The Locality of Distributed Symmetry Breaking

Symmetry breaking problems are among the most well studied in the field of distributed computing and yet the most fundamental questions about their complexity remain open. In this paper we work in the LOCAL model (where the input graph and underlying distributed network are identical) and study the randomized complexity of four fundamental symmetry breaking problems on graphs: computing MISs (maximal independent sets), maximal matchings, vertex colorings, and ruling sets. A small sample of our results includes - An MIS algorithm running in $O(\log^2\Delta + 2^{{O(\sqrt{\log\log} n})})$ time, where $\Delta$ is the maximum degree. This is the first MIS algorithm to improve on the 1986 algorithms of Luby and Alon, Babai, and Itai, when $\log n \ll \Delta \ll 2^{\sqrt{\log n}}$, and comes close to the $\Omega(\log \Delta)$ lower bound of Kuhn, Moscibroda, and Wattenhofer. - A maximal matching algorithm running in $O(\log\Delta + \log^4\log n)$ time. This is the first significant improvement to the 1986 algorithm of Israeli and Itai. Moreover, its dependence on $\Delta$ is provably optimal. - A method for reducing symmetry breaking problems in low arboricity/degeneracy graphs to low degree graphs. (Roughly speaking, the arboricity or degeneracy of a graph bounds the density of any subgraph.) Corollaries of this reduction include an $O(\sqrt{\log n})$-time maximal matching algorithm for graphs with arboricity up to $2^{\sqrt{\log n}}$ and an $O(\log^{2/3} n)$-time MIS algorithm for graphs with arboricity up to $2^{(\log n)^{1/3}}$. Each of our algorithms is based on a simple, but powerful technique for reducing a randomized symmetry breaking task to a corresponding deterministic one on a poly$(\log n)$-size graph.