Towards an Optimal Distributed Algorithm for Maximal Independent Set

The Maximal Independent Set (MIS) problem is one of the basics in the study of locality in distributed graph algorithms. This paper presents an extremely simple randomized algorithm providing a near-optimal local complexity for this problem, which incidentally, when combined with some known techniques, also leads to a near-optimal global complexity. Classical MIS algorithms of Luby [STOC’85] and Alon, Babai and Itai [JALG’86] provide the global complexity guarantee that, with high probability, all nodes terminate afterO(log n) rounds. In contrast, our initial focus is on the local complexity, and our main contribution is to provide a very simple algorithm guaranteeing that each particular node v terminates after O(log deg(v) + log 1/ε) rounds, with probability at least 1 − ε. The guarantee holds even if the randomness outside 2-hops neighborhood of v is determined adversarially. This degree-dependency is optimal, due to a lower bound of Kuhn, Moscibroda, and Wattenhofer [PODC’04]. Interestingly, this local complexity smoothly transitions to a global complexity: by adding techniques of Barenboim, Elkin, Pettie, and Schneider [FOCS’12; arXiv: 1202.1983v3], we get a randomized MIS algorithm with a high probability global complexity of O(log∆) + 2 √ log log , where ∆ denotes the maximum degree. This improves over the O(log ∆)+2 √ log log n) result of Barenboim et al., and gets close to the Ω(min{log∆,√logn}) lower bound of Kuhn et al. Corollaries include improved algorithms for MIS in graphs of upper-bounded arboricity, or lower-bounded girth, for Ruling Sets, for MIS in the Local Computation Algorithms (LCA) model, and a faster distributed algorithm for the Lovász Local Lemma. As standard, we use the phrase with high probability to indicate that an event has probability at least 1− 1/n. quasi nanos, gigantium humeris insidentes