Parallel Algorithms for the Edge-Coloring and Edge-Coloring Update Problems

Abstract

LetG(V,E) be a simple undirected graph with a maximum vertex degree Δ(G) (or Δ for short), |V| =nand |E| =m. An edge-coloring ofGis an assignment to each edge inGa color such that all edges sharing a common vertex have different colors. The minimum number of colors needed is denoted by χ′(G) (called thechromatic index). For a simple graphG, it is known that Δ ≤ χ′(G) ≤ Δ + 1. This paper studies two edge-coloring problems. The first problem is to perform edge-coloring for an existing edge-colored graphGwith Δ + 1 colors stemming from the addition of a new vertex intoG. The proposed parallel algorithm for this problem runs inO(Δ3/2log3Δ + Δ logn) time usingO(max{nΔ, Δ3}) processors. The second problem is to color the edges of a given uncolored graphGwith Δ + 1 colors. For this problem, our first parallel algorithm requiresO(Δ5.5log3Δ logn+ Δ5log4n) time andO(max{n2Δ,nΔ3}) processors, which is a slight improvement on the algorithm by H. J. Karloff and D. B. Shmoys [J. Algorithms8 (1987), 39–52]. Their algorithm costsO(Δ6log4n) time andO(n2Δ) processors if we use the fastest known algorithm for finding maximal independent sets by M. Goldberg and T. Spencer [SIAM J. Discrete Math.2 (1989), 322–328]. Our second algorithm requiresO(Δ4.5log3Δ logn+ Δ4log4n) time andO(max{n2,nΔ3}) processors. Finally, we present our third algorithm by incorporating the second algorithm as a subroutine. This algorithm requiresO(Δ3.5log3Δ logn+ Δ3log4n) time andO(max{n2log Δ,nΔ3}) processors, which improves, by anO(Δ2.5) factor in time, on Karloff and Shmoys' algorithm. All of these algorithms run in the COMMON CRCW PRAM model.