The Locality of Distributed Symmetry Breaking

We present new bounds on the locality of several classical symmetry breaking tasks in distributed networks. A sampling of the results include 1) A randomized algorithm for computing a maximal matching (MM) in O(log Delta+(log log n)(4)) rounds, where Delta is the maximum degree. This improves a 25-year old randomized algorithm of Israeli and Itai that takes O(log n) rounds and is provably optimal for all log Delta in the range [(log log n)(4), root logn]. 2) A randomized maximal independent set (MIS) algorithm requiring O(log Delta root log n) rounds, for all Delta, and only 2(O(root log log n)) rounds when Delta = poly(logn). These improve on the 25-year old O(log n)-round randomized MIS algorithms of Luby and Alon, Babai, and Itai when log Delta << root log n. 3) A randomized (Delta + 1)-coloring algorithm requiring O(log Delta + 2(O(root log log n))) rounds, improving on an algorithm of Schneider and Wattenhofer that takes O(log Delta+root log n) rounds. This result implies that an O(Delta)-coloring can be computed in 2(O(root log log n)) rounds for all., improving on Kothapalli et al.'s O(root log n)-round algorithm. We also introduce a new technique for reducing symmetry breaking problems on low arboricity graphs to low degree graphs. Corollaries of this reduction include MM and MIS algorithms for low arboricity graphs (e. g., planar graphs and graphs that exclude any fixed minor) requiring O(root log n) and O(log(2/3) n) rounds w.h.p., respectively.