Abstract
Wireless ad hoc network can be modeled as a unit disk graph (UDG) in which there is an edge between two nodes if and only if their Euclidean distance is at most one unit. The size of UDG is in O(n 2), where n is the number of network nodes. In the literature, the geometric spanners like Relative Neighborhood Graph (RNG), Gabriel Graph (GG), Delaunay Triangulation (Del), Planarized Localized Delaunay Triangulation (PLDel) and Yao Graph are proposed, which are sparse subgraphs of UDG. In this paper, we propose a hierarchy of geometric spanners called the k th order RNG (k-RNG), k th order GG (k-GG), k th order Del (k-Del), and k th order Yao (k-Yao) to reduce the spanning ratio and control topology, sparseness and connectivity. We have simulated these spanners and compared with the existing spanners. The simulation results show that the proposed spanners have better properties in terms of spanning ratio and connectivity by controlling topology and sparseness.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press, New York (2007)
Cheng, X., Huang, X., Li, X.Y.: Applications of computational geometry in wireless networks (2003)
Toussaint, G.T.: The relative neighbourhood graph of a finite planar set. Pattern Recognition 12, 261–268 (1980)
Bose, P., Devroye, L., Evans, W., Kirkpatrick, D.: On the spanning ratio of gabriel graphs and beta-skeletons. SIAM J. Discret. Math. 20(2), 412–427 (2006)
Keil, J.M., Gutwin, C.A.: The Delaunay triangulation closely approximates the complete Euclidean graph. In: Dehne, F., Santoro, N., Sack, J.-R. (eds.) WADS 1989. LNCS, vol. 382, pp. 47–56. Springer, Heidelberg (1989)
Dobkin, D.P., Friedman, S.J., Supowit, K.J.: Delaunay graphs are almost as good as complete graphs. Discrete & Computational Geometry 5, 399–407 (1990)
Chew, P.: There is a planar graph almost as good as the complete graph. In: Proceedings of the Second Annual Symposium on Computational Geometry, SCG 1986, pp. 169–177. ACM, New York (1986)
Li, X.Y., Calinescu, G., Wan, P.J.: Distributed construction of planar spanner and routing for ad hoc wireless networks. In: INFOCOM (2002)
Chang, M.S., Tang, C.Y., Lee, R.C.T.: 20-relative neighborhood graphs are hamiltonian. In: Asano, T., Imai, H., Ibaraki, T., Nishizeki, T. (eds.) SIGAL 1990. LNCS, vol. 450, pp. 53–65. Springer, Heidelberg (1990)
Chang, M.S., Tang, C.Y., Lee, R.C.T.: Solving the euclidean bottleneck biconnected edge subgraph problem by 2-relative neighborhood graphs. Discrete Applied Mathematics 39, 1–12 (1992)
Chang, M., Tang, C., Lee, R.: Solving the euclidean bottleneck matching problem by k-relative neighborhood graphs. Algorithmica 8, 177–194 (1992), doi:10.1007/BF01758842
Rao, S.V., Mukhopadhyay, A.: Fast algorithms for computing beta-skeletons and their relatives. Pattern Recognition 34, 2163–2172 (2001)
Kiran, P.: kth order geometric spanners for wireless ad hoc networks. Master’s thesis, Indian Institute of Technology Guwahati (2010)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kiran, P., Rao, S.V. (2011). k th Order Geometric Spanners for Wireless Ad Hoc Networks. In: Natarajan, R., Ojo, A. (eds) Distributed Computing and Internet Technology. ICDCIT 2011. Lecture Notes in Computer Science, vol 6536. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19056-8_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-19056-8_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-19055-1
Online ISBN: 978-3-642-19056-8
eBook Packages: Computer ScienceComputer Science (R0)