Bichromatic 2-Center of Pairs of Points

  • Esther M. Arkin
  • José Miguel Díaz-Báñez
  • Ferran Hurtado
  • Piyush Kumar
  • Joseph S. B. Mitchell
  • Belén Palop
  • Pablo Pérez-Lantero
  • Maria Saumell
  • Rodrigo I. Silveira
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7256)


We study a class of geometric optimization problems closely related to the 2-center problem: Given a set S of n pairs of points, assign to each point a color (“red” or “blue”) so that each pair’s points are assigned different colors and a function of the radii of the minimum enclosing balls of the red points and the blue points, respectively, is optimized. In particular, we consider the problems of minimizing the maximum and minimizing the sum of the two radii. For each case, minmax and minsum, we consider distances measured in the L 2 and in the L  ∞  metrics. Our problems are motivated by a facility location problem in transportation system design, in which we are given origin/destination pairs of points for desired travel, and our goal is to locate an optimal road/flight segment in order to minimize the travel to/from the endpoints of the segment.


Binary Search Facility Location Problem Blue Point Sweep Event Blue Disk 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Abellanas, M., Hurtado, F., Icking, C., Klein, R., Langetepe, E., Ma, L., Palop, B., Sacristán, V.: Voronoi diagram for services neighboring a highway. Information Processing Letters 86, 283–288 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Agarwal, P.K., Procopiuc, C.M.: Exact and approximation algorithms for clustering. Algorithmica 33(2), 201–226 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Agarwal, P.K., Sharir, M.: Efficient algorithms for geometric optimization. ACM Computing Surveys 30(4), 412–458 (1998)CrossRefGoogle Scholar
  4. 4.
    Ahn, H.-K., Alt, H., Asano, T., Bae, S.W., Brass, P., Cheong, O., Knauer, C., Na, H.-S., Shin, C.-S., Wolff, A.: Constructing optimal highways. International Journal of Foundations of Computer Science 20(1), 3–23 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Alt, H., Scharf, L.: Computing the depth of an arrangement of axis-aligned rectangles in parallel. In: Abstracts 26th European Workshop on Computational Geometry, pp. 33–36 (2010)Google Scholar
  6. 6.
    Bespamyatnikh, S., Segal, M.: Rectilinear Static and Dynamic Discrete 2-Center Problems. In: Dehne, F., Gupta, A., Sack, J.-R., Tamassia, R. (eds.) WADS 1999. LNCS, vol. 1663, pp. 276–287. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  7. 7.
    Cardinal, J., Collette, S., Hurtado, F., Langerman, S., Palop, B.: Optimal location of transportation devices. Computational Geometry: Theory and Applications 41, 219–229 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Cardinal, J., Langerman, S.: Min-max-min geometric facility location problems. In: Abstracts 22nd European Workshop on Computational Geometry, pp. 149–152 (2006)Google Scholar
  9. 9.
    Díaz-Báñez, J.M., Korman, M., Pérez-Lantero, P., Ventura, I.: Locating a service facility and a rapid transit line. In: Proc. 14th Spanish Meeting on Computational Geometry, pp. 189–192 (2011)Google Scholar
  10. 10.
    Drezner, Z.: On the rectangular p-center problem. Naval Research Logistics 34(2), 229–234 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Eppstein, D.: Dynamic three-dimensional linear programming. INFORMS Journal on Computing 4(4), 360–368 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Eppstein, D.: Faster construction of planar two-centers. In: Proc. 8th ACM-SIAM Symposium on Discrete Algorithms, Philadelphia, PA, USA, pp. 131–138 (1997)Google Scholar
  13. 13.
    Korman, M., Tokuyama, T.: Optimal Insertion of a Segment Highway in a City Metric. In: Hu, X., Wang, J. (eds.) COCOON 2008. LNCS, vol. 5092, pp. 611–620. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Sheth, K., Islam, T., Kopardekar, P.: Analysis of airspace tube structures. In: 27th Digital Avionics Systems Conference, IEEE/AIAA (October 2008)Google Scholar
  15. 15.
    Sridhar, B., Grabbe, S., Sheth, K., Bilimoria, K.: Initial study of tube networks for flexible airspace utilization. In: AIAA Guidance, Navigation, and Control Conference, AIAA-2006-6768, Keystone, CO (August 2006)Google Scholar
  16. 16.
    Yousefi, A., Donohue, G., Sherry, L.: High volume tube shaped sectors (HTS): A network of high-capacity ribbons connecting congested city pairs. In: 23rd Digital Avionics Systems Conference, IEEE/AIAA, Salt Lake City, UT (2004)Google Scholar
  17. 17.
    Zadeh, A., Yousefi, A., Tafazzoli, A.A.: Dynamic allocation and benefit assessment of NextGen flow corridors. In: 4th International Conference on Research in Air Transportation, Budapest, Hungary (June 2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Esther M. Arkin
    • 1
  • José Miguel Díaz-Báñez
    • 2
  • Ferran Hurtado
    • 3
  • Piyush Kumar
    • 4
  • Joseph S. B. Mitchell
    • 1
  • Belén Palop
    • 5
  • Pablo Pérez-Lantero
    • 6
  • Maria Saumell
    • 7
  • Rodrigo I. Silveira
    • 3
  1. 1.Dept. of Appl. Math. and StatisticsState Univ. of NYStony BrookUSA
  2. 2.Departamento Matemática Aplicada IIUniversidad de SevillaSpain
  3. 3.Dept. de Matemàtica Aplicada IIUniversitat Politècnica de CatalunyaSpain
  4. 4.Dept. of Computer ScienceFlorida State UniversityTallahasseeUSA
  5. 5.Departamento de InformáticaUniversidad de ValladolidSpain
  6. 6.Esc. de Ingeniería Civil en InformáticaUniversidad de ValparaísoChile
  7. 7.Dept. of Appl. Math.Charles UniversityPragueCzech Republic

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