Skip to main content
SpringerLink
Log in

Computing Colourful Simplicial Depth and Median in ℝ2

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

The colourful simplicial depth (CSD) of a point \(x \in \mathbb {R}^{2}\) relative to a configuration P = (P1, P2, … , Pk) of n points in k colour classes is the number of closed simplices (triangles) with vertices from three different colour classes that contain x in their convex hull. We consider the problems of efficiently computing the colourful simplicial depth of a given point x, and of finding a point in \(\mathbb {R}^{2}\), called a median, that maximizes colourful simplicial depth. Our algorithm for colourful simplicial depth runs in \(O(n \log {n})\) time, and in O(n) time if the points are already sorted around x. This is optimal for sorted inputs. Our algorithm for computing the colourful median runs in O(n4) time. Both results extend known algorithms for the monochromatic versions of these problems, and match the corresponding time complexities.

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

Similar content being viewed by others

References

  1. Adiprasito, K.A., Brinkmann, P., Padrol, A., Paták, P., Patáková, Z., Sanyal, R.: Colorful Simplicial Depth, Minkowski Sums, and Generalized Gale Transforms. Int. Math. Res. Not. 2019(6), 1894–1919 (2017)

    Article  MathSciNet  Google Scholar 

  2. Afshani, P., Sheehy, D.R., Stein, Y.: Approximating the simplicial depth in high dimensions. The European Workshop on Computational Geometry (2016)

  3. Ajtai, M., Chvátal, V., Newborn, M., Szemerédi, E.: Crossing-free subgraphs. North-Holland Math. Stud. 60, 9–12 (1982)

    Article  MathSciNet  Google Scholar 

  4. Aloupis, G.: Geometric measures of data depth, Data depth: robust multivariate analysis, computational geometry and applications. DIMACS Ser. Discrete Math. Theoret. Comput. Sci., Amer. Math. Soc., Providence, RI 72, 147–158 (2006)

    Article  Google Scholar 

  5. Aloupis, G., Cortés, C., Gómez, F., Soss, M., Toussaint, G.: Lower bounds for computing statistical depth. Comput. Stat. Data Anal. 40 (2), 223–229 (2002)

    Article  MathSciNet  Google Scholar 

  6. Aloupis, G., Langerman, S., Soss, M., Toussaint, G.: Algorithms for bivariate medians and a Fermat-Torricelli problem for lines. Comput. Geom. 26(1), 69–79 (2003)

    Article  MathSciNet  Google Scholar 

  7. Bárány, I.: A generalization of Carathéodory’s theorem. Discrete Math. 40(2-3), 141–152 (1982)

    Article  MathSciNet  Google Scholar 

  8. Bárány, I., Onn, S.: Colourful linear programming and its relatives. Math. Oper. Res. 22(3), 550–567 (1997)

    Article  MathSciNet  Google Scholar 

  9. Cheng, A.Y., Ouyang, M.: On algorithms for simplicial depth. In: Proceedings of the 13th Canadian Conference on Computational Geometry, University of Waterloo, Ontario, Canada, August 13-15, 2001, pp 53–56 (2001)

  10. Deza, A., Huang, S., Stephen, T., Terlaky, T.: Colourful simplicial depth. Discrete Comput. Geom. 35(4), 597–615 (2006)

    Article  MathSciNet  Google Scholar 

  11. Deza, A., Huang, S., Stephen, T., Terlaky, T.: The colourful feasibility problem. Discrete Appl. Math. 156(11), 2166–2177 (2008)

    Article  MathSciNet  Google Scholar 

  12. Edelsbrunner, H., Guibas, L.J.: Topologically sweeping an arrangement. J. Comput. Syst. Sci. 38(1), 165–194 (1989). 18Th Annual ACM Symposium on Theory of Computing (Berkeley, CA, 1986)

    Article  MathSciNet  Google Scholar 

  13. Edelsbrunner, H., Mücke, E.P.: Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. In: Proceedings of the Fourth Annual Symposium on Computational Geometry (Urbana, IL, 1988), pp 118–133. ACM, New York (1988)

  14. Elbassioni, K., Elmasry, A., Makino, K.: Finding simplices containing the origin in two and three dimensions. Int. J. Comput. Geom. Appl. 21(5), 495–506 (2011)

    Article  MathSciNet  Google Scholar 

  15. Elmasry, A., Elbassioni, K.M.: Output-sensitive algorithms for enumerating and counting simplices containing a given point in th plane. In: 17th Canadian Conference on Computational Geometry (CCCG’05) (Windsor, Canada), no. 17, University of Windsor, pp 248–251 (2005)

  16. Fukuda, K., Rosta, V.: Data depth and maximum feasible subsystems, Graph theory and combinatorial optimization, GERAD 25th Anniv. Ser., vol. 8, pp 37–67. Springer, New York (2005)

    MATH  Google Scholar 

  17. Gethner, E., Hogben, L., Lidický, B., Pfender, F., Ruiz, A., Young, M.: On crossing numbers of complete tripartite and balanced complete multipartite graphs. J. Graph Theory 84(4), 552–565 (2016)

    Article  MathSciNet  Google Scholar 

  18. Gil, J., Steiger, W., Wigderson, A.: Geometric medians. Discrete Math. 108(1-3), 37–51 (1992)

    Article  MathSciNet  Google Scholar 

  19. Khuller, S., Mitchell, J.S.B.: On a triangle counting problem. Inform. Process. Lett. 33(6), 319–321 (1990)

    Article  MathSciNet  Google Scholar 

  20. Lee, D.T., Ching, Y.T.: The power of geometric duality revisited. Inform. Process. Lett. 21(3), 117–122 (1985)

    Article  MathSciNet  Google Scholar 

  21. Leighton, F.T.: Complexity issues in VLSI, Foundations of Computing Series, MIT Press (1983)

  22. Liu, R.Y.: On a notion of data depth based on random simplices. Ann. Statist. 18(1), 405–414 (1990)

    Article  MathSciNet  Google Scholar 

  23. Matoušek, J., Wagner, U.: On Gromov’s method of selecting heavily covered points. Discrete Comput. Geom. 52(1), 1–33 (2014)

    Article  MathSciNet  Google Scholar 

  24. Meunier, F., Sarrabezolles, P.: Colorful linear programming, Nash equilibrium, and pivots. Discret. Appl. Math. 240, 78–91 (2018)

    Article  MathSciNet  Google Scholar 

  25. Mulzer, W., Stein, Y.: Computational aspects of the colorful Carathéodory theorem. Discret. Comput. Geom. 60(3), 720–755 (2018)

    Article  Google Scholar 

  26. Pilz, A., Welzl, E., Wettstein, M.: From crossing-free graphs on wheel sets to embracing simplices and polytopes with few vertices. In: 33rd International Symposium on Computational Geometry (SoCG 2017), vol. 77, pp 54:1–54:16 (2017)

  27. Rafalin, E., Souvaine, D.L.: Topological sweep of the complete graph. Discrete Appl. Math. 156(17), 3276–3290 (2008)

    Article  MathSciNet  Google Scholar 

  28. Rousseeuw, P.J., Ruts, I.: Bivariate location depth. Appl. Stat. J. Royal Stat. Soc. Series C 45(4), 516–526 (1996)

    MATH  Google Scholar 

  29. Sarrabezolles, P.: The colourful simplicial depth conjecture. J. Combin. Theory Ser. A 130, 119–128 (2015)

    Article  MathSciNet  Google Scholar 

  30. Zasenko, O.: Colourful simplicial depth in the plane, 2016, Java code, available at http://github.com/olgazasenko/ColourfulSimplicialDepthInThePlane Accessed February 2nd (2017)

  31. Zasenko, O., Stephen, T.: Algorithms for colourful simplicial depth and medians in the plane. In: Combinatorial Optimization and Applications - 10th International Conference, COCOA 2016, Hong Kong, China, December 16-18, 2016, Proceedings, Lecture Notes in Comput. Sci, vol. 10043, pp 378–392. Springer (2016)

Download references

Acknowledgments

This research was partially supported by an NSERC Discovery Grant to T. Stephen and by an SFU Graduate Fellowship to O. Zasenko. We thank A. Deza for comments on the presentation, and the anonymous referees for helpful comments.

A preliminary version of this work can be found in the proceedings of COCOA 2016 [31].

Author information

Authors and Affiliations

  1. New York University, Tandon School of Engineering, Brooklyn, NY, 11201, USA

    Greg Aloupis

  2. Department of Mathematics, Simon Fraser University, British Columbia, Canada

    Tamon Stephen & Olga Zasenko

Authors
  1. Greg Aloupis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tamon Stephen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Olga Zasenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tamon Stephen.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aloupis, G., Stephen, T. & Zasenko, O. Computing Colourful Simplicial Depth and Median in ℝ2. Theory Comput Syst 66, 417–431 (2022). https://doi.org/10.1007/s00224-021-10067-4

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-021-10067-4

Keywords

Search

Navigation