Abstract
Gabriel graph is one of the well-studied proximity graphs which has a wide range of applications in various research areas such as wireless sensor network, gene flow analysis, geographical variation analysis, facility location, cluster analysis, and so on. In numerous applications, an important query is to find a specific location in a Gabriel graph at where a given set of adjacent vertices can be obtained if a new point is inserted. To efficiently compute the answer of this query, our proposed solution is to subdivide the plane associated with the Gabriel graph into smaller subregions with the property that if a new point is inserted anywhere into a specific subregion then the set of adjacent vertices in the Gabriel graph remains constant for that point, regardless of the exact location inside the subregion. In this paper, we examine these planar subregions, named redundant cells, including some essential properties and sketch an algorithm of running time \({\mathcal {O}}(n^2)\) to construct the arrangement that yields these redundant cells.
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
Baumgart, M.: Partitioning bispanning graphs into spanning trees. Mathematics in Computer Science 3(1), 3–15 (2010)
Berg, M.D., Cheong, O., Kreveld, M.V., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer-Verlag TELOS, Santa (2008)
Bose, P., Cardinal, J., Collette, S., Demaine, E.D., Palop, B., Taslakian, P., Zeh, N.: Relaxed gabriel graphs. In: Proceedings of the 21st Annual Canadian Conference on Computational Geometry, CCCG, pp. 169–172, British Columbia, Canada (2009)
Bose, P., Collette, S., Hurtado, F., Korman, M., Langerman, S., Sacristan, V., Saumell, M.: Some properties of higher order delaunay and gabriel graphs. In: Proceedings of the 22nd Annual Canadian Conference on Computational Geometry, CCCG, pp. 13–16, Manitoba, Canada (2010)
Devillers, O., Teillaud, M., Yvinec, M.: Dynamic location in an arrangement of line segments in the plane. Algorithms Review 2(3), 89–103 (1992)
Di Concilio, A.: Point-free geometries: Proximities and quasi-metrics. Mathematics in Computer Science 7(1), 31–42 (2013)
Gabriel, K.R., Sokal, R.R.: A new statistical approach to geographic variation analysis. Systematic Zoology (Society of Systematic Biologists) 18(3), 259–278 (1969)
Guibas, L., Stolfi, J.: Primitives for the manipulation of general subdivisions and the computation of voronoi. ACM Transactions on Graphics 4(2), 74–123 (1985)
Hossain, M.Z., Wahid, M.A., Hasan, M., Amin, M.A.: Analysis of spatial subdivision on gabriel graph. http://www.cvcrbd.org/publications
Howe, S.E.: Estimating Regions and Clustering Spatial Data: Analysis and Implementation of Methods Using the Voronoi Diagram. Ph.D. thesis, Brown University, Providence, R.I. (1978)
Kao, T., Mount, D.M., Saalfeld, A.: Dynamic maintenance of delaunay triangulations. Tech. rep., College Park, MD, USA (1991)
Laloë, D., Moazami-Goudarzi, K., Lenstra, J.A., Marsan, P.A., Azor, P., Baumung, R., Bradley, D.G., Bruford, M.W., Cañón, J., Dolf, G., Dunner, S., Erhardt, G., Hewitt, G., Kantanen, J., Obexer-Ruff, G., Olsaker, I., Rodellar, C., Valentini, A., Wiener, P.: Spatial trends of genetic variation of domestic ruminants in europe. Diversity 2, 932–945 (2010)
Matula, D.W., Sokal, R.R.: Properties of gabriel graphs relevant to geographic variation research and the clustering of points in the plane. Geographical Analysis 12(3), 205–222 (1980)
O’Rourke, J.: Computational geometry in C, 2nd edn. Cambridge University Press, New York (1998)
Seidel, R.: A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons. Computational Geometry: Theory and Applications 1(1), 51–64 (1991)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Hossain, M.Z., Wahid, M.A., Hasan, M., Amin, M.A. (2015). Spatial Subdivision of Gabriel Graph. In: Tan, Y., Shi, Y., Buarque, F., Gelbukh, A., Das, S., Engelbrecht, A. (eds) Advances in Swarm and Computational Intelligence. ICSI 2015. Lecture Notes in Computer Science(), vol 9142. Springer, Cham. https://doi.org/10.1007/978-3-319-20469-7_34
Download citation
DOI: https://doi.org/10.1007/978-3-319-20469-7_34
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-20468-0
Online ISBN: 978-3-319-20469-7
eBook Packages: Computer ScienceComputer Science (R0)