Distributed Maximal Matching (MM) and Maximal Independent Set (MIS) are among the most fundamental and well-studied problems in Distributed Graph Algorithms. Quarter a century ago Israeli and Itai [II86] devised a randomized algorithm with running time O(log n) for the MM problem. Around the same time Luby [L86] and Alon,Babai and Itai [ABI86] devised randomized algorithms with running time O(log n) for the MIS problem as well. These algorithms are the state-of-the art to this day. Kuhn, Moscibroda and Wattenhofer [KMW10] showed that computing MM or MIS on graphs with maximum degree Delta = 2^{Theta(\sqrt{log n})} requires Omega(\sqrt{log n}) time. Their lower bound implies that these problems require Omega(min{log Delta, \sqrt{log n}}) time. Closing the gaps between these upper and lower bounds has become a central open problem in Distributed Graph Algorithms. In this paper we devise an algorithm for MM with running time O(max {log Delta, \sqrt{log n}}), and an algorithm for MIS with running time O(log Delta \cdot \sqrt{log n}). In view of the lower bound of [KMW10] our algorithm for the MM problem is tight up to constant factors when Delta = 2^{Theta(\sqrt {log n})}. Also, our algorithms are the first ones to provide a sublogarithmic running time for these problems for a very wide range of Delta. Specifically, so far sublogarithmic time algorithms for these problems were known only for Delta = o(log n). On the other hand, the running time of our algorithm for computing MM (respectively, MIS) is sublogarithmic as long as Delta = n^{o(1)} (resp., Delta = 2^{o(\sqrt{log n})}). We also extend our algorithms to graphs with arboricity at most lambda. Specifically, the running time of our algorithm for computing MM (resp., MIS) on such graphs is O(log lambda + \sqrt{\log n}) (resp., O(log lambda \cdot \sqrt{log n} + log^{3/4} n)).
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