The MST of Symmetric Disk Graphs

Consider an n-point metric space M = (V, δ), and a transmission range assignment r : V → R+ that maps each point v ∈ V to the disk of radius r(v) around it. The symmetric disk graph (henceforth, SDG) that corresponds to M and r is the undirected graph over V whose edge set includes an edge (u, v) if both r(u) and r(v) are no smaller than δ(u, v). SDGs are often used to model wireless communication networks. Abu-Affash, Aschner, Carmi and Katz (SWAT 2010, [1]) showed that for any n-point 2-dimensional Euclidean space M , the weight of the MST of every connected SDG for M is O(logn) ·w(MST (M)), and that this bound is tight. However, the upper bound proof of [1] relies heavily on basic geometric properties of constant-dimensional Euclidean spaces, and does not extend to Euclidean spaces of super-constant dimension. A natural question that arises is whether this surprising upper bound of [1] can be generalized for wider families of metric spaces, such as high-dimensional Euclidean spaces. In this paper we generalize the upper bound of Abu-Affash et al. [1] for Euclidean spaces of any dimension. Furthermore, our upper bound extends to arbitrary metric spaces and, in particular, it applies to any of the normed spaces lp. Specifically, we demonstrate that for any n-point metric space M , the weight of the MST of every connected SDG for M is O(logn) · w(MST (M)). A preliminary version of this paper is to appear in Proceedings of the 12th International Symposium on Algorithms and Data Structures (WADS), 2011. Department of Computer Science, Ben-Gurion University of the Negev, POB 653, Beer-Sheva 84105, Israel. E-mail: {shayso}@cs.bgu.ac.il This research has been supported by the Clore Fellowship grant No. 81265410 and by BSF grant No. 2008430. Partially supported by the Lynn and William Frankel Center for Computer Sciences.