Results 1 
9 of
9
Hash Based Adaptive Parallel Multilevel Methods with SpaceFilling Curves
 NIC Series
, 2002
"... this paper a parallelisable and cheap method based on spacefilling curves is proposed. The partitioning is embedded into the parallel solution algorithm using multilevel iterative solvers and adaptive grid refinement. Numerical experiments on two massively parallel computers prove the efficienc ..."
Abstract

Cited by 8 (0 self)
 Add to MetaCart
this paper a parallelisable and cheap method based on spacefilling curves is proposed. The partitioning is embedded into the parallel solution algorithm using multilevel iterative solvers and adaptive grid refinement. Numerical experiments on two massively parallel computers prove the efficiency of this approach
FINITE ENTROPY FOR MULTIDIMENSIONAL CELLULAR AUTOMATA
, 2007
"... Abstract. Let X = S G where G is a countable group and S is a finite set. A cellular automaton (CA) is an endomorphism T: X → X (continuous, commuting with the action of G). Shereshevsky [14] proved that for G = Z d with d> 1 no CA can be forward expansive, raising the following conjecture: For G = ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
Abstract. Let X = S G where G is a countable group and S is a finite set. A cellular automaton (CA) is an endomorphism T: X → X (continuous, commuting with the action of G). Shereshevsky [14] proved that for G = Z d with d> 1 no CA can be forward expansive, raising the following conjecture: For G = Z d, d> 1 the topological entropy of any CA is either zero or infinite. Morris and Ward [11], proved this for linear CA’s, leaving the original conjecture open. We show that this conjecture is false, proving that for any d there exist a ddimensional CA with finite, nonzero topological entropy. We also discuss a measuretheoretic counterpart of this question for measurepreserving CA’s. 1.
FourDimensional Hilbert Curves for RTrees
"... Twodimensional Rtrees are a class of spatial index structures in which objects are arranged to enable fast window queries: report all objects that intersect a given query window. One of the most successful methods of arranging the objects in the index structure is based on sorting the objects acco ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
Twodimensional Rtrees are a class of spatial index structures in which objects are arranged to enable fast window queries: report all objects that intersect a given query window. One of the most successful methods of arranging the objects in the index structure is based on sorting the objects according to the positions of their centres along a twodimensional Hilbert spacefilling curve. Alternatively one may use the coordinates of the objects ’ bounding boxes to represent each object by a fourdimensional point, and sort these points along a fourdimensional Hilberttype curve. In experiments by Kamel and Faloutsos and by Arge et al. the first solution consistently outperformed the latter when applied to point data, while the latter solution clearly outperformed the first on certain artificial rectangle data. These authors did not specify which fourdimensional Hilberttype curve was used; many exist. In this paper we show that the results of the previous papers can be explained by the choice of the fourdimensional Hilberttype curve that was used and by the way it was rotated in fourdimensional space. By selecting a curve that has certain properties and choosing the right rotation one can combine the strengths of the twodimensional and the fourdimensional approach into one, while avoiding their apparent weaknesses. The effectiveness of our approach is demonstrated with experiments on various data sets. For real data taken from VLSI design, our new curve yields Rtrees with query times that are better than those of Rtrees that were obtained with previously used curves. 1
Locality and boundingbox quality of twodimensional spacefilling curves
 in: ESA
"... Spacefilling curves can be used to organise points in the plane into boundingbox hierarchies (such as Rtrees). We develop measures of the boundingbox quality of spacefilling curves that express how effective different spacefilling curves are for this purpose. We give general lower bounds on th ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
Spacefilling curves can be used to organise points in the plane into boundingbox hierarchies (such as Rtrees). We develop measures of the boundingbox quality of spacefilling curves that express how effective different spacefilling curves are for this purpose. We give general lower bounds on the boundingbox quality measures and on locality according to Gotsman and Lindenbaum for a large class of spacefilling curves. We describe a generic algorithm to approximate these and similar quality measures for any given curve. Using our algorithm we find good approximations of the locality and the boundingbox quality of several known and new spacefilling curves. Surprisingly, some curves with relatively bad locality by Gotsman and Lindenbaum’s measure, have good boundingbox quality, while the curve with the bestknown locality has relatively bad boundingbox quality. 1
Approximation and Analytical Studies of Interclustering Performances of SpaceFilling Curves
"... A discrete spacefilling curve provides a linear traversal/indexing of a multidimensional grid space. This paper presents an application of random walk to the study of interclustering of spacefilling curves and an analytical study on the interclustering performances of 2dimensional Hilbert and ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
A discrete spacefilling curve provides a linear traversal/indexing of a multidimensional grid space. This paper presents an application of random walk to the study of interclustering of spacefilling curves and an analytical study on the interclustering performances of 2dimensional Hilbert and zorder curve families. Two underlying measures are employed: the mean intercluster distance over all intercluster gaps and the mean total intercluster distance over all subgrids. We show how approximating the mean intercluster distance statistics of continuous multidimensional spacefilling curves fits into the formalism of random walk, and derive the exact formulas for the two statistics for both curve families. The excellent agreement in the approximate and true mean intercluster distance statistics suggests that the random walk may furnish an effective model to develop approximations to clustering and locality statistics for spacefilling curves. Based upon the analytical results, the asymptotic comparisons indicate that zorder curve family performs better than Hilbert curve family with respect to both statistics.
www.elsevier.com/locate/entcs Locality of Corner Transformation for Multidimensional Spatial Access Methods
"... The geometric structural complexity of spatial objects does not render an intuitive distance metric on the data space that measures spatial proximity. However, such a metric provides a formal basis for analytical work in transformationbased multidimensional spatial access methods, including localit ..."
Abstract
 Add to MetaCart
The geometric structural complexity of spatial objects does not render an intuitive distance metric on the data space that measures spatial proximity. However, such a metric provides a formal basis for analytical work in transformationbased multidimensional spatial access methods, including locality preservation of the underlying transformation and distancebased spatial queries. We study the Hausdorff distance metric on the space of multidimensional polytopes, and prove a tight relationship between the metric on the original space of kdimensional hyperrectangles and the standard pnormed metric on the transform space of 2kdimensional points under the corner transformation, which justifies the effectiveness of the transformationbased technique in preserving spatial locality. Keywords: databases, multidimensional spatial access methods, corner transformation, locality
ENTROPY OF CELLULAR AUTOMATA
, 2007
"... Abstract. Let X = S G where G is a countable group and S is a finite set. A cellular automaton (CA) is an endomorphism T: X → X (continuous, commuting with the action of G). Shereshevsky [13] proved that for G = Z d with d> 1 no CA can be forward expansive, raising the following conjecture: For G = ..."
Abstract
 Add to MetaCart
Abstract. Let X = S G where G is a countable group and S is a finite set. A cellular automaton (CA) is an endomorphism T: X → X (continuous, commuting with the action of G). Shereshevsky [13] proved that for G = Z d with d> 1 no CA can be forward expansive, raising the following conjecture: For G = Z d, d> 1 the topological entropy of any CA is either zero or infinite. Morris and Ward [10], proved this for linear CA’s, leaving the original conjecture open. We show that this conjecture is false, proving that for any d there exist a ddimensional CA with finite, nonzero topological entropy. We also discuss a measuretheoretic counterpart of this question for measurepreserving CA’s. 1.
Reordering Columns for Smaller Indexes
, 909
"... Columnoriented indexes—such as projection or bitmap indexes—are compressed by runlength encoding to reduce storage and increase speed. Sorting the tables improves compression. On realistic data sets, permuting the columns in the right order before sorting can reduce the number of runs by a factor ..."
Abstract
 Add to MetaCart
Columnoriented indexes—such as projection or bitmap indexes—are compressed by runlength encoding to reduce storage and increase speed. Sorting the tables improves compression. On realistic data sets, permuting the columns in the right order before sorting can reduce the number of runs by a factor of two or more. For many cases, we prove that the number of runs in table columns is minimized if we sort columns by increasing cardinality. Yet—maybe surprisingly—we must sometimes maximize the number of runs to minimize the index size. Experimentally, sorting based on Hilbert spacefilling curves is poor at minimizing the number of runs. Key words:
Netherlands. and
"... Twodimensional Rtrees are a class of spatial index structures in which objects are arranged to enable fast window queries: report all objects that intersect a given query window. One of the most successful methods of arranging the objects in the index structure is based on sorting the objects acco ..."
Abstract
 Add to MetaCart
Twodimensional Rtrees are a class of spatial index structures in which objects are arranged to enable fast window queries: report all objects that intersect a given query window. One of the most successful methods of arranging the objects in the index structure is based on sorting the objects according to the positions of their centres along a twodimensional Hilbert spacefilling curve. Alternatively one may use the coordinates of the objects ’ bounding boxes to represent each object by a fourdimensional point, and sort these points along a fourdimensional Hilberttype curve. In experiments by Kamel and Faloutsos and by Arge et al. the first solution consistently outperformed the latter when applied to point data, while the latter solution clearly outperformed the first on certain artificial rectangle data. These authors did not specify which fourdimensional Hilberttype curve was used; many exist. In this paper we show that the results of the previous papers can be explained by the choice of the fourdimensional Hilberttype curve that was used and by the way it was rotated in fourdimensional space. By selecting a curve that has certain properties and choosing the right rotation one can combine the strengths of the twodimensional and the fourdimensional approach into one, while avoiding their apparent weaknesses. The effectiveness of our approach is demonstrated with