Ultrafast Distributed Coloring of High Degree Graphs

We give a new randomized distributed algorithm for the $\mathrm{\Delta }+1$-list coloring problem. The algorithm and its analysis dramatically simplify the previous best result known of Chang, Li, and Pettie [SICOMP 2020]. This allows for numerous refinements, and in particular, we can color all $n$-node graphs of maximum degree $\mathrm{\Delta }\ge {\log }^{2+\mathrm{\Omega }(1)}n$ in $O({\log }^{\ast }n)$ rounds. The algorithm works in the CONGEST model, i.e., it uses only $O(\log n)$ bits per message for communication. On low-degree graphs, the algorithm shatters the graph into components of size $\mathrm{poly}(\log n)$ in $O(\log^* \Delta)$rounds,showingthattherandomizedcomplexityof$\Delta+1$-list coloring in CONGEST depends inherently on the deterministic complexity of related coloring problems.