Simple and Near-Optimal Distributed Coloring for Sparse Graphs

Graph coloring is one of the central problems in distributed graph algorithms. Much of the research on this topic has focused on coloring with $\Delta+1$ colors, where $\Delta$ denotes the maximum degree. Using $\Delta+1$ colors may be unsatisfactory in sparse graphs, where not all nodes have such a high degree; it would be more desirable to use a number of colors that improves with sparsity. A standard measure that captures sparsity is arboricity, which is the smallest number of forests into which the edges of the graph can be partitioned. We present simple randomized distributed algorithms that, with high probability, color any $n$-node $\alpha$-arboricity graph: - using $(2+\varepsilon)\cdot \alpha$ colors, for constant $\varepsilon>0$, in $O(\log n)$ rounds, if $\alpha=\tilde{\Omega}(\log n)$, or - using $O(\alpha \log \alpha )$ colors, in $O(\log n)$ rounds, or - using $O(\alpha)$ colors, in $O(\log n \cdot \min{\log\log n,\; \log \alpha})$ rounds. These algorithms are nearly-optimal, as it is known by results of Linial [FOCS'87] and Barenboim and Elkin [PODC'08] that coloring with $\Theta(\alpha)$ colors, or even poly$(\alpha)$ colors, requires $\Omega(\log_{\alpha} n)$ rounds. The previously best-known $O(\log n)$-time result was a deterministic algorithm due to Barenboim and Elkin [PODC'08], which uses $\Theta(\alpha ^2)$ colors. Barenboim and Elkin stated improving this number of colors as an open problem in their Distributed Graph Coloring Book.