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Proof That Hilbert Space Filling Curve Converges to a Continuous Curve

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Clustering Analyses of Two-Dimensional Space-Filling Curves: Hilbert and z-Order Curves

SN Computer Science volume 4, Article number:8 (2023) Cite this article

Abstract

A discrete space-filling curve provides a linear traversal or indexing of a multi-dimensional grid space. This paper presents two analytical studies on clustering analyses of the 2-dimensional Hilbert and z-order curve families. The underlying measure is the mean number of cluster over all identically shaped subgrids. We derive the exact formulas for the clustering statistics for the 2-dimensional Hilbert and z-order curve families. The two exact analytical results yield: (1) a comparison of their relative performances with respect to this measure: when the grid-order is sufficiently larger than the subgrid-order (typical scenario for most applications), Hilbert curve family performs significantly better than z-order curve family, and (2) good agreements with asymptotic analyses of continuous space-filling curves and of (non-continuous) z-order curve in the literature.

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Authors and Affiliations

  1. Computer Science Department, Oklahoma State University, Stillwater, OK, 74078, USA

    H. K. Dai

  2. Department of Computer Science, Arkansas State University, Jonesboro, AR, 72401, USA

    H. C. Su

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Correspondence to H. K. Dai.

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This article is part of the topical collection "Future Data and Security Engineering 2021" guest edited by Tran Khanh Dang.

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Dai, H.K., Su, H.C. Clustering Analyses of Two-Dimensional Space-Filling Curves: Hilbert and z-Order Curves. SN COMPUT. SCI. 4, 8 (2023). https://doi.org/10.1007/s42979-022-01320-9

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  • DOI : https://doi.org/10.1007/s42979-022-01320-9

Keywords

  • Index structure
  • Space-filling curve
  • Hilbert curve
  • z-Order curve
  • Clustering

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