Low-interference networks in metric spaces of bounded doubling dimension

Let S be a set of n points in a metric space, let R = {Rp : p ∈ S} be a set of positive real numbers, and let GR be the undirected graph with vertex set S in which {p, q} is an edge if and only if |pq| ≤ min(Rp, Rq). The interference of a point q of S is defined to be the number of points p ∈ S \ {q} with |pq| ≤ Rp. For the case when S is a subset of the Euclidean space R, Halldórsson and Tokuyama have shown how to compute a set R such that the graph GR is connected and the maximum interference of any point of S is 2O(d)(1+log(R/D)), where D is the closest-pair distance in S and R is the maximum length of any edge in a minimum spanning tree of S. In this paper, it is shown that the same result holds in any metric space of bounded doubling dimension. Moreover, it is shown that such a set can be computed in O(n log n) expected time. It is shown, by an example of a metric space of doubling dimension one, that the upper bound on the maximum interference is optimal. In fact, this example shows that the maximum interference can be as large as n − 1; in contrast, Halldórsson and Tokuyama have shown that, in R where d ≥ 1 is a constant, there always exists a set R such that GR is connected and each point has interference o(n2). School of Computer Science, Carleton University, Ottawa, Canada. Research supported by NSERC. Faculty of Computer Science, Dalhousie University, Halifax, Canada. Research supported by NSERC and the Canada Research Chairs programme.