The Locality of Distributed Symmetry Breaking

• Authors: Leonid Barenboim;Michael Elkin;Seth Pettie;Johannes Schneider
• Venue: FOCS '12 Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
• Year: 2012

We present new bounds on the locality of several classical symmetry breaking tasks in distributed networks. A sampling of the results include \begin{enumerate} \item 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 {\em provably optimal} for all $\log\Delta$ in the range $[(\log\log n)^4, \sqrt{\log n}]$. \item A randomized maximal independent set (MIS) algorithm requiring $O(\log\Delta\sqrt{\log n})$ rounds, for all $\Delta$, and only $2^{O(\sqrt{\log\log n})}$ rounds when $\Delta=\poly(\log n)$. These improve on the 25-year old $O(\log n)$-round randomized MIS algorithms of Luby and Alon, Babai, and Itai~when $\log\Delta \ll \sqrt{\log n}$. \item A randomized $(\Delta+1)$-coloring algorithm requiring $O(\log\Delta + 2^{O(\sqrt{\log\log n})})$ rounds, improving on an algorithm of Schneider and Wattenhofer that takes $O(\log\Delta + \sqrt{\log n})$ rounds. This result implies that an $O(\Delta)$-coloring can be computed in $2^{O(\sqrt{\log\log n})}$ rounds for all $\Delta$, improving on Kothapalli et al.'s $O(\sqrt{\log n})$-round algorithm. \end{enumerate} 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 $O(\sqrt{\log n})$ and $O(\log^{2/3} n)$ rounds w.h.p., respectively.