Patterns from nature: Distributed greedy colouring with simple messages and minimal graph knowledge

Abstract

A well-established problem in global optimization is the problem of colouring the vertices of an arbitrary graph using the minimal number of colours, such that adjacent vertices are assigned different colours. One way to restrict the number of colours used is to allow only greedy colourings. A greedy colouring is an assignment of colours to the vertices of a graph that can be obtained by an algorithm that considers each vertex in turn and assigns the first colour that is not already assigned to some neighbour. An optimal colouring can always be obtained in this way, by choosing an appropriate order on the vertices.Recently, a new bio-inspired approach to distributed pattern formation has been proposed, based on modelling the neurological development of the fruit fly. Building on that approach, we propose a new simple randomised algorithm for distributed greedy colouring using only local processing at the vertices and messages along the edges. In our approach the processors exchange only simple messages representing potential colour values and each processor has minimal graph knowledge. We discuss two variations of this algorithm, and investigate their time complexity and message complexity both theoretically and experimentally.In addition, we show experimentally that the number of colours used turns out to be optimal or near-optimal for many standard graph colouring benchmarks. Thus, for distributed networks, our algorithm serves as an effective heuristic approach to computing a colouring with a small number of colours.