We give a new randomized distributed algorithm for (Δ + 1)-coloring in the LOCAL model, running in O ( log Δ) + 2 O ( √ log log n) rounds in a graph of maximum degree Δ. This implies that the (Δ + 1)-coloring problem is easier than the maximal independent set problem and the maximal matching problem, due to their lower bounds of Ω(min( log n log log n , log Δ log log Δ )) by Kuhn, Moscibroda, and Wattenhofer [PODC'04]. Our algorithm also extends to list-coloring where the palette of each node contains Δ + 1 colors. We extend the set of distributed symmetry-breaking techniques by performing a decomposition of graphs into dense and sparse parts. output. An r -round algorithm in the LOCAL model implies that each vertex only uses information in its r -neighborhood to compute the output, and vice versa. Graph coloring is one of the central problems of graph theory, with numerous applications to algorithms and combinatorics. The (Δ + 1)-coloring problem is the most celebrated case, because for every graph there exists a (Δ + 1)-coloring, which can be found via a simple (sequential) greedy algorithm. Furthermore, even if Δ-colorings exist for some graphs, there are examples where it cannot be solved locally; for example, 2-coloring a ring requires Ω(n) rounds. Vertex-coloring in the distributed model, in particular, has applications as a subroutine in other distributed algorithms (e.g., the Lovász Local Lemma [19] ) and scheduling problems (e.g., radio network broadcasts [13, 15, 18, 46] ). It is typical in such applications that solution quality depends on the number of colors used. Thus, (Δ + 1)-coloring is a natural problem, as it leads to the optimal schedules that one can hope to guarantee as a function of Δ.
√ log log n) rounds with high probability REF .
Distributed (Δ +1)-Coloring in Sublogarithmic Rounds
JACM
Computer Science Mathematics