Abstract
The cover time of graphs has much relevance to algorithmic applications and has been extensively investigated. Recently, with the advent of ad-hoc and sensor networks, an interesting class of random graphs, namely random geometric graphs, has gained new relevance and its properties have been the subject of much study. A random geometric graph \({\mathcal G}(n,r)\) is obtained by placing n points uniformly at random on the unit square and connecting two points iff their Euclidean distance is at most r. The phase transition behavior with respect to the radius r of such graphs has been of special interest. We show that there exists a critical radius r opt such that for any \(r \geq r_{\rm opt} {\mathcal G}(n,r)\) has optimal cover time of Θ(n log n) with high probability, and, importantly, r opt = Θ(r con) where r con denotes the critical radius guaranteeing asymptotic connectivity. Moreover, since a disconnected graph has infinite cover time, there is a phase transition and the corresponding threshold width is O(r con). We are able to draw our results by giving a tight bound on the electrical resistance of \({\mathcal G}(n,r)\) via the power of certain constructed flows.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Kempe, D., Dobra, A., Gehrke, J.: Gossip-based computation of aggregate information. In: Proc. of the 44th Annual IEEE Symposium on Foundations of Computer Science, pp. 482–491 (2003)
Gkantsidis, C., Mihail, M., Saberi, A.: Random walks in peer-to-peer networks. In: Proc. 23 Annual Joint Conference of the IEEE Computer and Communications Societies, INFOCOM (2004)
Wagner, I.A., Lindenbaum, M., Bruckstein, A.M.: Robotic exploration, brownian motion and electrical resistance. In: Rolim, J.D.P., Serna, M., Luby, M. (eds.) RANDOM 1998. LNCS, vol. 1518, pp. 116–130. Springer, Heidelberg (1998)
Jerrum, M., Sinclair, A.: The markov chain monte carlo method: an approach to approximate counting and integration. In: Hochbaum, D. (ed.) Approximations for NP-hard Problems, pp. 482–520. PWS Publishing, Boston (1997)
Avin, C., Brito, C.: Efficient and robust query processing in dynamic environments using random walk techniques. In: Proc. of the third international symposium on Information processing in sensor networks, pp. 277–286 (2004)
Matthews, P.: Covering problems for Brownian motion on spheres. Ann. Probab. 16, 189–199 (1988)
Aldous, D.J.: Lower bounds for covering times for reversible Markov chains and random walks on graphs. J. Theoret. Probab. 2, 91–100 (1989)
Chandra, A.K., Raghavan, P., Ruzzo, W.L., Smolensky, R.: The electrical resistance of a graph captures its commute and cover times. In: Proc. of the 21st annual ACM symposium on Theory of computing, pp. 574–586 (1989)
Broder, A., Karlin, A.: Bounds on the cover time. J. Theoret. Probab. 2, 101–120 (1989)
Zuckerman, D.: A technique for lower bounding the cover time. In: Proc. of the twenty-second annual ACM symposium on Theory of computing, pp. 254–259 (1990)
Aleliunas, R., Karp, R.M., Lipton, R.J., Lovász, L., Rackoff, C.: Random walks, universal traversal sequences, and the complexity of maze problems. In: 20th Annual Symposium on Foundations of Computer Science (San Juan, Puerto Rico, 1979), pp. 218–223. IEEE, New York (1979)
Jonasson, J.: On the cover time for random walks on random graphs. Comb. Probab. Comput. 7, 265–279 (1998)
Jonasson, J., Schramm, O.: On the cover time of planar graphs. Electronic Communications in Probability 5, 85–90 (2000)
Cooper, C., Frieze, A.: The cover time of sparse random graphs. In: Proc. of the fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2003), pp. 140–147 (2003)
Penrose, M.D.: Random Geometric Graphs. In: Volume 5 of Oxford Studies in Probability. Oxford University Press, Oxford (2003)
Estrin, D., Govindan, R., Heidemann, J., Kumar, S.: Next century challenges: Scalable coordination in sensor networks. In: Proc. of the ACM/IEEE International Conference on Mobile Computing and Networking., pp. 263–270 (1999)
Pottie, G.J., Kaiser, W.J.: Wireless integrated network sensors. Communications of the ACM 43, 51–58 (2000)
Goel, A., Rai, S., Krishnamachari, B.: Monotone properties have sharp thresholds in random geometric graphs. STOC slides (2004), http://www.stanford.edu/sanat/slides/thresholdsstoc.pdf
Avin, C., Ercal, G.: Bounds on the mixing time and partial cover of ad-hoc and sensor networks. In: Proceedings of the 2nd European Workshop on Wireless Sensor Networks (EWSN 2005), pp. 1–12 (2005)
Boyd, S., Ghosh, A., Prabhakar, B., Shah, D.: Gossip and mixing times of random walks on random graphs. Unpublished. (2004), http://www.stanford.edu/~boyd/reports/gossip_gnr.pdf
Sadagopan, N., Krishnamachari, B., Helmy, A.: Active query forwarding in sensor networks (acquire). To appear Elsevier journal on Ad Hoc Networks (2003)
Braginsky, D., Estrin, D.: Rumor routing algorthim for sensor networks. In: Proc. of the 1st ACM Int. workshop on Wireless sensor networks and applications, pp. 22–31 (2002)
Servetto, S.D., Barrenechea, G.: Constrained random walks on random graphs: routing algorithms for large scale wireless sensor networks. In: Proc. of the 1st Int. workshop on Wireless sensor networks and applications, pp. 12–21 (2002)
Penrose, M.D.: The longest edge of the random minimal spanning tree. The Annals of Applied Probability 7, 340–361 (1997)
Gupta, P., Kumar, P.R.: Critical power for asymptotic connectivity in wireless networks. In: Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W. H. Fleming, pp. 547–566 (1998)
Feige, U.: A tight lower bound on the cover time for random walks on graphs. Random Structures and Algorithms 6, 433–438 (1995)
Rai, S.: The spectrum of a random geometric graph is concentrated (2004), http://arxiv.org/PS_cache/math/pdf/0408/0408103.pdf
Winkler, P., Zuckerman, D.: Multiple cover time. Random Structures and Algorithms 9, 403–411 (1996)
Dolev, S., Schiller, E., Welch, J.: Random walk for self-stabilizing group communication in ad-hoc networks. In: Proc. of the 21st IEEE Symposium on Reliable Distributed Systems (SRDS 2002), p. 70. IEEE Computer Society, Los Alamitos (2002)
Goel, A., Rai, S., Krishnamachari, B.: Sharp thresholds for monotone properties in random geometric graphs. In: Proc. of the thirty-sixth annual ACM symposium on Theory of computing, pp. 580–586 (2004)
Avin, C., Ercal, G.: On the cover time of random geometric graphs. Technical Report 040050, UCLA (2004), ftp://ftp.cs.ucla.edu/tech-report/2004-reports/040050.pdf
Doyle, P.G., Snell, J.L.: Random Walks and Electric Networks. In: The Mathematical Association of America, vol. 22 (1984)
Bollobás, B.: Random Graphs. Academic Press, Orlando (1985)
Muthukrishnan, S., Pandurangan, G.: The bin-covering technique for thresholding random geometric graph properties. In: Proc. of the ACM-SIAM Symposium on Discrete Algorithms (2005) (to appear)
Synge, J.L.: The fundamental theorem of electrical networks. Quarterly of Applied Math., 113–127 (1951)
Aldous, D., Fill, J.: Reversible markov chains and random walks on graphs. Unpublished (1997), http://stat-www.berkeley.edu/users/aldous/RWG/book.html
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Avin, C., Ercal, G. (2005). On the Cover Time of Random Geometric Graphs. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds) Automata, Languages and Programming. ICALP 2005. Lecture Notes in Computer Science, vol 3580. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11523468_55
Download citation
DOI: https://doi.org/10.1007/11523468_55
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-27580-0
Online ISBN: 978-3-540-31691-6
eBook Packages: Computer ScienceComputer Science (R0)