Deterministic (Δ + 1)-Coloring in sublinear (in Δ) Time in static, dynamic and faulty networks

In the distributed message passing model a communication network is represented by an n-vertex graph G = (V,E) of maximum degree Δ. Computation proceeds in discrete synchronous rounds consisting of sending and receiving messages and performing local computations. The running time of an algorithm is the number of rounds it requires. In the static setting the network remains unchanged throughout the entire execution. In the dynamic setting the topology of the network changes, and a new solution has to be computed after each change. In the faulty setting the network is static, but some vertices or edges may lose the computed solution as a result of faults. The goal of an algorithm in this setting is fixing the solution. The problems of (Δ + 1)-vertex-coloring and (2Δ a 1)- edge-coloring are among the most important and intensively studied problems in distributed computing. Despite a very intensive research in the last 30 years, no deterministic algorithms for these problems with sublinear (in Δ) time have been known so far. Moreover, for more restricted scenarios and some related problems there are lower bounds of Ω(Δ) [13, 14, 20, 27]. The question of the possibility to devise algorithms that overcome this challenging barrier is one of the most fundamental questions in distributed symmetry breaking [4, 6, 13, 14, 19, 24]. In this paper we settle this question for (Δ+1)-vertex-coloring and (2Δa1)- edge-coloring by devising deterministic algorithms that require O(Δ3=4 logΔ+log ∗ n) time in the static, dynamic and faulty settings. (The term log ∗ n is unavoidable in view of the lower bound of Linial [21].) Moreover, for (1+o(1))Δ- vertex-coloring and (2+o(1))Δ-edge-coloring we devise algorithms with O( √ Δ + log ∗ n) deterministic time. This is roughly a quadratic improvement comparing to the state- of-the-art that requires O(Δ+log ∗ n) time [4, 19, 24]. Our results are actually more general than that since they apply also to a variant of the list-coloring problem that generalizes ordinary coloring. Our results are obtained using a novel technique for coloring partially-colored graphs (also known as fixing). We partition the uncolored parts into a small number of subgraphs with certain helpful properties. Then we color these subgraphs gradually using a technique that employs con- structions of polynomials in a novel way. Our construction is inspired by the algorithm of Linial [21] for ordinary O(Δ2)-coloring. However, it is a more sophisticated construction that differs from [21] in several important respects. These new insights in using systems of polynomials allow us to significantly speed up the O(Δ)-coloring algorithms. Moreover, they allow us to devise algorithms with the same running time also in the more complicated settings of dynamic and faulty networks.