Rectilinear Convex Hull with Minimum Area

  • Carlos Alegría-Galicia
  • Tzolkin Garduño
  • Areli Rosas-Navarrete
  • Carlos Seara
  • Jorge Urrutia
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7579)


Let P be a set of n points in the plane. We solve the problem of computing an orientation of the plane for which the rectilinear convex hull of P has minimum area in optimal Θ(nlogn) time and O(n) space.


rectilinear convex hull ortho-convexity optimization 


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  1. 1.
    Avis, D., Beresford-Smith, B., Devroye, L., Elgindy, H., Guévremont, E., Hurtado, F., Zhu, B.: Unoriented Θ-maxima in the plane: complexity and algorithms. SIAM J. Comput. 28(1), 278–296 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bae, S.W., Lee, C., Ahn, H.-K., Choi, S., Chwa, K.-Y.: Computing minimum-area rectilinear convex hull and L-shape. Computational Geometry: Theory and Applications 42(3), 903–912 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bentley, J., Ottmann, T.: Algorithms for reporting and counting geometric intersections. IEEE Trans. Computers C-28, 643–647 (1979)CrossRefzbMATHGoogle Scholar
  4. 4.
    Biedl, T., Genç, B.: Reconstructing orthogonal polyhedra from putative vertex sets. Computational Geometry: Theory and Applications 44(8), 409–417 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Biswas, A., Bhowmick, P., Sarkar, M., Bhattacharya, B.B.: Finding the Orthogonal Hull of a Digital Object: A Combinatorial Approach. In: Brimkov, V.E., Barneva, R.P., Hauptman, H.A. (eds.) IWCIA 2008. LNCS, vol. 4958, pp. 124–135. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Díaz-Báñez, J.M., López, M.A., Mora, M., Seara, C., Ventura, I.: Fitting a two-joint orthogonal chain to a point set. Computational Geometry: Theory and Applications 44(3), 135–147 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Fink, E., Wood, D.: Strong restricted-orientation convexity. Geometriae Dedicata 69, 35–51 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Franěk, V., Matousěk, J.: Computing D-convex hulls in the plane. Computational Geometry: Theory and Applications 42, 81–89 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Khamsi, M.A., Kirk, W.A.: An introduction to metric spaces and fixed point theory. Wiley-Interscience (2001)Google Scholar
  10. 10.
    Kung, H.T., Luccio, F., Preparata, F.P.: On finding the maxima of a set of vectors. Journal of the ACM 22, 469–476 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Martynchik, V., Metelski, N., Wood, D.: O-convexity: computing hulls, approximations, and orientation sets. In: Canadian Conference on Computational Geometry, pp. 2–7 (1996)Google Scholar
  12. 12.
    Matoušek, J., Plecháč, P.: On functional separately convex hulls. Discrete and Computational Geometry 19, 105–130 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ottman, T., Soisalon-Soisinen, E., Wood, D.: On the definition and computation of rectilinear convex hulls. Information Sciences 33, 157–171 (1984)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer (1985)Google Scholar
  15. 15.
    Rawlins, G.J.E., Wood, D.: Ortho-convexity and its generalizations. In: Computational Morphology: A Computational Geometric Approach to the Analysis of Form, pp. 137–152. Elseiver Science Publishers B.V., North-Holland (1988)CrossRefGoogle Scholar
  16. 16.
    Uchoa, E., De Aragão, M.P., Ribeiro, C.C.: Preprocessing Steiner problems from VLSI layout. Networks 40(1), 38–50 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Carlos Alegría-Galicia
    • 3
  • Tzolkin Garduño
    • 3
  • Areli Rosas-Navarrete
    • 3
  • Carlos Seara
    • 2
  • Jorge Urrutia
    • 1
  1. 1.Instituto de MatemáticasUniversidad Nacional Autónoma de MéxicoMéxico
  2. 2.Universitat Politècnica de CatalunyaSpain
  3. 3.Posgrado en Ciencia e, Ingeniería de la ComputaciónUniversidad Nacional Autónoma de MéxicoMéxico

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