby’s algorithm takes O(m) messages on m-edge graphs and this is the best known bound with respect to messages. Our work is motivated by the following central question: can we break the Θ(logn) time complexity barrier and the Θ(m) message complexity barrier in the Congest model for MIS or closely-related symmetry breaking problems? This paper presents progress towards this question for the distributed ruling set problem in the Congest model. A β-ruling set is an independent set such that every node in the graph is at most β hops from a node in the independent set. We present the following results: Time Complexity: We show that we can break the O(logn) “barrier” for 2and 3-ruling sets. We compute 3-ruling sets in O ( logn log logn ) rounds with high probability (whp). More generally we show that 2-ruling sets can be computed in O ( log ∆ · (logn)1/2+ε + logn log logn ) rounds for any ε > 0, which is o(logn) for a wide range of ∆ values (e.g., ∆ = 2(logn)). These are the first 2and 3-ruling set algorithms to improve over the O(logn)-round complexity of Luby’s algorithm in the Congest model. Message Complexity: We show an Ω(n2) lower bound on the message complexity of computing an MIS (i.e., 1-ruling set) which holds also for randomized algorithms and present a contrast to this by showing a randomized algorithm for 2-ruling sets that, whp, uses only O(n log2 n) messages and runs in O(∆ logn) rounds. This is the first message-efficient algorithm known for ruling sets, which has message complexity nearly linear in n (which is optimal up to a polylogarithmic factor). Our results are a step toward understanding the time and message complexity of symmetry breaking problems in the Congest model. 1998 ACM Subject Classification C.2.4 Distributed Systems, F.1.2 Modes of Computation, F.2.2 Nonnumerical Algorithms and Problems, G.2.2 Graph Theory © Shreyas Pai, Gopal Pandurangan, Sriram V. Pemmaraju, Talal Riaz, Peter Robinson; licensed under Creative Commons License CC-BY Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany ar X iv :1 70 5. 07 86 1v 1 [ cs .D C ] 2 2 M ay 2 01 7 2 Symmetry Breaking in the Congest Model