Advertisement

A faster dual algorithm for the Euclidean minimum covering ball problem

  • Marta Cavaleiro
  • Farid Alizadeh
S.I.: Stochastic Modeling and Optimization, in memory of András Prékopa

Abstract

Dearing and Zeck (Oper Res Lett 37(3):171–175, 2009) presented a dual algorithm for the problem of the minimum covering ball in \({\mathbb {R}}^n\). Each iteration of their algorithm has a computational complexity of at least \({\mathscr {O}}(n^3)\). In this paper we propose a modification to their algorithm that, together with an implementation that uses updates to the QR factorization of a suitable matrix, achieves a \({\mathscr {O}}(n^2)\) iteration.

Keywords

Minimum covering ball Smallest enclosing ball 1-Center Minmax location Computational geometry 

Notes

References

  1. Agarwal, P. K., & Sharathkumar, R. (2015). Streaming algorithms for extent problems in high dimensions. Algorithmica, 72(1), 83–98.CrossRefGoogle Scholar
  2. Bâdoiu, M., & Clarkson, KL. (2003). Smaller core-sets for balls. In Proceedings of the 14th annual ACM-SIAM symposium on discrete algorithms (pp. 801–802). SIAM, Philadelphia, PA, USA, SODA ’03.Google Scholar
  3. Bâdoiu, M., Har-Peled, S., & Indyk, P. (2002). Approximate clustering via core-sets. In Proceedings of the 34th annual ACM symposium on theory of computing (pp. 250–257). ACM, New York, NY, USA, STOC ’02.Google Scholar
  4. Chan, T. M., & Pathak, V. (2011). Streaming and dynamic algorithms for minimum enclosing balls in high dimensions (pp. 195–206). Berlin: Springer.Google Scholar
  5. Dearing, P., & Zeck, C. R. (2009). A dual algorithm for the minimum covering ball problem in \({\mathbb{R}}^n\). Operations Research Letters, 37(3), 171–175.CrossRefGoogle Scholar
  6. Dyer, M., Megiddo, N., & Welzl, E. (2004). Linear programming, Chap. 45 (2nd ed.). Boca Raton, FL: Chapman and Hall/CRC.Google Scholar
  7. Fischer, K., & Gärtner, B. (2004). The smallest enclosing ball of balls: Combinatorial structure and algorithms. International Journal of Computational Geometry and Applications, 14(4–5), 341–387.CrossRefGoogle Scholar
  8. Fischer, K., Gärtner, B., & Kutz, M. (2003). Fast smallest-enclosing-ball computation in high dimensions. In: Proceedings of the Algorithms—ESA 2003: 11th Annual European Symposium (pp. 630–641). Budapest, Hungary, September 16–19, 2003, Springer, Berlin.Google Scholar
  9. Gärtner, B. (1999). Fast and robust smallest enclosing balls (pp. 325–338). Berlin: Springer.Google Scholar
  10. Gärtner, B., & Schönherr, S. (2000). An efficient, exact, and generic quadratic programming solver for geometric optimization. In Proceedings of the 16th annual ACM symposium on computational geometry (SCG) (pp. 110–118).Google Scholar
  11. Golub, G. H., & Van Loan, C. (1996). Matrix computations (3rd ed.). Baltimore, MD: Johns Hopkins University Press.Google Scholar
  12. Hale, T. S., & Moberg, C. R. (2003). Location science research: A review. The Annals of Operations Research, 123(1), 21–35.CrossRefGoogle Scholar
  13. Hopp, T. H., & Reeve, C. P. (1996). An algorithm for computing the minimum covering sphere in any dimension. Technical Report NISTIR 5831, National Institute of Standards and Technology, Gaithersburg, MD, USAGoogle Scholar
  14. Hubbard, P. M. (1996). Approximating polyhedra with spheres for time-critical collision detection. ACM Transactions on Graphics, 15(3), 179–210.CrossRefGoogle Scholar
  15. Kumar, P., Mitchell, J. S. B, & Yildirim, E. A. (2003). Computing core-sets and approximate smallest enclosing hyperspheres in high dimensions. In Proceedings of the 5th workshop on algorithm engineering and experiments (pp. 44–55). Springer, Heidelberg, ALENEX03.Google Scholar
  16. Larsson, T., & Källberg, L. (2013). Fast and robust approximation of smallest enclosing balls in arbitrary dimensions. In: Proceedings of the 11th Eurographics/ACMSIGGRAPH symposium on geometry processing (pp. 93–101). Eurographics Association, Aire-la-Ville, Switzerland, Switzerland, SGP ’13.Google Scholar
  17. Larsson, T., Capannini, G., & Kllberg, L. (2016). Parallel computation of optimal enclosing balls by iterative orthant scan. Computers and Graphics, 56, 1–10.CrossRefGoogle Scholar
  18. Megiddo, N. (1983). Linear-time algorithms for linear programming in \({\mathbb{R}}^3\) and related problems. SIAM Journal on Computing, 12(4), 759–776.CrossRefGoogle Scholar
  19. Megiddo, N. (1984). Linear programming in linear time when the dimension is fixed. Journal of the ACM, 31(1), 114–127.CrossRefGoogle Scholar
  20. Moradi, E., & Bidkhori, M. (2009). Single facility location problem. In R. Zanjirani Farahani & M. Hekmatfar (Eds.), Facility location: Concepts, models, algorithms and case studies (pp. 37–68). Heidelberg: Physica-Verlag HD.CrossRefGoogle Scholar
  21. Nielsen, F., & Nock, R. (2009). Approximating smallest enclosing balls with applications to machine learning. International Journal of Computational Geometry and Applications, 19(05), 389–414.CrossRefGoogle Scholar
  22. Plastria, F. (2002). Continuous covering location problems. In Z. Drezner & H. W. Hamacher (Eds.), Facility location: Applications and theory (pp. 37–79). Berlin: Springer.CrossRefGoogle Scholar
  23. Sylvester, J. J. (1857). A question in the geometry of situation. Quaterly Journal of Pure and Applied Mathematics, 1, 1–79.Google Scholar
  24. Welzl, E. (1991). Smallest enclosing disks (balls and ellipsoids) (pp. 359–370). Berlin: Springer.Google Scholar
  25. Yildirim, E. A. (2008). Two algorithms for the minimum enclosing ball problem. SIAM Journal on Optimization, 19(3), 1368–1391.CrossRefGoogle Scholar
  26. Zarrabi-Zadeh, H., & Chan, T. M. (2006). A simple streaming algorithm for minimum enclosing balls. In Proceedings of the 18th Canadian conference on computational geometry (pp. 139–142).Google Scholar
  27. Zhou, G., Tohemail, K. C., & Sun, J. (2005). Efficient algorithms for the smallest enclosing ball problem. Computational Optimization and Applications, 30(2), 147–160.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.MSIS Department and RUTCORRutgers UniversityPiscatawayUSA

Personalised recommendations