Skip to main content
SpringerLink
Log in

Approximated algorithms for the minimum dilation triangulation problem

  • Published:
Journal of Heuristics Aims and scope Submit manuscript

Abstract

The complexity status of the minimum dilation triangulation (MDT) problem for a general point set is unknown. Therefore, we focus on the development of approximated algorithms to find high quality triangulations of minimum dilation. For an initial approach, we design a greedy strategy able to obtain approximate solutions to the optimal ones in a simple way. We also propose an operator to generate the neighborhood which is used in different algorithms: Local Search, Iterated Local Search, and Simulated Annealing. Besides, we present an algorithm called Random Local Search where good and bad solutions are accepted using the previous mentioned operator. For the experimental study we have created a set of problem instances since no reference to benchmarks for these problems were found in the literature. We use the sequential parameter optimization toolbox for tuning the parameters of the SA algorithm. We compare our results with those obtained by the OV-MDT algorithm that uses the obstacle value to sort the edges in the constructive process. This is the only available algorithm found in the literature. Through the experimental evaluation and statistical analysis, we assess the performance of the proposed algorithms using this operator.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. For a given triangulation, the stabbing number is the maximum number of edges that are encountered (in their interior or at an endpoint) by any infinite line.

  2. http://www.cgal.org.

  3. http://CRAN.R-project.org/package=SPOT.

References

  • Aarts, E., Lenstra, J.: Local Search in Combinatorial Optimization. Wiley, New York (1997)

    MATH  Google Scholar 

  • Bartz-Beielstein, T.: Experimental Research in Evolutionary Computation: The New Experimentalism (Natural Computing Series). Springer-Verlag New York Inc., Secaucus (2006)

    Google Scholar 

  • Bartz-Beielstein, T.: SPOT: an R package for automatic and interactive tuning of optimization algorithms by sequential parameter optimization. CoRR (2010)

  • Beirouti, R., Snoeyink, J.: Implementations of the LMT heuristic for minimum weight triangulation. In: Proceedings of the Fourteenth Annual Symposium on Computational Geometry, SCG ’98, pp. 96–105. ACM, New York (1998)

  • Bokka, V., Gurla, H.: Constant-time algorithms for constrained triangulations on reconfigurable meshes. IEEE Trans. Parallel Distrib. Syst. 9(11), 1057–1072 (1998)

    Article  Google Scholar 

  • Bonichon, N., Gavoille, C., Hanusse, N., Perkovic, L.: The stretch factor of \(l_{1}\)- and \(l_{inf}\)-delaunay triangulations. CoRR abs/1202.5127 (2012)

  • Černý, V.: Thermodynamical approach to the traveling salesman problem: an efficient simulation algorithm. J. Optim. Theory Appl. 45, 41–51 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  • Chen, Q., Medioni, G.: Image synthesis from a sparse set of views. In: Proceedings of the 8th Conference on Visualization ’97, VIS ’97, p. 269-ff. IEEE Computer Society Press (1997)

  • Cheong, O., Haverkort, H., Lee, M.: Computing a minimum-dilation spanning tree is NP-hard. In: Proceedings of the Thirteenth Australasian Symposium on Theory of Computing, CATS ’07, vol. 65, pp. 15–24 (2007)

  • 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 ’86, pp. 169–177. ACM, New York (1986)

  • de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications, 2nd edn. Springer-Verlag, Heidelberg (2000)

    Book  Google Scholar 

  • Demšar, J.: Statistical comparisons of classifiers over multiple data sets. J. Mach. Learn. Res. 7, 1–30 (2006)

    MATH  MathSciNet  Google Scholar 

  • Derrac, J., García, S., Molina, D., Herrera, F.: A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evolut. Comput. 1(1), 3–18 (2011)

    Article  Google Scholar 

  • Dobkin, D., Friedman, S., Supowit, K.: Delaunay graphs are almost as good as complete graphs. Discret. Comput. Geom. 5(4), 399–407 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  • Drysdale, R., Rote, G., Aichholzer, O.: A simple linear time greedy triangulation algorithm for uniformly distributed points. In: Report IIG-408, Institutes for Information Processing, Technische Universit at Graz (1995)

  • Edmonds, J.: Matroids and the greedy algorithm. Math. Progr. 1(1), 127–136 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  • Eppstein, D.: Dilation-free planar graphs. http://www.ics.uci.edu/~eppstein/junkyard/dilation-free/ (1997). Accessed 4 Feb 2014

  • Eppstein, D.: Spanning trees and spanners. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 425–461. Elsevier, Amsterdam (2000)

    Chapter  Google Scholar 

  • Eppstein, D., Wortman, K.: Minimum dilation stars. In: Proceedings of the 21st ACM Symposium on Computational Geometry, pp. 321–326, cs.CG/0412025. ACM (2005)

  • Giannopoulos, P., Knauer, C., Marx, D.: Minimum-dilation tour (and path) is NP-hard. In: EWCG 07: Proceedings of the 23rd European Workshop on Computational Geometry, pp. 18–21 (2007)

  • Keil, J., Gutwin, C.: The delaunay triangulation closely approximates the complete euclidean graph. In: Dehne, F., Sack, J.R., Santoro, N. (eds.) LNCS, vol. 382, pp. 47–56. Springer, Berlin (1989)

    Google Scholar 

  • Keil, J., Gutwin, C.: Classes of graphs which approximate the complete euclidean graph. Discret. Comput. Geom. 7(1), 13–28 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  • Kirkpatrick, S., Gelatt, C., Vecchi, M.: Optimization by simulated annealing. Science 220, 671–680 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  • Klein, A.: Effiziente berechnung einer dilationsminimalen triangulierung. PhD thesis (2006a)

  • Klein, A.: Minimum dilation triangulation. Geometry Lab. http://www.geometrylab.de/ (2006b). Accessed 4 Feb 2014

  • Klein, R., Kutz, M.: Computing geometric minimum-dilation graphs is NP-hard. In: Kaufmann, M., Wagner, D. (eds.) Graph Drawing, pp. 556–567. Springer, Berlin (2006)

    Google Scholar 

  • Knauer, C., Mulzer, W.: An exclusion region for minimum dilation triangulations. In: EuroCG, Technische Universiteit Eindhoven, pp. 33–36 (2005a)

  • Knauer, C., Mulzer, W.: Minimum dilation triangulations. In: Freie Universitt Berlin, Fachbereich Mathematik und Informatik, Technical Report (2005b)

  • Kolingerová, I., Ferko, A.: Multicriteria-optimized triangulations. Vis. Comput. 17(6), 380–395 (2001)

    Article  MATH  Google Scholar 

  • Lourenco, H., Martin, O., Stützle, T.: Iterated Local Search, pp. 321–353. Kluwer Academic Publishers, Norwell (2002)

    Google Scholar 

  • Markatou, M., Tian, H., Biswas, S., Hripcsak, G.: Analysis of variance of cross-validation estimators of the generalization error. J. Mach. Learn. Res. 6, 1127–1168 (2005)

    MATH  MathSciNet  Google Scholar 

  • Martin, O., Otto, S., Felten, E.: Large-step Markov chains for the traveling salesman problem. Complex Syst. 5, 299–326 (1991)

    MATH  MathSciNet  Google Scholar 

  • Michalewicz, Z., Fogel, D.: How to Solve It: Modern Heuristics. Springer, Berlin (2004)

    Book  Google Scholar 

  • Narasimhan, G., Smid, M.: Approximating the stretch factor of euclidean graphs. SIAM J. Comput. 30, 2000 (1999)

    MathSciNet  Google Scholar 

  • Narasimhan, G., Smid, M.: Geometric Spanner Networks. Cambridge University Press, New York (2007)

    Book  MATH  Google Scholar 

  • Nielson, G.: Tools for triangulations and tetrahedrizations. In: Scientific Visualization, Overviews, Methodologies, and Techniques, IEEE Computer Society, pp. 429–525 (1997)

  • Papadimitriou, C.: The complexity of combinatorial optimization problems. PhD thesis, Princeton, NJ, aAI7704795 (1976)

  • Vite-Silva, I., Cruz-Cortés, N., Toscano-Pulido, G., Fraga, L.: Optimal triangulation in 3d computer vision using a multi-objective evolutionary algorithm. In: Giacobini, M. (ed.) Applications of Evolutionary Computing. Lecture Notes in Computer Science, vol. 4448, pp. 330–339. Springer, Berlin (2007)

    Google Scholar 

  • Xia, G.: Improved upper bound on the stretch factor of delaunay triangulations. In: Proceedings of the 27th Annual ACM Symposium on Computational Geometry, SoCG ’11, pp. 264–273. ACM, New York (2011)

  • Xia, G., Zhang, L.: Improved lower bound for the stretch factor of delaunay triangulations manuscript. Poster in fwcg10: 20th Annual Fall Workshop on Computational Geometry (2010)

  • Zhou, H., Wu, H., Xia, S., Jin, M., Ding, N.: A distributed triangulation algorithm for wireless sensor networks on 2d and 3d surface. In: INFOCOM, 2011 Proceedings IEEE, pp. 1053–1061 (2011)

Download references

Acknowledgments

The authors would like to thank to Research Project Tecnologías Avanzadas de Bases de Datos 22/F014 financed by Universidad Nacional de San Luis, San Luis, Argentina; CONICET (Consejo Nacional de Investigaciones Científicas y Técnicas), Argentina; and ESF EUROCORES programme EuroGIGA, CRP ComPoSe: grant EUI-EURC-2011-4306, Spain. The authors acknowledge support from MINCyT-CONACyT through bilateral project No. MX/11/03 -164626.

Author information

Authors and Affiliations

  1. Universidad Nacional de San Luis, Ejército de Los Andes 950, D5700HHW , San Luis, Argentina

    Maria Gisela Dorzán & Mario Guillermo Leguizamón

  2. Departamento de Inteligencia Artificial, Universidad Veracruzana, Sebastián Camacho 5, Centro, 91000 , Xalapa, Veracruz, Mexico

    Efrén Mezura-Montes

  3. Facultad de Informática, Universidad Politécnica de Madrid, Campus de Montegancedo Boadilla del Monte, 28660 , Madrid, Spain

    Gregorio Hernández-Peñalver

Authors
  1. Maria Gisela Dorzán
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mario Guillermo Leguizamón
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Efrén Mezura-Montes
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Gregorio Hernández-Peñalver
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maria Gisela Dorzán.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dorzán, M.G., Leguizamón, M.G., Mezura-Montes, E. et al. Approximated algorithms for the minimum dilation triangulation problem. J Heuristics 20, 189–209 (2014). https://doi.org/10.1007/s10732-014-9237-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10732-014-9237-2

Keywords

Search

Navigation