Norm-Based Locality Measures of Two-Dimensional Hilbert Curves

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9778)


A discrete space-filling curve provides a 1-dimensional indexing or traversal of a multi-dimensional grid space. Applications of space-filling curves include multi-dimensional indexing methods, parallel computing, and image compression. Common goodness-measures for the applicability of space-filling curve families are locality and clustering. Locality reflects proximity preservation that close-by grid points are mapped to close-by indices or vice versa. We present an analytical study on the locality property of the 2-dimensional Hilbert curve family. The underlying locality measure, based on the p-normed metric \(d_{p}\), is the maximum ratio of \(d_{p}(u, v)^{m}\) to \(d_{p}(\tilde{u}, \tilde{v})\) over all corresponding point-pairs (uv) and \((\tilde{u}, \tilde{v})\) in the m-dimensional grid space and 1-dimensional index space, respectively. Our analytical results identify all candidate representative grid-point pairs (realizing the locality-measure values) for all real norm-parameters in the unit interval [1, 2] and grid-orders. Together with the known results for other norm-parameter values, we have almost complete knowledge of the locality measure of 2-dimensional Hilbert curves over the entire spectrum of possible norm-parameter values.


Space-filling curves Hilbert curves z-order curves Locality 


  1. 1.
    Alber, J., Niedermeier, R.: On multi-dimensional curves with Hilbert property. Theory Comput. Syst. 33(4), 295–312 (2000)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bauman, K.E.: The dilation factor of the Peano-Hilbert curve. Math. Notes 80(5), 609–620 (2006)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Chochia, G., Cole, M., Heywood, T.: Implementing the hierarchical PRAM on the 2D mesh: Analyses and experiments. In: Proceedings of the Seventh IEEE Symposium on Parallel and Distributeed Processing, pp. 587–595. IEEE Computer Society, Washington, October 1995Google Scholar
  4. 4.
    Dai, H.K., Su, H.C.: Approximation and analytical studies of inter-clustering performances of space-filling curves. In: Proceedings of the International Conference on Discrete Random Walks (Discrete Mathematics and Theoretical Computer Science, vol. AC (2003)), pp. 53–68, September 2003Google Scholar
  5. 5.
    Dai, H.K., Su, H.C.: On the locality properties of space-filling curves. In: Ibaraki, T., Katoh, N., Ono, H. (eds.) ISAAC 2003. LNCS, vol. 2906, pp. 385–394. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Dai, H.K., Su, H.C.: Clustering performance of 3-dimensional Hilbert curves. In: Gu, Q., Hell, P., Yang, B. (eds.) AAIM 2014. LNCS, vol. 8546, pp. 299–311. Springer, Heidelberg (2014)Google Scholar
  7. 7.
    Gotsman, C., Lindenbaum, M.: On the metric properties of discrete space-filling curves. IEEE Trans. Image Process. 5(5), 794–797 (1996)CrossRefGoogle Scholar
  8. 8.
    Niedermeier, R., Reinhardt, K., Sanders, P.: Towards optimal locality in mesh-indexings. Discrete Appl. Math. 117(1–3), 211–237 (2002)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Computer Science DepartmentOklahoma State UniversityStillwaterUSA
  2. 2.Department of Computer ScienceArkansas State UniversityJonesboroUSA

Personalised recommendations