We provide a tight analysis which settles the round complexity of the well-studied parallel randomized greedy MIS algorithm, thus answering the main open question of Blelloch, Fineman, and Shun [SPAA'12]. The parallel distributed randomized greedy Maximal Independent Set (MIS) algorithm works as follows. An order of the vertices is chosen uniformly at random. Then, in each round, all vertices that appear before their neighbors in the order are added to the independent set and removed from the graph along with their neighbors. The main question of interest is the number of rounds it takes until the graph is empty. This algorithm has been studied since 1987, initiated by Coppersmith, Raghavan, and Tompa [FOCS'87], and the previously best known bounds were @math rounds in expectation for Erd o s-R ' e nyi random graphs by Calkin and Frieze [Random Struc. & Alg. '90] and @math rounds with high probability for general graphs by Blelloch, Fineman, and Shun [SPAA'12]. We prove a high probability upper bound of @math on the round complexity of this algorithm in general graphs, and that this bound is tight. This also shows that parallel randomized greedy MIS is as fast as the celebrated algorithm of Luby [STOC'85, JALG'86].
We consider the parallel greedy algorithm of Coppersmith, Raghavan, and Tompa (Proc. of 28th Annual IEEE Symp. on Foundations of Computer Science, pp. 260–269, 1987) for finding the lexicographically first maximal independent set of a graph. We prove an Ω(log n) bound on the expected number of iterations for most edge densities. This complements the O(log n) bound proved in Calkin and Frieze (Random Structures and Algorithms, Vol. 1, pp. 39–50, 1990). © 1992 Wiley Periodicals, Inc.