scholarly journals Tag-systems for the Hilbert curve

2007 ◽  
Vol Vol. 9 no. 2 ◽  
Author(s):  
Patrice Séébold
Keyword(s):  
Hilbert Space ◽  
Hilbert Curve ◽  
Space Filling ◽  
Space Filling Curve ◽  
Finite Approximations ◽  
Infinite Word ◽  
International Audience ◽  
Filling Curve

International audience Hilbert words correspond to finite approximations of the Hilbert space filling curve. The Hilbert infinite word H is obtained as the limit of these words. It gives a description of the Hilbert (infinite) curve. We give a uniform tag-system to generate automatically H and, by showing that it is almost cube-free, we prove that it cannot be obtained by simply iterating a morphism.

The LBF R-Tree

2010 ◽  
pp. 1-27
Author(s):  
Todd Eavis
Keyword(s):  
Hilbert Space ◽  
Data Warehousing ◽  
Experimental Results ◽  
Data Sets ◽  
Space Filling ◽  
Fully Integrated ◽  
Space Filling Curve ◽  
Filling Curve ◽  
Common User ◽  
Tree Performance

In multi-dimensional database environments, such as those typically associated with contemporary data warehousing, we generally require effective indexing mechanisms for all but the smallest data sets. While numerous such methods have been proposed, the R-tree has emerged as one of the most common and reliable indexing models. Nevertheless, as user queries grow in terms of both size and dimensionality, R-tree performance can deteriorate significantly. Moreover, in the multi-terabyte spaces of today’s enterprise warehouses, the combination of data and indexes ? R-tree or otherwise ? can produce unacceptably large storage requirements. In this chapter, the authors present a framework that addresses both of these concerns. First, they propose a variation of the classic R-tree that specifically targets data warehousing architectures. Their new LBF R-tree not only improves performance on common user-defined range queries, but gracefully degrades to a linear scan of the data on pathologically large queries. Experimental results demonstrate a reduction in disk seeks of more than 50% relative to more conventional R-tree designs. Second, the authors present a fully integrated, block-oriented compression model that reduces the storage footprint of both data and indexes. It does so by exploiting the same Hilbert space filling curve that is used to construct the LBF R-tree itself. Extensive testing demonstrates compression rates of more than 90% for multi-dimensional data, and up to 98% for the associated indexes.

On the Behaviour of p-Adic Scaled Space Filling Curve Indices for High-Dimensional Data

2020 ◽  
Author(s):  
Patrick Erik Bradley ◽  
Markus Wilhelm Jahn
Keyword(s):  
Gray Code ◽  
High Dimensional ◽  
Affine Transformations ◽  
Hilbert Curve ◽  
Space Filling ◽  
Real World Data ◽  
Space Filling Curve ◽  
Indexing Method ◽  
Filling Curve ◽  
Hilbert Curves

Abstract Space filling curves are widely used in computer science. In particular, Hilbert curves and their generalizations to higher dimension are used as an indexing method because of their nice locality properties. This article generalizes this concept to the systematic construction of $p$-adic versions of Hilbert curves based on special affine transformations of the $p$-adic Gray code and develops a scaled indexing method for data taken from high-dimensional spaces based on these new curves, which with increasing dimension is shown to be less space consuming than the optimal standard static Hilbert curve index. A measure is derived, which allows to assess the local sparsity of a dataset, and is tested on some real-world data.

Querying multi-dimensional data indexed using the Hilbert space-filling curve

2001 ◽  
Vol 30 (1) ◽  
pp. 19-24 ◽  
Author(s):  
J. K. Lawder ◽  
P. J. H. King
Keyword(s):  
Hilbert Space ◽  
Space Filling ◽  
Space Filling Curve ◽  
Filling Curve
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