The MST of symmetric disk graphs is light

Abstract

Symmetric disk graphs are often used to model wireless communication networks. Given a set S of n points in R d (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 ) ⩾ | u v | and r ( v ) ⩾ | u v | } , where | u v | 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 ( log n ) 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 2 k 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 ( k n log n ) . We also present two applications of our main theorem, including an alternative 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 ) ⩾ | u v | } ), 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.