Near-Optimal, Distributed Edge Colouring via the Nibble Method

Abstract

We give a distributed randomized algorithm to edge colour a network. Let G be a graph with n nodes and maximum degree ∆. Here we prove:• If ∆ = Ω(log 1+δ n) for some δ > 0 and λ > 0 is fixed, the algorithm almost always colours G with (1 + λ)∆ colours in time O(log n).• If s > 0 is fixed, there exists a positive constant k such that if ∆ = Ω(log k n), the algorithm almost always colours G with ∆ + ∆/ log s n = (1 + o(1))∆ colours in time O(log n + log s n log log n).By "almost always" we mean that the algorithm may fail, but the failure probability can be made arbitrarily close to 0. The algorithm is based on the nibble method, a probabilistic strategy introduced by Vojtěch Rödl. The analysis makes use of a powerful large deviation inequality for functions of independent random variables.

Distributed deterministic edge coloring using bounded neighborhood independence

“…al. [11] used the Rődl nibble method to improve this to a randomized (1 + ϵ)∆-edge-coloring in time O(log n), as long as ∆ = ω(log n). Grable and Panconesi [14] showed that if for every edge e = (u, w), the degree of either u or w is sufficiently large (at least 2 Ω( log n log log n ) ), then a (1 + ϵ)∆-edge-coloring, for an arbitrarily small constant ϵ > 0, can be computed in O(log log n) time by a randomized algorithm.…”

Talagrand’s Inequality and Locality in Distributed Computing

“…We shall lay down conditions on f and on algorithms computing f locally that will enable the methods in the previous section to be extended to to derive a sharp concentration result on f . In particular, we indicate how the edge colouring algorithm above [8] as well as the edge and vertex colouring algorithms from [2,3] follow directly as well as the analysis of a vertex colouring process in [7]. Suppose that f is a function determined by each vertex v of the graph assigning labels (v) to itself and labels (v, e), e ∈ N(v) to its incident edges by some randomised process.…”

“…So, a partition of the edges into a small number of colour classes -i.e. a "good" edge colouring-gives an efficient schedule to perform data transfers (for more details, see [8,2]). The analysis of edge colouring algorithms published in the literature is extremely long and difficult and that in [8] is moreover, based an a certain ad hoc extension of the Chernoff-Hoeffding bounds.…”

Distributed Degree Splitting, Edge Coloring, and Orientations

“…Panconesi and Srinivasan [PS97] give the first such result. Dubhashi, Grable and Panconesi [DGP98] later improve this to (1 + ε)∆-edge-coloring in O(log n) rounds, when ∆ = Ω(log 1+Ω(1) n). This was later refined and extended to graphs of degree ∆ ≥ C 0 , for a constant C 0 depending on ε.…”