The MST of Symmetric Disk Graphs Is Light

Symmetric disk graphs are often used to model wireless communication networks. Given a set S of n points in R (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(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 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.