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On MultiDimensional Hilbert Indexings
 Theory of Computing Systems
, 1998
"... Indexing schemes for grids based on spacefilling curves (e.g., Hilbert indexings) find applications in numerous fields, ranging from parallel processing over data structures to image processing. Because of an increasing interest in discrete multidimensional spaces, indexing schemes for them hav ..."
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Cited by 13 (1 self)
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Indexing schemes for grids based on spacefilling curves (e.g., Hilbert indexings) find applications in numerous fields, ranging from parallel processing over data structures to image processing. Because of an increasing interest in discrete multidimensional spaces, indexing schemes for them have won considerable interest. Hilbert curves are the most simple and popular spacefilling indexing scheme. We extend the concept of curves with Hilbert property to arbitrary dimensions and present first results concerning their structural analysis that also simplify their applicability. We define and analyze in a precise mathematical way rdimensional Hilbert indexings for arbitrary r 2. Moreover, we generalize and simplify previous work and clarify the concept of Hilbert curves for multidimensional grids. As we show, Hilbert indexings can be completely described and analyzed by "generating elements of order 1", thus, in comparison with previous work, reducing their structural comp...
On the Quality of Partitions based on SpaceFilling Curves
, 2002
"... This paper presents bounds on the quality of partitions induced by spacefilling curves. We compare the surface that surrounds an arbitrary index range with the optimal partition in the grid, i. e. the square. It is shown that partitions induced by Lebesgue and Hilbert curves behave about 1.85 times ..."
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Cited by 6 (1 self)
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This paper presents bounds on the quality of partitions induced by spacefilling curves. We compare the surface that surrounds an arbitrary index range with the optimal partition in the grid, i. e. the square. It is shown that partitions induced by Lebesgue and Hilbert curves behave about 1.85 times worse with respect to the length of the surface. The Lebesgue indexing gives better results than the Hilbert indexing in worst case analysis. Furthermore, the surface of partitions based on the Lebesgue indexing are at most 3 times larger than the optimal in average case.
Average Case Quality of Partitions Induced by the Lebesgue Indexing
, 2001
"... This paper presents the quality of partitions induced by the Lebesgue curve in average case. The surface that surrounds an arbitrary index range is compared with the optimal partition in the grid, i. e. the square. The upper bound on the surface is asymptotically 3 times the optimal size. ..."
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Cited by 2 (2 self)
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This paper presents the quality of partitions induced by the Lebesgue curve in average case. The surface that surrounds an arbitrary index range is compared with the optimal partition in the grid, i. e. the square. The upper bound on the surface is asymptotically 3 times the optimal size.
Logarithmic PathLength in SpaceFilling Curves
 14TH CANADIAN CONFERENCE ON COMPUTATIONAL GEOMETRY
"... Data structures based on spacefilling curves have shown to be a good approach in several applications. For the monitoring of moving objects, e. g. necessary for the contact detection in finiteelement simulations, we need a special metrics to compare the quality of different curves. This paper ..."
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Cited by 2 (0 self)
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Data structures based on spacefilling curves have shown to be a good approach in several applications. For the monitoring of moving objects, e. g. necessary for the contact detection in finiteelement simulations, we need a special metrics to compare the quality of different curves. This paper
Definition of a New Circular SpaceFilling Curve  βΩIndexing
"... This technical report presents the definition of a circular Hilbertlike spacefilling curve. Preliminary evaluations in a simulation environment have shown good locality preserving properties. The results are compared with known bounds for other indexing schemes: Hilbert, Lebesgue, and HInde ..."
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This technical report presents the definition of a circular Hilbertlike spacefilling curve. Preliminary evaluations in a simulation environment have shown good locality preserving properties. The results are compared with known bounds for other indexing schemes: Hilbert, Lebesgue, and HIndexing. We evaluated partitions induced by the indexing schemes and uses the diameter and the surface as measures. For both we present worst case and average case results.