Consider an n-point metric space M = (V, δ) and a transmission range assignment r : V → R + that maps each point v ε V to the disk of radius r(v) around it. The symmetric disk graph (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 d(u, v). SDGs are often used to model wireless communication networks. Abu-Affash et al. [Lecture Notes in Comput. Sci. 6139, Springer, Heidelberg, 2010, pp. 236-247] showed that for any n-point 2-dimensional Euclidean space M, the weight of the minimum spanning tree (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 Abu-Affash et al. 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 Abu-Affash et al. can be generalized for wider families of metric spaces, such as highdimensional Euclidean spaces. In this paper we generalize the upper bound of Abu-Affash et al. 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(log n) · w(MST(M)).
- Finite metric space
- Minimum spanning tree
- Symmetric disk graph
- Wireless communication network