The MST of Symmetric Disk Graphs (in Arbitrary Metrics) is Light

Abstract

Consider an n-point metric M = (V,delta), and a transmission range assignment r: V \rightarrow \mathbb R^+ that maps each point v in 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 delta(u,v). SDGs are often used to model wireless communication networks. Abu-Affash, Aschner, Carmi and Katz (SWAT 2010, \cite{AACK10}) showed that for any {2-dimensional Euclidean} n-point metric M, the weight of the MST of every {connected} SDG for M is O(log n) w(MST(M)), and that this bound is tight. However, the upper bound proof of \cite{AACK10} relies heavily on basic geometric properties of 2-dimensional Euclidean metrics, and does not extend to higher dimensions. A natural question that arises is whether this surprising upper bound of \cite{AACK10} can be generalized for wider families of metrics, such as 3-dimensional Euclidean metrics. In this paper we generalize the upper bound of Abu-Affash et al. \cite{AACK10} for Euclidean metrics of any dimension. Furthermore, our upper bound extends to {arbitrary metrics} and, in particular, it applies to any of the normed spaces ell_p. Specifically, we demonstrate that for {any} n-point metric M, the weight of the MST of every connected SDG for M is O(log n) w(MST(M)).