Advertisement

Neighbourhood Graphs and Locally Minimal Triangulations

  • Ivana Kolingerová
  • Tomáš Vomáčka
  • Martin Maňák
  • Andrej Ferko
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10990)

Abstract

Neighbourhood (or proximity) graphs, such as nearest neighbour graph, closest pairs, relative neighbourhood graph and k-nearest neighbour graph are useful tools in many tasks inspecting mutual relations, similarity and closeness of objects. Some of neighbourhood graphs are subsets of Delaunay triangulation (DT) and this relation can be used for efficient computation of these graphs. This paper concentrates on relation of neighbourhood graphs to the locally minimal triangulation (LMT) and shows that, although generally these graphs are not LMT subgraphs, in most cases LMT contains all or many edges of these graphs. This fact can also be used for the neighbourhood graphs computation, namely in kinetic problems, because LMT computation is easier.

Keywords

Nearest neighbour graph K-nearest neighbour graph Locally minimal triangulation Delaunay triangulation Kinetic problem 

Notes

Acknowledgements

This work was supported by the Czech Science Foundation, the project number 17-07690S, and by the Ministry of Education, Youth and Sports of the Czech Republic, project number LO1506 (PUNTIS). We would like to thank to T. Bayer from the Charles University in Prague, Czech Republic for supplying us the real terrain data for the experiments.

References

  1. 1.
    Aichholzer, O., et al.: Triangulations intersect nicely. Discrete Comput. Geom. 16(4), 339–359 (1996)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Basch, J., Guibas, L.J., Hershberger, J.: Data structures for mobile data. J. Algorithms 31(1), 1–28 (1999)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bayer, T.: Department of Applied Geoinformatics and Cartography, Faculty of Science, Charles University, Prague, Czech Republic. https://web.natur.cuni.cz/~bayertom. Accessed 15 May 2017
  4. 4.
    Beirouti, R., Snoeyink, J.: Implementations of the LMT heuristic for minimum weight triangulation. In: Proceedings of the Fourteenth Annual Symposium on Computational Geometry, SCG 1998, pp. 96–105. ACM, New York (1998)Google Scholar
  5. 5.
    Bose, P., Devroye, L., Evans, W.: Diamonds are not a minimum weight triangulation’s best friend. Int. J. Comput. Geom. Appl. 12(06), 445–453 (2002)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cho, H.G.: On the expected number of common edges in Delaunay and greedy triangulation. J. WSCG 5(1–3), 50–59 (1997)Google Scholar
  7. 7.
    Dey, T.K.: Curve and Surface Reconstruction: Algorithms with Mathematical Analysis (Cambridge Monographs on Applied and Computational Mathematics). Cambridge University Press, New York (2006)Google Scholar
  8. 8.
    Dickerson, M.T., Keil, J.M., Montague, M.H.: A large subgraph of the minimum weight triangulation. Discrete Comput. Geom. 18(3), 289–304 (1997)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Dickerson, M.T., Montague, M.H.: A (usually?) connected subgraph of the minimum weight triangulation. In: Proceedings of the Twelfth Annual Symposium on Computational Geometry, SCG 1996, pp. 204–213. ACM, New York (1996)Google Scholar
  10. 10.
    Gavrilova, M., Rokne, J.: Swap conditions for dynamic Voronoi diagrams for circles and line segments. Comput. Aided Geom. Des. 16(2), 89–106 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gavrilova, M., Rokne, J.: Updating the topology of the dynamic Voronoi diagram for spheres in Euclidean d-dimensional space. Comput. Aided Geom. Des. 20(4), 231–242 (2003)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Guibas, L., Russel, D.: An empirical comparison of techniques for updating Delaunay triangulations. In: Proceedings of the Twentieth Annual Symposium on Computational Geometry, SCG 2004, pp. 170–179. ACM, New York (2004)Google Scholar
  13. 13.
    Kim, Y.S., Park, D.G., Jung, H.Y., Cho, H.G., Dong, J.J., Ku, K.J.: An improved TIN compression using Delaunay triangulation. In: Proceedings of Seventh Pacific Conference on Computer Graphics and Applications (Cat. No.PR00293), pp. 118–125 (1999)Google Scholar
  14. 14.
    Maus, A., Drange, J.M.: All closest neighbors are proper delaunay edges generalized, and its application to parallel algorithms. In: Proceedings of Norwegian informatikkonferanse, pp. 1–12 (2010)Google Scholar
  15. 15.
    Okabe, A., Boots, B., Sugihara, K., Chiu, S.N.: Spatial tessellations: concepts and applications of Voronoi diagrams. Probability and Statistics, 2nd edn. Wiley, NYC (2000)Google Scholar
  16. 16.
    Preparata, F.P., Shamos, M.: Computational Geometry: An Introduction. Springer, New York (1985).  https://doi.org/10.1007/978-1-4612-1098-6CrossRefzbMATHGoogle Scholar
  17. 17.
    Spelič, D., Novak, F., Žalik, B.: Delaunay triangulation benchmarks. J. Electr. Eng. 59(1), 49–52 (2008)Google Scholar
  18. 18.
    Su, P., Drysdale, R.L.S.: A comparison of sequential Delaunay triangulation algorithms. Comput. Geom. 7(5), 361–385 (1997)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Toussaint, G.T.: The relative neighbourhood graph of a finite planar set. Pattern Recogn. 12(4), 261–268 (1980)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Veltkamp, R.C.: The \(\gamma \)-neighborhood graph. Comput. Geom. 1(4), 227–246 (1992)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Ivana Kolingerová
    • 1
    • 2
  • Tomáš Vomáčka
    • 1
  • Martin Maňák
    • 1
    • 2
  • Andrej Ferko
    • 3
  1. 1.Department of Computer Science, Faculty of Applied SciencesUniversity of West BohemiaPilsenCzech Republic
  2. 2.New Technologies for the Information Society, Faculty of Applied SciencesUniversity of West BohemiaPilsenCzech Republic
  3. 3.Department of Algebra, Geometry and Didactics of Mathematics, Faculty of Mathematics, Physics and InformaticsComenius UniversityBratislavaSlovakia

Personalised recommendations