We present an $O(\log^3\log n)$-round distributed algorithm for the $(\Delta+1)$-coloring problem, where each node broadcasts only one $O(\log n)$-bit message per round to its neighbors. Previously, the best such broadcast-based algorithm required $O(\log n)$ rounds. If $\Delta \in \Omega(\log^{3} n)$, our algorithm runs in $O(\log^* n)$ rounds. Our algorithm's round complexity matches state-of-the-art in the much more powerful CONGEST model [Halld\'orsson et al., STOC'21 & PODC'22], where each node sends one different message to each of its neighbors, thus sending up to $\Theta(n\log n)$ bits per round. This is the best complexity known, even if message sizes are unbounded. Our algorithm is simple enough to be implemented in even weaker models: we can achieve the same $O(\log^3\log n)$ round complexity if each node reads its received messages in a streaming fashion, using only $O(\log^3 n)$-bit memory. Therefore, we hope that our algorithm opens the road for adopting the recent exciting progress on sublogarithmic-time distributed $(\Delta+1)$-coloring algorithms in a wider range of (theoretical or practical) settings.

翻译：我们提出了一种用于$(\Delta+1)$着色问题的$O(\log^3\log n)$轮分布式算法，其中每个节点每轮仅向邻居广播一条$O(\log n)$比特的消息。此前，基于广播的最佳算法需要$O(\log n)$轮。若$\Delta \in \Omega(\log^{3} n)$，我们的算法可在$O(\log^* n)$轮内运行。该算法的轮复杂度与功能更强大的CONGEST模型中的最优算法持平[Halldórsson等人，STOC'21 & PODC'22]，该模型允许节点向每个邻居发送不同的消息，从而每轮可发送多达$\Theta(n\log n)$比特。即便在消息规模无限制的情况下，这也是已知的最佳复杂度。我们的算法足够简单，甚至可在更弱的模型中实现：若每个节点以流式方式读取接收到的消息，仅需$O(\log^3 n)$比特内存，即可达到相同的$O(\log^3\log n)$轮复杂度。因此，我们希望该算法能为将次对数轮分布式$(\Delta+1)$着色算法的最新进展推广至更广泛（理论或实践）场景铺平道路。