TY - GEN

T1 - The MST of symmetric disk graphs is light

AU - Abu-Affash, A. Karim

AU - Aschner, Rom

AU - Carmi, Paz

AU - Katz, Matthew J.

PY - 2010/7/21

Y1 - 2010/7/21

N2 - Symmetric disk graphs are often used to model wireless communication networks. Given a set S of n points in ℝd (representing n transceivers) and a transmission range assignment r : S → ℝ, the symmetric disk graph of S (denoted SDG(S)) is the undirected graph over S whose set of edges is E = {(u,v) | r(u) ≥ |uv| and r(v) ≥ |uv|}, where |uv| denotes the Euclidean distance between points u and v. We prove that the weight of the MST of any connected symmetric disk graph over a set S of n points in the plane, is only O(logn) times the weight of the MST of the complete Euclidean graph over S. We then show that this bound is tight, even for points on a line. Next, we prove that if the number of different ranges assigned to the points of S is only k, k ≪ n, then the weight of the MST of SDG(S) is at most 2k times the weight of the MST of the complete Euclidean graph. Moreover, in this case, the MST of SDG(S) can be computed efficiently in time O(kn log n). We also present two applications of our main theorem, including an alternative and simpler proof of the Gap Theorem, and a result concerning range assignment in wireless networks. Finally, we show that in the non-symmetric model (where E = {(u,v) | r(u) ≥ |uv|}), the weight of a minimum spanning subgraph might be as big as Ω(n) times the weight of the MST of the complete Euclidean graph.

AB - Symmetric disk graphs are often used to model wireless communication networks. Given a set S of n points in ℝd (representing n transceivers) and a transmission range assignment r : S → ℝ, the symmetric disk graph of S (denoted SDG(S)) is the undirected graph over S whose set of edges is E = {(u,v) | r(u) ≥ |uv| and r(v) ≥ |uv|}, where |uv| denotes the Euclidean distance between points u and v. We prove that the weight of the MST of any connected symmetric disk graph over a set S of n points in the plane, is only O(logn) times the weight of the MST of the complete Euclidean graph over S. We then show that this bound is tight, even for points on a line. Next, we prove that if the number of different ranges assigned to the points of S is only k, k ≪ n, then the weight of the MST of SDG(S) is at most 2k times the weight of the MST of the complete Euclidean graph. Moreover, in this case, the MST of SDG(S) can be computed efficiently in time O(kn log n). We also present two applications of our main theorem, including an alternative and simpler proof of the Gap Theorem, and a result concerning range assignment in wireless networks. Finally, we show that in the non-symmetric model (where E = {(u,v) | r(u) ≥ |uv|}), the weight of a minimum spanning subgraph might be as big as Ω(n) times the weight of the MST of the complete Euclidean graph.

UR - http://www.scopus.com/inward/record.url?scp=77954630163&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-13731-0_23

DO - 10.1007/978-3-642-13731-0_23

M3 - Conference contribution

AN - SCOPUS:77954630163

SN - 364213730X

SN - 9783642137303

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

SP - 236

EP - 247

BT - Algorithm Theory - SWAT 2010 - 12th Scandinavian Symposium and Workshops on Algorithm Theory, Proceedings

T2 - 12th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2010

Y2 - 21 June 2010 through 23 June 2010

ER -