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, \(\{H^m_k | k = 1,2,...\}\) and \(\{Z^m_k | k = 1,2,...\}\), 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\(_{k\rightarrow\infty}\frac{L_\delta(H^m_k)}{L_\delta(Z^m_k)} > 1\), which suggests that the z-order curve family performs better than the Hilbert curve family over the considered parameter ranges.
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© 2003 Springer-Verlag Berlin Heidelberg
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Dai, H.K., Su, H.C. (2003). On the Locality Properties of Space-Filling Curves. In: Ibaraki, T., Katoh, N., Ono, H. (eds) Algorithms and Computation. ISAAC 2003. Lecture Notes in Computer Science, vol 2906. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24587-2_40
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DOI: https://doi.org/10.1007/978-3-540-24587-2_40
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20695-8
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