Deterministic dominating set construction in networks with bounded degree

Abstract

This paper considers the problem of calculating dominating sets in networks with bounded degree. In these networks, the maximal degree of any node is bounded by Δ, which is usually significantly smaller than n, the total number of nodes in the system. Such networks arise in various settings of wireless and peer-to-peer communication. A trivial approach of choosing all nodes into the dominating set yields an algorithm with the approximation ratio of Δ+1. We show that any deterministic algorithm with non-trivial approximation ratio requires Ω(log* n) rounds, meaning effectively that no o(Δ)-approximation deterministic algorithm with a running time independent of the size of the system may ever exist. On the positive side, we show two deterministic algorithms that achieve logΔ and 2logΔ-approximation in O(Δ3+log* n) and O(Δ2logΔ+log* n) time, respectively. These algorithms rely on coloring rather than node IDs to break symmetry. © 2011 Springer-Verlag Berlin Heidelberg.