We study local symmetry breaking problems in the Congest model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. Our work is motivated by the following central question: can we break the long-standing Θ(log n) 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(log n) "barrier" for 2- and 3-ruling sets. We compute 3-ruling sets in O((log n)/(log log n)) rounds with high probability (whp). More generally we show that 2-ruling sets can be computed in O(log Δ · (log n)1/2 + ε + (log n)/(log log n)) rounds for any ε 0, which is o(log n) for a wide range of Δ values (e.g., Δ = 2(log n)1/2-ε). These are the first 2- and 3-ruling set algorithms to improve over the O(log n)-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(Δ log n) 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.
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