Very fast parallel algorithms for approximate edge coloring

Abstract

This paper presents very fast parallel algorithms for approximate edge coloring. Let log(1)n=logn,log(k)n=log(log(k-1)n), and log*(n)=min{k|log(k)n<1}. It is shown that a graph with n vertices and m edges can be edge colored with (2⌈log1/4log*(n)⌉) c·(⌈Δ/logc/4log*(n)⌉) 2 colors in O(loglog*(n)) time using O(m+n) processors on the EREW PRAM, where Δ is the maximum vertex degree of the graph and c is an arbitrarily large constant. It is also shown that the graph can be edge colored using at most ⌈4Δ1+4/logloglog*(Δ)log 1/2log*(Δ)⌉ colors in O(logΔloglog*(Δ)/logloglog*(Δ)+loglog*(n)) time using O(m+n) processors on the same model. © 2001 Elsevier Science B.V.