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

• Authors: Shay Solomon
• Venue: SIAM Journal on Discrete Mathematics
• Year: 2012

Consider an $n$-point metric space $M = (V,\delta)$ 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 (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 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 $O(\log n) \cdot w(MST(M))$, 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 high-dimensional 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 $\ell_p$. Specifically, we demonstrate that for any $n$-point metric space $M$, the weight of the MST of every connected SDG for $M$ is $O(\log n) \cdot w(MST(M))$.