Randomised algorithms for finding small weakly-connected dominating sets of regular graphs

Authors:
William Duckworth;Bernard Mans
Affiliations:
Department of Computing, Macquarie University, Sydney, Australia;Department of Computing, Macquarie University, Sydney, Australia
Venue:
CIAC'03 Proceedings of the 5th Italian conference on Algorithms and complexity
Year:
2003

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Abstract

A weakly-connected dominating set, W, of a graph, G, is a dominating set such that the subgraph consisting of V(G)and all edges incident with vertices in W is connected. Finding a small weakly-connected dominating set of a graph has important applications in clustering mobile ad-hoc networks. In this paper we introduce several new randomised greedy algorithms for finding small weakly-connected dominating sets of regular graphs. These heuristics proceed by growing a weakly-connected dominating set in the graph. We analyse the averagecase performance of the simplest such heuristic on random regular graphs using differential equations. This introduces upper bounds on the size of a smallest weakly-connected dominating set of a random regular graph. We then show that for random regular graphs, other "growing" greedy strategies have exactly the same average-case performance as the simple heuristic.