Distributed Coloring Depending on the Chromatic Number or the Neighborhood Growth

Abstract

We deterministically compute a Δ + 1 coloring in time O(Δ_5c + 2·(Δ₅)^2/c/(Δ₁)^ε + (Δ₁)^ε + log^* n) and O(Δ_5c + 2·(Δ₅)^1/c/Δ^ε + Δ^ε + (Δ₅)^dlogΔ₅logn) for arbitrary constants d,ε and arbitrary constant integer c, where Δ _i is defined as the maximal number of nodes within distance i for a node and Δ: = Δ₁. Our greedy algorithm improves the state-of-the-art Δ + 1 coloring algorithms for a large class of graphs, e.g. graphs of moderate neighborhood growth. We also state and analyze a randomized coloring algorithm in terms of the chromatic number, the run time and the used colors. If $\Delta \in \Omega(\log^{1+1/\log^* n} n)$ and $\chi\in O(\Delta/\log^{1+1/\log^* n} n)$ then our algorithm executes in time O(logχ + log^* n) with high probability. For graphs of polylogarithmic chromatic number the analysis reveals an exponential gap compared to the fastest Δ + 1 coloring algorithm running in time $O(\log \Delta+\sqrt{\log n})$. The algorithm works without knowledge of χ and uses less than Δ colors, i.e., (1 − 1/O(χ))Δ with high probability. To the best of our knowledge this is the first distributed algorithm for (such) general graphs taking the chromatic number χ into account.