Conference Paper

On the Locality Properties of Space-Filling Curves.

DOI: 10.1007/978-3-540-24587-2_40 Conference: Algorithms and Computation, 14th International Symposium, ISAAC 2003, Kyoto, Japan, December 15-17, 2003, Proceedings
Source: DBLP

ABSTRACT A discrete space-filling curve provides a linear traversal or indexing of a multi-dimensional grid space. We present an analytical
study of the locality properties of the m-dimensional k-order discrete Hilbert and z-order curve families, {Hmk | k = 1,2,...}\{H^m_k | k = 1,2,...\} and {Zmk | k = 1,2,...}\{Z^m_k | k = 1,2,...\}, respectively, based on the locality measure L

δ
that cumulates all index-differences of point-pairs at a common 1-normed distance δ. We derive the exact formulas for L

δ
(H

k

m
) and L

δ
(Z

k

m
) for m = 2 and arbitrary δ that is an integral power of 2, and m = 3 and δ = 1. The results yield a constant asymptotic ratio lim$_{k\rightarrow\infty}\frac{L_\delta(H^m_k)}{L_\delta(Z^m_k)} > 1$_{k\rightarrow\infty}\frac{L_\delta(H^m_k)}{L_\delta(Z^m_k)} > 1, which suggests that the z-order curve family performs better than the Hilbert curve family over the considered parameter
ranges.

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    ABSTRACT: A discrete space-filling curve provides a linear traversal/indexing of a multi-dimensional grid space. This paper presents an application of random walk to the study of inter-clustering of space-filling curves and an analytical study on the inter-clustering performances of 2-dimensional Hilbert and z-order curve families. Two underlying measures are employed: the mean inter-cluster distance over all inter-cluster gaps and the mean total inter-cluster distance over all subgrids. We show how approximating the mean inter-cluster distance statistics of continuous multi-dimensional space-filling curves fits into the formalism of random walk, and derive the exact formulas for the two statistics for both curve families. The excellent agreement in the approximate and true mean inter-cluster distance statistics suggests that the random walk may furnish an effective model to develop approximations to clustering and locality statistics for space-filling curves. Based upon the analytical results, the asymptotic comparisons indicate that z-order curve family performs better than Hilbert curve family with respect to both statistics.
    Discrete Random Walks, DRW'03, Paris, France, September 1-5, 2003; 01/2003