Skip to main content
SpringerLink
Log in

Computing the Center Region and Its Variants

  • Conference paper
  • First Online:
WALCOM: Algorithms and Computation (WALCOM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10167))

Included in the following conference series:

  • 725 Accesses

Abstract

We present an \(O(n^2\log ^4 n)\)-time algorithm for computing the center region of a set of n points in the three-dimensional Euclidean space. This improves the previously best known algorithm by Agarwal, Sharir and Welzl, which takes \(O(n^{2+\epsilon })\) time for any \(\epsilon > 0\). It is known that the complexity of the center region is \(\varOmega (n^2)\), thus our algorithm is almost tight.

The second problem we consider is computing a colored version of the center region in the two-dimensional Euclidean space. We present an \(O(n\log ^4 n)\)-time algorithm for this problem.

This work was supported by the NRF grant 2011-0030044 (SRC-GAIA) funded by the government of Korea.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The authors in [3] roughly analyze this procedure and mention that this procedure takes \(O(m\text { polylog} (m+s))\) time. We analyze the running time of this procedure to give a more tight bound.

References

  1. Abellanas, M., Hurtado, F., Icking, C., Klein, R., Langetepe, E., Ma, L., Palop, B., Sacristán, V.: Smallest color-spanning objects. In: Heide, F.M. (ed.) ESA 2001. LNCS, vol. 2161, pp. 278–289. Springer, Heidelberg (2001). doi:10.1007/3-540-44676-1_23

    Chapter  Google Scholar 

  2. Abellanas, M., Hurtado, F., Icking, C., Klein, R., Langetepe, E., Ma, L., Palop, B., Sacristán, V.: The farthest color Voronoi diagram and related problems. Technical report, University of Bonn (2006)

    Google Scholar 

  3. Agarwal, P.K., Sharir, M., Welzl, E.: Algorithms for center and tverberg points. ACM Trans. Algorithms 5(1), 1–20 (2008)

    MathSciNet  Google Scholar 

  4. Chan, T.M.: An optimal randomized algorithm for maximum Tukey depth. In: Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2004), pp. 430–436 (2004)

    Google Scholar 

  5. Jadhav, S., Mukhopadhyay, A.: Computing a centerpoint of a finite planar set of points in linear time. Discrete Comput. Geom. 12(3), 291–312 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Khanteimouri, P., Mohades, A., Abam, M.A., Kazemi, M.R.: Computing the smallest color-spanning axis-parallel square. In: Cai, L., Cheng, S.-W., Lam, T.-W. (eds.) ISAAC 2013. LNCS, vol. 8283, pp. 634–643. Springer, Heidelberg (2013). doi:10.1007/978-3-642-45030-3_59

    Chapter  Google Scholar 

  7. Langerman, S., Steiger, W.: Optimization in arrangements. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 50–61. Springer, Heidelberg (2003). doi:10.1007/3-540-36494-3_6

    Chapter  Google Scholar 

  8. Matousek, J.: Computing the center of a planar point set. In: Discrete and Computational Geometry: Papers from the DIMACS Special Year. American Mathematical Society (1991)

    Google Scholar 

  9. Megiddo, N.: Applying parallel computation algorithms in the design of serial algorithms. J. ACM 30(4), 852–865 (1983)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, POSTECH, Pohang, South Korea

    Eunjin Oh & Hee-Kap Ahn

Authors
  1. Eunjin Oh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hee-Kap Ahn
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eunjin Oh .

Editor information

Editors and Affiliations

  1. Universiti Teknologi Brunei, Bandar Seri Begawan, Brunei Darussalam

    Sheung-Hung Poon

  2. Bangladesh University of Engineering and Technology, Dhaka, Bangladesh

    Md. Saidur Rahman

  3. National Taiwan University, Taipei, Taiwan

    Hsu-Chun Yen

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Oh, E., Ahn, HK. (2017). Computing the Center Region and Its Variants. In: Poon, SH., Rahman, M., Yen, HC. (eds) WALCOM: Algorithms and Computation. WALCOM 2017. Lecture Notes in Computer Science(), vol 10167. Springer, Cham. https://doi.org/10.1007/978-3-319-53925-6_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-53925-6_20

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-53924-9

  • Online ISBN: 978-3-319-53925-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation