对称圆盘图的MST（任意图）是轻图
The MST of Symmetric Disk Graphs (in Arbitrary Metrics) is Light
摘要：
考虑一个n点度量M =（V，delta），以及一个传输范围分配r：V \rightarrow \mathbb R^+，它将V中的每个点v映射到其周围半径为r（v）的圆盘上。对应于M和r的{对称圆盘图}（以下简称SDG）是V上的无向图，如果r（u）和r（v）都不小于delta（u，v），则其边集包括一条边（u，v）。SDG通常用于对无线通信网络进行建模。Abu-Affash，Aschner，Carmi and Katz（SWAT 2010，\cite{AACK10}）证明了对于任何{2维欧几里德} n点度量M，M的每个{连通} SDG的MST的权重是O（log n）w（MST（M）），并且这个界是紧的。然而，\cite{AACK10}的上界证明严重依赖于2维欧几里德度量的基本几何性质，并且没有扩展到更高的维度。一个自然出现的问题是，这个令人惊讶的上界\cite{AACK10}可以推广到更广泛的家庭的度量，如三维欧几里德度量。本文推广了Abu-Affash等人的\cite{AACK10}关于任意维欧氏度量的上界.此外，我们的上界扩展到{任意度量}，特别是，它适用于任何赋范空间ell_p。具体来说，我们证明了对于{任何} n点度量M，M的每个连通SDG的MST的权重是O（log n）w（MST（M））。
英文摘要：
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)).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
