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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