$(Δ+1)$ Coloring in the Congested Clique Model

In this paper, we present improved algorithms for the $(Δ+1)$ (vertex) coloring problem in the Congested-Clique model of distributed computing. In this model, the input is a graph on $n$ nodes, initially each node knows only its incident edges, and per round each two nodes can exchange $O(\log n)$ bits of information. Our key result is a randomized $(Δ+1)$ vertex coloring algorithm that works in $O(\log\log Δ\cdot \log^* Δ)$-rounds. This is achieved by combining the recent breakthrough result of [Chang-Li-Pettie, STOC'18] in the \local\ model and a degree reduction technique. We also get the following results with high probability: (1) $(Δ+1)$-coloring for $Δ=O((n/\log n)^{1-ε})$ for any $ε\in (0,1)$, within $O(\log(1/ε)\log^* Δ)$ rounds, and (2) $(Δ+Δ^{1/2+o(1)})$-coloring within $O(\log^* Δ)$ rounds. Turning to deterministic algorithms, we show a $(Δ+1)$-coloring algorithm that works in $O(\log Δ)$ rounds.