We provide a O(log 6 log n)-round randomized algorithm for distance-2 coloring in CONGEST with ∆ 2 +1 colors. For ∆ ≫ poly log n, this improves exponentially on the O(log ∆+poly log log n) algorithm of [Halldórsson, Kuhn, Maus, Nolin, DISC'20].Our study is motivated by the ubiquity and hardness of local reductions in CONGEST. For instance, algorithms for the Local Lovász Lemma [Moser, Tardos, JACM'10; Fischer, Ghaffari, DISC'17; Davies, SODA'23] usually assume communication on the conflict graph, which can be simulated in LOCAL with only constant overhead, while this may be prohibitively expensive in CONGEST. We hope our techniques help tackle in CONGEST other coloring problems defined by local relations.

Is fully decentralized graph streaming possible? We consider this question in the context of the ∆ `1-coloring problem. With the celebrated distributed sketching technique of palette sparsification [Assadi, Chen, and Khanna SODA'19], nodes limit themselves to Oplog nq independently sampled colors. They showed that it suffices to color the resulting sparsified graph with edges between nodes that sampled a common color. To compute the actual coloring, however, that information must be gathered at a single server for centralized processing. We seek instead a local algorithm to compute such a coloring in the sparsified graph. The question is if this can be achieved in polyplog nq distributed rounds with small messages.Our main result is an algorithm that computes a ∆ `1-coloring after palette sparsification with poly log n random colors per node and runs in Oplog 2 ∆ `log 3 log nq rounds on the sparsified graph, using Oplog nq-bit messages. We show that this is close to the best possible: any distributed ∆`1-coloring algorithm that runs in the LOCAL model on the sparsified graph given by palette sparsification requires Ωplog ∆{ log log nq rounds.Our result has implications beyond streaming, as space efficiency also leads to low message complexity. In particular, our algorithm yields the first polyplog nq-round algorithms for ∆ `1coloring in two previously studied distributed models: the Node Capacitated Clique, and the cluster graph model.