Fast Distributed Brooks' Theorem
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We give a randomized Δ-coloring algorithm in the LOCAL model that runs in polyloglog n rounds, where n is the number of nodes of the input graph and Δ is its maximum degree. This means that randomized Δ-coloring is a rare distributed coloring problem with an upper and lower bound in the same ballpark, polyloglog n, given the known Ω(log_Δlog n) lower bound [Brandt et al., STOC '16]. Our main technical contribution is a constant time reduction to a constant number of (deg+1)-list coloring instances, for Δ = ω(log^4 n), resulting in a polyloglog n-round CONGEST algorithm for such graphs. This reduction is of independent interest for other settings, including providing a new proof of Brooks' theorem for high degree graphs, and leading to a constant-round Congested Clique algorithm in such graphs. When Δ=ω(log^21 n), our algorithm even runs in O(log^* n) rounds, showing that the base in the Ω(log_Δlog n) lower bound is unavoidable. Previously, the best LOCAL algorithm for all considered settings used a logarithmic number of rounds. Our result is the first CONGEST algorithm for Δ-coloring non-constant degree graphs.
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