Abstract
A localized Delaunay triangulation owns the following interesting properties for sensor and wireless ad hoc networks: it can be built with localized information, the communication cost imposed by control information is limited, and it supports geographical routing algorithms that offer guaranteed convergence. This paper presents two localized algorithms, fast localized Delaunay triangulation 1 (FLDT1) and fast localized Delaunay triangulation 2 (FLDT2), that build a graph called planar localized Delaunay triangulation, PLDel, known to be a good spanner of the Unit Disk Graph, UDG. Our algorithms improve previous algorithms with similar theoretical bounds in the following aspects: unlike previous work, FLDT1 and FLDT2 build PLDel in a single communication step, maintaining a communication cost of O(n log n), which is within a constant of the optimal. Additionally, we show that FLDT1 is more robust than previous triangulation algorithms, because it does not require the strict UDG connectivity model to work. The small signaling cost of our algorithms allows us to improve routing performance, by efficiently using the PLDel graph instead of sparser graphs, like the Gabriel or the Relative Neighborhood graphs.
Similar content being viewed by others
References
Araujo, F., & Rodrigues, L. (2004). Fast localized delaunay triangulation. In The 8th International Conference on Principles of Distributed Systems (OPODIS 2004) (pp. 81–93). Grenoble: Springer-Verlag, LNCS 3544.
Araujo, F., & Rodrigues, L. (2006). Single-step creation of localized delaunay triangulations. Technical Report TR 06/03, Centre of Informatics and Systems of the University of Coimbra, ISSN 0874-338X.
Avin, C. (2005). Fast and efficient restricted delaunay triangulation in random geometric graphs. In Workshop on Combinatorial and Algorithmic Aspects of Networking (CAAN 2005).
Bhardwaj, M., Chandrakasan, A., & Garnett, T. (2001). Upper bounds on the lifetime of sensor networks. In IEEE International Conference on Communications (pp. 785–790).
Boissonnat, J.-D., & Teillaud, M. (1993). On the randomized construction of the Delaunay tree. Theoretical Computer Science, 112(2), 339–354.
Bondy, J. A., & Murty, U. S. R. (1976). Graph Theory with Applications. North-Holland: Elsevier.
Bose, P., & Morin, P. (1999). Online routing in triangulations. In 10th Annual Internation Symposium on Algorithms and Computation (ISAAC).
Bose, P., Morin, P., Stojmenovic, I., & Urrutia, J. (1999). Routing with guaranteed delivery in ad hoc wireless networks. In International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications (DIALM) (pp. 48–55).
Dobkin, D., Friedman, S. J., & Supowit, K. J. (1990). Delaunay graphs are almost as good as complete graphs. Discrete Computational Geometry, 5(1), 399–407.
Eppstein, D. (2000). Spanning trees and spanners. In Handbook of Computational Geometry (pp. 425–461). North-Holland: Elsevier.
Finn, G. (1987). Routing and addressing problems in large metropolitan-scale internetworks. Technical Report ISU/RR-87-180, Institute for Scientific Information, March.
Fortune, S. (1987). A sweepline algorithm for Voronoi diagrams. Algorithmica, 2, 153–174.
Frey, H., & Stojmenovic, I. (2006). On delivery guarantees of face and combined greedy-face routing in ad hoc and sensor networks. In MobiCom ’06: Proceedings of the 12th Annual International Conference on Mobile Computing and Networking (pp. 390–401). New York: ACM Press.
Gao, J., Guibas, L., Hershberger, J., Zhang, L., & Zhu, A. (2001). Geometric spanners for routing in mobile networks. In 2nd ACM Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc 01).
Karp, B., & Kung, H. (2000). GPRS: Greedy perimeter stateless routing for wireless networks. In ACM/IEEE International Conference on Mobile Computing and Networking.
Kim, Y.-J., Govindan, R., Karp, B., & Shenker, S. (2005). On the pitfalls of geographic face routing. In DIALM-POMC ’05: Proceedings of the 2005 Joint Workshop on Foundations of Mobile Computing (pp. 34–43). New York: ACM Press.
Kozma, G., Lotker, Z., Sharir, M., & Stupp, G. (2004). Geometrically aware communication in random wireless networks. In PODC ’04: Proceedings of the Twenty-third Annual ACM Symposium on Principles of Distributed Computing (pp. 310–319). New York: ACM Press.
Kranakis, E., Singh, H., & Urrutia, J. (1999). Compass routing on geometric networks. In 11th Canadian Conference on Computation Geometry (CCCG 99).
Kuhn, F., Wattenhofer, R., Zhang, Y., & Zollinger, A. (2003). Geometric ad-hoc routing: Of theory and practice. In 22nd ACM Symposium on the Principles of Distributed Computing (PODC 2003), Boston, July.
Kuhn, F., Wattenhofer, R., & Zollinger, A. (2002). Asymptotically optimal geometric mobile ad-hoc routing. In 6th International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications (DIALM’02).
Lan, L., & Wen-Jing, H. (2002). Localized Delaunay triangulation for topological construction and routing on manets. In 2nd ACM Workshop on Principles of Mobile Computing (POMC’02).
Lee, D.-T., & Schachter, B. (1980). Two algorithms for constructing a Delaunay triangulation. International Journal of Computer and Information Sciences, 9(3), 219–242
Li, X.-Y., Calinescu, G., & Wan, P.-J. (2002). Distributed construction of a planar spanner and routing for ad hoc wireless networks. In The 21st Annual Joint Conference of the IEEE Computer and Communications Societies (INFOCOM).
Li, X.-Y., Calinescu, G., Wan, P.-Jun, & Wang, Y. (2003). Localized delaunay triangulation with application in ad hoc wireless networks. IEEE Transactions on Parallel and Distributed Systems, 14(9), 1035–1047.
Li, X.-Y., Stojmenovic, I., & Wang, Y. (2004). Partial delaunay triangulation and degree limited localized bluetooth scatternet formation. IEEE Transactions on Parallel and Distributed Systems, 15(4), 350–361.
Liebeherr, J., Nahas, M., & Si, W. (2001). Application-layer multicasting with Delaunay triangulation overlays. Technical Report CS-2001-26, University of Virginia, Department of Computer Science, Charlottesville, VA 22904, 5.
Lynch, N. (1996). Distributed algorithms. In Data Link Protocols (Chap. 16, pp. 691–732). Morgan-Kaufmann.
Preparata, F. P., & Shamos, M. I. (1985). Computational Geometry: An Introduction. New York: Springer-Verlag.
Rodoplu, V., & Meng, T. (1998). Minimum energy mobile wireless networks. In 1998 IEEE International Conference on Communications, ICC’98 (Vol. 3, pp. 1633–1639). Atlanta, June.
Sibson, R. (1977). Locally equiangular triangulations. The Computer Journal, 21(3), 243–245.
Stojmenovic, I., & Lin, X. (2001). Power-aware localized routing in wireless networks. IEEE Transactions on Parallel and Distributed Systems, 12(11), 1122–1133.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Araujo, F., Rodrigues, L. Single-step creation of localized Delaunay triangulations. Wireless Netw 15, 845–858 (2009). https://doi.org/10.1007/s11276-007-0078-x
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11276-007-0078-x