Fast Distributed Brooks' Theorem

We give a randomized $Δ$-coloring algorithm in the LOCAL model that runs in $\text{poly} \log \log n$ rounds, where $n$ is the number of nodes of the input graph and $Δ$ is its maximum degree. This means that randomized $Δ$-coloring is a rare distributed coloring problem with an upper and lower bound in the same ballpark, $\text{poly}\log\log n$, given the known $Ω(\log_Δ\log n)$ lower bound [Brandt et al., STOC '16]. Our main technical contribution is a constant time reduction to a constant number of $(\text{deg}+1)$-list coloring instances, for $Δ= ω(\log^4 n)$, resulting in a $\text{poly} \log\log n$-round CONGEST algorithm for such graphs. This reduction is of independent interest for other settings, including providing a new proof of Brooks' theorem for high degree graphs, and leading to a constant-round Congested Clique algorithm in such graphs. When $Δ=ω(\log^{21} n)$, our algorithm even runs in $O(\log^* n)$ rounds, showing that the base in the $Ω(\log_Δ\log n)$ lower bound is unavoidable. Previously, the best LOCAL algorithm for all considered settings used a logarithmic number of rounds. Our result is the first CONGEST algorithm for $Δ$-coloring non-constant degree graphs.