Conference Paper
On the Locality Properties of SpaceFilling Curves
DOI: 10.1007/9783540245872_40 Conference: Algorithms and Computation, 14th International Symposium, ISAAC 2003, Kyoto, Japan, December 1517, 2003, Proceedings
Source: DBLP
ABSTRACT
A discrete spacefilling curve provides a linear traversal or indexing of a multidimensional grid space. We present an analytical
study of the locality properties of the mdimensional korder discrete Hilbert and zorder 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 indexdifferences of pointpairs at a common 1normed 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 zorder curve family performs better than the Hilbert curve family over the considered parameter
ranges.
study of the locality properties of the mdimensional korder discrete Hilbert and zorder 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 indexdifferences of pointpairs at a common 1normed 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 zorder curve family performs better than the Hilbert curve family over the considered parameter
ranges.

 "Thus, this work and our work are concerned with different metrics. Dai and Su [7] [8] present upper bounds on the stretch of specific SFCs including the Hilbert curve, but do not consider lower bounds on the stretch, like we do here. "
Conference Paper: A Lower Bound On Proximity Preservation by Space Filling Curves
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ABSTRACT: A space filling curve (SFC) is a proximity preserving mapping from a high dimensional space to a single dimensional space. SFCs have been used extensively in dealing with multidimensional data in parallel computing, scientific computing, and databases. The general goal of an SFC is that points that are close to each other in highdimensional space are also close to each other in the single dimensional space. While SFCs have been used widely, the extent to which proximity can be preserved by an SFC is not precisely understood yet. We consider natural metrics, including the "nearestneighbor stretch" of an SFC, which measure the extent to which an SFC preserves proximity. We first show a powerful negative result, that there is an inherent lower bound on the stretch of any SFC. We then show that the stretch of the commonly used Z curve is within a factor of 1.5 from the optimal, irrespective of the number of dimensions. Further we show that a very simple SFC also achieves the same stretch as the Z curve. Our results apply to SFCs in any dimension d such that d is a constant. 

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