Distributed Maximal Independent Set using Small Messages.

Maximal Independent Set (MIS) is one of the central problems in distributed graph algorithms. The celebrated works of Luby [STOC'85] and Alon, Babai, and Itai [JALG'86] provide O(log n)-round randomized distributed MIS algorithms, which work with O(log n)-bit messages. This round complexity was improved to [MATH HERE] in a breakthrough of Barenboim, Elkin, Pettie, and Schneider [FOCS'11; JACM'16] and then to [MATH HERE] by Ghaffari [SODA'16], where Δ denotes the maximum degree. However, these improvements have one drawback: they require much larger messages, up to poly(Δ log n) bits. Indeed, the question of improving the O(log n) round complexity using small messages has remained open for three decades, for essentially all values of Δ, except for Δ = o(log n) where there are O(Δ + log* n)-round deterministic algorithms. We present a randomized distributed MIS algorithm, with O(log n)-bit messages, that achieves a round complexity of [MATH HERE]. This is the first algorithm with small messages that improves on the O(log n) round complexity of Luby and Alon et al. for a wide range of Δ, and its complexity almost matches that of the best known algorithm using unbounded message sizes. As applications of this MIS algorithm or along the way to it, we obtain improved distributed algorithms with small messages for some other well-studied problems including network decompositions, (Δ + 1)-vertex coloring, and ruling sets.