Tight analysis of parallel randomized greedy MIS

Abstract

We provide a tight analysis which settles the round com-plexity 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 Maxi-mal 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 neigh-bors. The main question of interest is the number of rounds it takes until the graph is empty. This algo-rithm has been studied since 1987, initiated by Cop-persmith, Raghavan, and Tompa [FOCS'87], and the previously best known bounds were O(log n) rounds in expectation for Erd}os-Rényi random graphs by Calkin and Frieze [Random Struc. & Alg. '90] and O(log2 n) rounds with high probability for general graphs by Blel-loch, Fineman, and Shun [SPAA'12]. We prove a high probability upper bound of O(log n) 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].