ISAAC 2006: Algorithms and Computation pp 527-536 | Cite as
Balanced Cut Approximation in Random Geometric Graphs
Abstract
A random geometric graph \({\mathcal{G}}(n,r)\) is obtained by spreading n points uniformly at random in a unit square, and by associating a vertex to each point and an edge to each pair of points at Euclidian distance at most r. Such graphs are extensively used to model wireless ad-hoc networks, and in particular sensor networks. It is well known that, over a critical value of r, the graph is connected with high probability.
In this paper we study the robustness of the connectivity of random geometric graphs in the supercritical phase, under deletion of edges. In particular, we show that, for a sufficiently large r, any cut which separates two components of Θ(n) vertices each contains Ω(n 2 r 3) edges with high probability. We also present a simple algorithm that, again with high probability, computes one such cut of size O(n 2 r 3). From these two results we derive a constant expected approximation algorithm for the β-balanced cut problem on random geometric graphs: find an edge cut of minimum size whose two sides contain at least βn vertices each.
Keywords
ad-hoc networks sensor networks random geometric graphs balanced cut approximation algorithmsPreview
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References
- 1.Akyildiz, I., Su, W., Sankarasubramaniam, Y., Cayirci, E.: Wireless sensor networks: a survey. Computer Networks 38, 393–422 (2002)CrossRefGoogle Scholar
- 2.Arora, S., Karger, D., Karpinski, M.: Polynomial time approximation schemes for dense instances of NP-hard problems. In: ACM Symposium on the Theory of Computing (STOC), pp. 284–293 (1995)Google Scholar
- 3.Díaz, J., Penrose, M., Petit, J., Serna, M.: Approximating layout problems on random geometric graphs. Journal of Algorithms 39, 78–116 (2001)MATHCrossRefGoogle Scholar
- 4.Bui, T.N., Jones, C.: Finding good approximate vertex and edge partitions is NP-hard. Information Processing Letters 42, 153–159 (1992)MATHCrossRefMathSciNetGoogle Scholar
- 5.Díaz, J., Petit, J., Serna, M.: Evaluation of basic protocols for optical smart dust networks. IEEE Transactions on Mobile Networks 2, 189–196 (2003)Google Scholar
- 6.Feige, U., Krauthgamer, R.: A polylogarithmic approximation of the minimum bisection. SIAM Journal on Computing 31(3), 1090–1119 (2002)MATHCrossRefMathSciNetGoogle Scholar
- 7.Garg, G., Saran, H., Vazirani, V.: Finding separator cuts in planar graphs within twice the optimal. In: IEEE Symposium on Foundations of Computer Science (FOCS), pp. 14–23 (1994)Google Scholar
- 8.Garey, M., Johnson, D.: Computers and Intractability. W.H. Freeman, New York (1979)MATHGoogle Scholar
- 9.Gilbert, E.: Random plane networks. Journal of the Society for Industrial and Applied Mathematics 9, 533–543 (1961)MATHCrossRefGoogle Scholar
- 10.Gerez, S.H.: Algorithms for VLSI design automation. Wiley, Chichester (2003)Google Scholar
- 11.Goel, A., Rai, S., Krishnamachari, V.: Sharp thresholds for monotone properties in random geometric graphs. In: ACM Symposium on Foundations of Computer Science (FOCS), pp. 13–23 (2004)Google Scholar
- 12.Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)MATHGoogle Scholar
- 13.Muthukrishnan, S., Pandurangan, G.: The Bin-covering technique for thresholding random geometric graph properties graphs. In: ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 989–998 (2005)Google Scholar
- 14.Penrose, M.: The longest edge of the random minimal spanning tree. The Annals of Applied Probability 7(2), 340–361 (1997)MATHCrossRefMathSciNetGoogle Scholar
- 15.Penrose, M.: Random Geometric Graphs, Oxford Studies in Probability. Oxford U.P., Oxford (2003)MATHCrossRefGoogle Scholar