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Line-of-Sight Networks

Published online by Cambridge University Press:  01 March 2009

ALAN FRIEZE ,
JON KLEINBERG ,
R. RAVI  and
WARREN DEBANY
ALAN FRIEZE
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA 15213, USA (e-mail: [email protected])
JON KLEINBERG
Affiliation:
Department of Computer Science, Cornell University, Ithaca NY 14853, USA (e-mail: [email protected])
R. RAVI
Affiliation:
Tepper School of Business, Carnegie Mellon University, Pittsburgh PA 15213, USA (e-mail: [email protected])
WARREN DEBANY
Affiliation:
Information Grid Division, Air Force Research Laboratory/RIG, 525 Brooks Road, Rome, NY 13441-4505, USA (e-mail: [email protected])
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Abstract

Random geometric graphs have been one of the fundamental models for reasoning about wireless networks: one places n points at random in a region of the plane (typically a square or circle), and then connects pairs of points by an edge if they are within a fixed distance of one another. In addition to giving rise to a range of basic theoretical questions, this class of random graphs has been a central analytical tool in the wireless networking community.

For many of the primary applications of wireless networks, however, the underlying environment has a large number of obstacles, and communication can only take place among nodes when they are close in space and when they have line-of-sight access to one another – consider, for example, urban settings or large indoor environments. In such domains, the standard model of random geometric graphs is not a good approximation of the true constraints, since it is not designed to capture the line-of-sight restrictions.

Here we propose a random-graph model incorporating both range limitations and line-of-sight constraints, and we prove asymptotically tight results for k-connectivity. Specifically, we consider points placed randomly on a grid (or torus), such that each node can see up to a fixed distance along the row and column it belongs to. (We think of the rows and columns as ‘streets’ and ‘avenues’ among a regularly spaced array of obstructions.) Further, we show that when the probability of node placement is a constant factor larger than the threshold for connectivity, near-shortest paths between pairs of nodes can be found, with high probability, by an algorithm using only local information. In addition to analysing connectivity and k-connectivity, we also study the emergence of a giant component, as well an approximation question, in which we seek to connect a set of given nodes in such an environment by adding a small set of additional ‘relay’ nodes.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 1-2: Papers from The 2006 Oberwolfach Meeting on Combinatorics, Probability and Computing , March 2009 , pp. 145 - 163
Copyright
Copyright © Cambridge University Press 2008

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References

[1]Alon, N. and Spencer, J. H. (2000) The Probabilistic Method, 2nd edn, Wiley.Google Scholar
[2]Bettstetter, C. (2002) On the minimum node degree and connectivity of a wireless multihop network. In Proc. 3rd ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc '02), pp. 80–91.Google Scholar
[3]Bettstetter, C. (2004) On the connectivity of ad hoc networks. The Computer Journal (special issue on Mobile and Pervasive Computing) 47 432–447.Google Scholar
[4]Bollobás, B. (2001) Random Graphs, 2nd edn, Cambridge University Press.Google Scholar
[5]Bollobás, B., Janson, S. and Riordan, O. Line-of-sight percolation. Combin. Probab. Comput., this issue.Google Scholar
[6]Chrobak, M., Naor, J. and Novick, M.Using bounded degree spanning trees in the design of efficient algorithms on claw-free graphs. In Proc. Workshop on Algorithms and Data Structures, Vol. 382 of Lecture Notes in Computer Science, pp. 147–162.Google Scholar
[7]Deuschel, J. and Pisztora, A. (1996) Surface order large deviations for high-density percolation. Probab. Theory Rel. Fields 104 467–482.Google Scholar
[8]Efrat, A. and Har-Peled, S. (2002) Locating guards in art galleries. In 2nd IFIP International Conference on Theoretical Computer Science, pp. 181–192.Google Scholar
[9]Efrat, A., Har-Peled, S. and Mitchell, J. S. B. (2005) Approximation algorithms for two optimal location problems in sensor networks. In Broadnets 2005, pp. 767–776.Google Scholar
[10]Falconer, K. J. and Grimmett, G. R. (1992) On the geometry of random Cantor sets and fractal percolation. J. Theoret. Probab. 5 465–485.Google Scholar
[11]Frieze, A. M., Kleinberg, J., Ravi, R. and Debany, W. (2007) Line-of-sight networks. In Proc. 18th ACM–SIAM Symposium on Discrete Algorithms, pp. 968–977.Google Scholar
[12]Frieze, A. M. and Molloy, M. (1999) Splitting an expander graph. J. Algorithms 33 166–172.Google Scholar
[13]Goel, A., Rai, S. and Krishnamachari, B. (2004) Sharp thresholds for monotone properties in random geometric graphs. In Proc. ACM Symposium on Theory of Computing, pp. 580–586.CrossRefGoogle Scholar
[14]Grimmett, G. (1999) Percolation, 2nd edn, Springer.Google Scholar
[15]Gupta, P. and Kumar, P. (1998) Critical power for asymptotic connectivity in wireless networks. In Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W. H. Fleming (McEneaney, W. M., Yin, G. and Zhang, Q., eds), Birkhäuser, Boston, pp. 547–566.Google Scholar
[16]Gupta, P. and Kumar, P. (2000) The capacity of wireless networks. In IEEE Trans. Inform. Theory 46 388–404. Corrigendum: A correction to the proof of a lemma in ‘The capacity of wireless networks’, IEEE Trans. Inform. Theory 49 3117.Google Scholar
[17]Kalai, G. and Matousek, J. (1997) Guarding galleries where every point sees a large area. Israel J. Math. 101 125–139.Google Scholar
[18]Klein, P. N. and Ravi, R. (1995) A nearly best-possible approximation algorithm for node-weighted Steiner trees. J. Algorithms 19 104–115.Google Scholar
[19]Mauve, M., Widmer, J. and Hartenstein, H. (2001) A survey on position-based routing in mobile ad hoc networks. IEEE Network Magazine 15 (6)30–39.Google Scholar
[20]Mobile ad-hoc networks (MANET) charter. http://www.ietf.org/html.charters/manet-charter.htmlGoogle Scholar
[21]Penrose, M. D. (1999) On k-connectivity for a geometric random graph. Random Struct. Alg. 15 145–164.Google Scholar
[22]Penrose, M. D. (2003) Random Geometric Graphs, Vol. 5 of Oxford Studies in Probability, Oxford University Press.Google Scholar
[23]Robins, G. and Zelikovsky, A. (2000) Improved Steiner tree approximation in graphs. In Proc. 10th Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 770–779.Google Scholar
[24]Royer, E. and Toh, C.-K. (1999) A review of current routing protocols for ad-hoc mobile wireless networks. IEEE Personal Communications, April 1999.Google Scholar
[25]Valtr, P. (1998) Guarding galleries where no point sees a small area. Israel J. Math. 104 1–16.Google Scholar