Locality and Bounding-Box Quality of Two-Dimensional Space-Filling Curves

  • Herman Haverkort
  • Freek van Walderveen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5193)

Abstract

Space-filling curves can be used to organise points in the plane into bounding-box hierarchies (such as R-trees). We develop measures of the bounding-box quality of space-filling curves that express how effective different curves are for this purpose. We give general lower bounds on the bounding-box quality and on locality according to Gotsman and Lindenbaum for a large class of curves. We describe a generic algorithm to approximate these and similar quality measures for any given curve. Using our algorithm we find good approximations of the locality and bounding-box quality of several known and new space-filling curves. Surprisingly, some curves with bad locality by Gotsman and Lindenbaum’s measure, have good bounding-box quality, while the curve with the best-known locality has relatively bad bounding-box quality.

References

  1. 1.
    Alber, J., Niedermeier, R.: On multidimensional curves with Hilbert property. Theory of Computing Systems 33(4), 295–312 (2000)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Asano, T., Ranjan, D., Roos, T., Welzl, E., Widmayer, P.: Space-Filling Curves and Their Use in the Design of Geometric Data Structures. Theor. Comput. Sci. 181(1), 3–15 (1997)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bauman, K.E.: The dilation factor of the Peano-Hilbert curve. Math. Notes 80(5), 609–620 (2006)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chochia, G., Cole, M., Heywood, T.: Implementing the hierarchical PRAM on the 2D mesh: Analyses and experiments. In: Symp. on Parallel and Distributed Processing, pp. 587–595 (1995)Google Scholar
  5. 5.
    Gardner, M.: Mathematical Games—In which “monster” curves force redefinition of the word “curve”. Scientific American 235(6), 124–133 (1976)CrossRefGoogle Scholar
  6. 6.
    Gotsman, C., Lindenbaum, M.: On the metric properties of discrete space-filling curves. IEEE Trans. Image Processing 5(5), 794–797 (1996)CrossRefGoogle Scholar
  7. 7.
    Haverkort, H., van Walderveen, F.: Locality and bounding-box quality of two-dimensional space-filling curves (manuscript, 2008) arXiv:0806.4787 [cs.CG]Google Scholar
  8. 8.
    Hilbert, D.: Über die stetige Abbildung einer Linie auf ein Flächenstück. Math. Ann. 38(3), 459–460 (1891)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Kamel, I., Faloutsos, C.: On packing R-trees. In: Conf. on Information and Knowledge Management, pp. 490–499 (1993)Google Scholar
  10. 10.
    Lebesgue, H.L.: Leçons sur l’intégration et la recherche des fonctions primitives, pp. 44–45. Gauthier-Villars (1904)Google Scholar
  11. 11.
    von Luxburg, U.: Lokalitätsmaße von Peanokurven. Student project report, Universität Tübingen, Wilhelm-Schickard-Institut für Informatik (1998)Google Scholar
  12. 12.
    Manolopoulos, Y., Nanopoulos, A., Papadopoulos, A.N., Theodoridis, Y.: R-trees: Theory and Applications. Springer, Heidelberg (2005)Google Scholar
  13. 13.
    Niedermeier, R., Reinhardt, K., Sanders, P.: Towards optimal locality in mesh-indexings. Discrete Applied Mathematics 117, 211–237 (2002)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Niedermeier, R., Sanders, P.: On the Manhattan-distance between points on space-filling mesh-indexings. Technical Report IB 18/96, Karlsruhe University, Dept. of Computer Science (1996)Google Scholar
  15. 15.
    Peano, G.: Sur une courbe, qui remplit toute une aire plane. Math. Ann. 36(1), 157–160 (1890)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Sagan, H.: Space-Filling Curves. Universitext series. Springer, Heidelberg (1994)MATHGoogle Scholar
  17. 17.
    Wierum, J.-M.: Definition of a new circular space-filling curve: βΩ-indexing. Technical Report TR-001-02, Paderborn Center for Parallel Computing (PC2) (2002)Google Scholar
  18. 18.
    Wunderlich, W.: Über Peano-Kurven. Elemente der Mathematik 28(1), 1–10 (1973)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Herman Haverkort
    • 1
  • Freek van Walderveen
    • 1
  1. 1.Dept. of Computer ScienceEindhoven University of Technologythe Netherlands

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