Distributed $(Δ+1)$-Coloring in Sublogarithmic Rounds

The $(\Delta+1)$-coloring problem is a fundamental symmetry breaking problem in distributed computing. We give a new randomized coloring algorithm for $(\Delta+1)$-coloring running in $O(\sqrt{\log \Delta})+ 2^{O(\sqrt{\log \log n})}$ rounds with probability $1-1/n^{\Omega(1)}$ in a graph with $n$ nodes and maximum degree $\Delta$. This implies that the $(\Delta+1)$-coloring problem is easier than the maximal independent set problem and the maximal matching problem, due to their lower bounds by Kuhn, Moscibroda, and Wattenhofer [PODC'04]. Our algorithm also extends to the list-coloring problem where the palette of each node contains $\Delta+1$ colors.