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 1 , all nodes terminate after O(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], we 2 get a randomized MIS algorithm with a high probability global complexity of O(log ∆) + 2 O( √ log log n) , where ∆ denotes the maximum degree. This improves over the O(log 2 ∆) + 2 O( √ log log n) result of Barenboim et al., and gets close to the Ω(min{log ∆, √ log n}) 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. 1 As standard, we use the phrase with high probability to indicate that an event has probability at least 1 − 1/n. 2 quasi nanos, gigantium humeris insidentes
A recent breakthrough by Ghaffari REF obtained a randomized algorithm in O(log ∆) + 2 O( √ log log n) rounds.
An Improved Distributed Algorithm for Maximal Independent Set
arXiv: Data Structures and Algorithms
Mathematics Computer Science