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The goal of this paper is to understand the complexity of symmetry breaking problems, specifically maximal independent set (MIS) and the closely related β-ruling set problem, in two computational models suited for large-scale graph processing, namely the k-machine model and the graph streaming model. We present a number of results. For MIS in the k-machine model, we improve the Õ(m/k2 + ∆/k)-round upper bound of Klauck et al. (SODA 2015) by presenting an Õ(m/k2)-round algorithm. We also present an Ω̃(n/k2) round lower bound for MIS, the first lower bound for a symmetry breaking problem in the k-machine model. For β-ruling sets, we use hierarchical sampling to obtain more efficient algorithms in the k-machine model and also in the graph streaming model. More specifically, we obtain a k-machine algorithm that runs in Õ(βn∆1/β/k2) rounds and, by using a similar hierarchical sampling technique, we obtain one-pass algorithms for both insertion-only and insertion-deletion streams that use O(β · n1+1/2 β−1 ) space. The latter result establishes a clear separation between MIS, which is known to require Ω(n2) space (Cormode et al., ICALP 2019), and β-ruling sets, even for β = 2. Finally, we present an even faster 2-ruling set algorithm in the k-machine model, one that runs in Õ(n/k2− + k1− ) rounds for any , 0 ≤ ≤ 1. For a wide range of values of k this round complexity simplifies to Õ(n/k2) rounds, which we conjecture is optimal. Our results use a variety of techniques. For our upper bounds, we prove and use simulation theorems for beeping algorithms, hierarchical sampling, and L0-sampling, whereas for our lower bounds we use information-theoretic arguments and reductions to 2-party communication complexity problems. 2012 ACM Subject Classification Theory of computation → Distributed algorithms

The Complexity of Symmetry Breaking in Massive Graphs.