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

TL;DR

This paper proves that the minimum spanning tree weight of symmetric disk graphs in any metric space, including higher-dimensional Euclidean and normed spaces, is bounded by a logarithmic factor of the MST weight, generalizing previous 2D results.

Contribution

It extends the known upper bound on MST weight of symmetric disk graphs from 2D Euclidean metrics to arbitrary metric spaces, including higher dimensions and normed spaces.

Findings

MST weight of SDGs is O(log n) times the MST weight in any metric.

The bound applies to higher-dimensional Euclidean and normed spaces.

Generalizes previous 2D Euclidean results to broader metric families.

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)).