Efficient parallel algorithms for edge coloring problems

Abstract

We present efficient parallel algorithms for several edge coloring problems. For each of the problems that we considered all of the known efficient algorithms were highly sequential. We develop several techniques for exploiting the power of parallel processing in spite of the apparent “inherently” sequential nature of these problems. Classical graph theory results, due to Vizing and Shannon, respectively, state that simple graphs can be edge colored with Δ + 1 colors, and multigraphs can be edge colored with [ 3Δ 2 ] colors. Furthermore, these results are tight, and there exist polynomial-time algorithms that find colorings using the prescribed number of colors. In this paper, we give NC algorithms for coloring multigraphs with 3[ Δ 2 ] colors and simple graphs which have Δ = O(log O(1) n) with Δ + 1 colors. In addition, we will present for each fixed positive ϵ ≤ 1 4 , a randomized NC algorithm that colors arbitrary simple graphs with Δ + O(Δ ( 1 2 +ϵ) ) colors.