Coloring Fast with Broadcasts

Summary

We present an O(log3 log n)-round distributed algorithm for the (Δ + 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 Δ ∈ Ω(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 [Halldórsson et al., STOC'21 & PODC'22], where each node sends one different message to each of its neighbors, thus sending up to Θ(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 (Δ + 1)-coloring algorithms in a wider range of (theoretical or practical) settings.