Parallel Algorithms for Colouring Vertices of Graph Using Bit Manipulation

Abstract

Mathematicians for the past two decades have addressed the problem of colouring graph vertices purely as an academic interest. Recently, it has drawn the attention of computer scientists as many resource allocation problems in computers are being modelled as graph colouring problem. Since, the problem of colouring graph vertices is known to be NP-complete, frequently, one has to resort to near-optimal solution. In this paper, we present an efficient scheme that partitions the vertices of a graph into k colour classes using logical operations on bit vectors. The algorithms based on largest and smallest vertex remaining degree are presented. These are extended using heuristics based on Min/Max principal. The algorithms have inherent parallelism and are implemented on loosely coupled parallel computer. Using bit manipulation, it takes a few seconds to obtain near-optimal solution to colour graphs with 200 vertices. It is also observed that one of the four algorithms presented gives optimal solution. The parallel algorithms have linear speedup.