Coloring Fast with Broadcasts

We present an O(log(3) logn)-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 is an element of Omega(log(3) n), our algorithm runs in O(log*n) rounds. Our algorithm's round complexity matches the state-of-the-art in the much more powerful CONGEST model [Halldorsson et al., STOC'21 & PODC'22], where each node sends one different message to each of its neighbors, thus sending up to Theta(n logn) 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) logn) 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.