Parallel symmetry-breaking in sparse graphs

We describe efficient deterministic techniques for breaking symmetry in parallel. The techniques work well on rooted trees and graphs of constant degree or genus. Our primary technique allows us to 3-color a rooted tree in &Ogr;(lg*n) time on an EREW PRAM using a linear number of processors. We apply these techniques to construct fast linear processor algorithms for several problems, including (&Dgr; + 1)-coloring constant-degree graphs, 5-coloring planar graphs, and finding depth-first-search trees in planar graphs. We also prove lower bounds for 2-coloring directed lists and for finding maximal independent sets in arbitrary graphs.