SWAT 2010: Algorithm Theory - SWAT 2010 pp 236-247

# The MST of Symmetric Disk Graphs Is Light

• A. Karim Abu-Affash
• Rom Aschner
• Paz Carmi
• Matthew J. Katz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6139)

## 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) ≥ |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(knlogn).

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.

## Keywords

Transmission Range Minimum Span Tree Steiner Point Unit Disk Graph Packet Radio Network
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Althöfer, I., Das, G., Dobkin, D., Joseph, D., Soares, J.: On sparse spanners of weighted graphs. Discrete and Computational Geometry 9(1), 81–100 (1993)
2. 2.
Blough, D.M., Leoncini, M., Resta, G., Santi, P.: On the symmetric range assignment problem in wireless ad hoc networks. In: TCS’02: Proc. of the IFIP 17th World Computer Congress – TC1 Stream / 2nd IFIP International Conference on Theoretical Computer Science, pp. 71–82 (2002)Google Scholar
3. 3.
Călinescu, G., Măndoiu, I.I., Zelikovsky, A.: Symmetric connectivity with minimum power consumption in radio networks. In: TCS’02: Proc. of the IFIP 17th World Computer Congress – TC1 Stream / 2nd IFIP International Conference on Theoretical Computer Science, pp. 119–130 (2002)Google Scholar
4. 4.
Chandra, B., Das, G., Narasimhan, G., Soares, J.: New sparseness results on graph spanners. Internat. J. of Computational Geometry and Applications 5, 125–144 (1995)
5. 5.
Clementi, A.E.F., Penna, P., Silvestri, R.: Hardness results for the power range assignment problem in packet radio networks. In: Hochbaum, D.S., Jansen, K., Rolim, J.D.P., Sinclair, A. (eds.) RANDOM 1999 and APPROX 1999. LNCS, vol. 1671, pp. 197–208. Springer, Heidelberg (1999)Google Scholar
6. 6.
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press, Cambridge (2009)
7. 7.
Kirousis, L., Kranakis, E., Krizanc, D., Pelc, A.: Power consumption in packet radio networks. Theoretical Computer Science 243(1-2), 289–305 (2000)
8. 8.
Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press, Cambridge (2007)

## Authors and Affiliations

• A. Karim Abu-Affash
• 1
• Rom Aschner
• 1
• Paz Carmi
• 1
• Matthew J. Katz
• 1
1. 1.Department of Computer ScienceBen-Gurion UniversityBeer-ShevaIsrael