Connected Rectilinear Graphs on Point Sets

  • Maarten Löffler
  • Elena Mumford
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5417)

Abstract

Given n points in d-dimensional space, we would like to connect the points with straight line segments to form a connected graph whose edges use d pairwise perpendicular directions. We prove that there exists at most one such set of directions. For d = 2 we present an algorithm for computing these directions (if they exist) in O (n 2) time.

References

  1. 1.
    Durocher, S., Kirkpatrick, D.: On the hardness of turn-angle-restricted rectilinear cycle cover problems. In: CCCG 2002, pp. 13–16 (2002)Google Scholar
  2. 2.
    Edelsbrunner, H., Guibas, L.: Topologically sweeping an arrangement. J. Comput. Syst. Sci. 38, 165–194 (1989)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Efrat, A., Erten, C., Kobourov, S.: Fixed-location circular-arc drawing of planar graphs. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 147–158. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  4. 4.
    Fekete, S., Woeginger, G.: Angle-restricted tours in the plane. Comput. Geom. Theory Appl. 8(4), 195–218 (1997)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Fredman, M.: How good is the information theory bound in sorting? Theoret. Comput. Sci. 1, 355–361 (1976)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Garg, A., Tamassia, R.: On the computational complexity of upward and rectilinear planarity testing. SIAM J. on Computing 31(2), 601–625 (2002)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Hoffman, F., Kriegel, K.: Embedding rectilinear graphs in linear time. Inf. Process. Lett. 29(2), 75–79 (1988)CrossRefGoogle Scholar
  8. 8.
    Jansen, K., Woeginger, G.: The complexity of detecting crossingfree configurations in the plane. BIT 33(4), 580–595 (1993)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Löffler, M., Mumford, E.: Connected rectilinear polygons on point sets (2008), http://www.cs.uu.nl/research/techreps/UU-CS-2008-028.html
  10. 10.
    O’Rourke, J.: Uniqueness of orthogonal connect-the-dots. In: Toussaint, G. (ed.) Computational Morphology, pp. 97–104 (1988)Google Scholar
  11. 11.
    Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. In: Whitesides, S.H. (ed.) GD 1998. LNCS, vol. 1547, pp. 263–274. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  12. 12.
    Rappaport, D.: On the complexity of computing orthogonal polygons from a set of points. Technical Report TR-SOCS-86.9, McGill Univ., Montreal, PQ (1986)Google Scholar
  13. 13.
    Vijayan, G., Wigderson, A.: Rectilinear graphs and their embeddings. SIAM J. on Computing 14(2), 355–372 (1985)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Maarten Löffler
    • 1
  • Elena Mumford
    • 2
  1. 1.Dept. Information and Computing SciencesUtrecht UniversityThe Netherlands
  2. 2.Dept. of Mathematics and Computer ScienceTU EindhovenThe Netherlands

Personalised recommendations