TY - GEN

T1 - The MST of symmetric disk graphs (in arbitrary metric spaces) is light

AU - Solomon, Shay

N1 - Funding Information: ★ This research has been supported by the Clore Fellowship grant No. 81265410, by the BSF grant No. 2008430, and by the Lynn and William Frankel Center for CS.

PY - 2011

Y1 - 2011

N2 - Consider an n-point metric space M = (V,δ), and a transmission range assignment r: V → ℝ+ that maps each point v ∈ 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 δ(u,v). SDGs are often used to model wireless communication networks. Abu-Affash, Aschner, Carmi and Katz (SWAT 2010, [1]) showed that for any n-point 2-dimensional Euclidean space M, the weight of the MST of every connected SDG for M is O(logn)·w(MST(M)), and that this bound is tight. However, the upper bound proof of [1] 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 [1] 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. [1] 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(logn)·w(MST(M)).

AB - Consider an n-point metric space M = (V,δ), and a transmission range assignment r: V → ℝ+ that maps each point v ∈ 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 δ(u,v). SDGs are often used to model wireless communication networks. Abu-Affash, Aschner, Carmi and Katz (SWAT 2010, [1]) showed that for any n-point 2-dimensional Euclidean space M, the weight of the MST of every connected SDG for M is O(logn)·w(MST(M)), and that this bound is tight. However, the upper bound proof of [1] 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 [1] 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. [1] 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(logn)·w(MST(M)).

UR - http://www.scopus.com/inward/record.url?scp=80052128092&partnerID=8YFLogxK

U2 - https://doi.org/10.1007/978-3-642-22300-6_59

DO - https://doi.org/10.1007/978-3-642-22300-6_59

M3 - منشور من مؤتمر

SN - 9783642222993

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 691

EP - 702

BT - Algorithms and Data Structures - 12th International Symposium, WADS 2011, Proceedings

T2 - 12th International Symposium on Algorithms and Data Structures, WADS 2011

Y2 - 15 August 2011 through 17 August 2011

ER -