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71
Analysis of the clustering properties of the Hilbert spacefilling curve
 IEEE Transactions on Knowledge and Data Engineering
, 2001
"... AbstractÐSeveral schemes for the linear mapping of a multidimensional space have been proposed for various applications, such as access methods for spatiotemporal databases and image compression. In these applications, one of the most desired properties from such linear mappings is clustering, whic ..."
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Cited by 141 (10 self)
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AbstractÐSeveral schemes for the linear mapping of a multidimensional space have been proposed for various applications, such as access methods for spatiotemporal databases and image compression. In these applications, one of the most desired properties from such linear mappings is clustering, which means the locality between objects in the multidimensional space being preserved in the linear space. It is widely believed that the Hilbert spacefilling curve achieves the best clustering [1], [14]. In this paper, we analyze the clustering property of the Hilbert spacefilling curve by deriving closedform formulas for the number of clusters in a given query region of an arbitrary shape (e.g., polygons and polyhedra). Both the asymptotic solution for the general case and the exact solution for a special case generalize previous work [14]. They agree with the empirical results that the number of clusters depends on the hypersurface area of the query region and not on its hypervolume. We also show that the Hilbert curve achieves better clustering than the z curve. From a practical point of view, the formulas given in this paper provide a simple measure that can be used to predict the required disk access behaviors and, hence, the total access time.
Designing pixeloriented visualization techniques: Theory and applications
 IEEE Transactions on Visualization and Computer Graphics
, 2000
"... AbstractÐVisualization techniques are ofincreasing importance in exploring and analyzing large amounts ofmultidimensional information. One important class of visualization techniques which is particularly interesting for visualizing very large multidimensional data sets is the class ofthe pixelorie ..."
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Cited by 85 (8 self)
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AbstractÐVisualization techniques are ofincreasing importance in exploring and analyzing large amounts ofmultidimensional information. One important class of visualization techniques which is particularly interesting for visualizing very large multidimensional data sets is the class ofthe pixeloriented techniques. The basic idea ofpixeloriented visualization techniques is to represent as many data objects as possible on the screen at the same time by mapping each data value to a pixel ofthe screen and arranging the pixels adequately. A number of different pixeloriented visualization techniques have been proposed in recent years and it has been shown that the techniques are useful for visual data exploration in a number of different application contexts. In this paper, we discuss a number ofissues which are ofhigh importance in developing pixeloriented visualization techniques. The major goal ofthis article is to provide a formal basis of pixeloriented visualization techniques and show that the design decisions in developing them can be seen as solutions ofwelldefined optimization problems. This is true for the mapping ofthe data values to colors, the arrangement ofpixels inside the subwindows, the shape ofthe subwindows, and the ordering ofthe dimension subwindows. The paper also discusses the design issues of special variants of pixeloriented techniques for visualizing large spatial data sets. The optimization functions for the mentioned design decisions are important for the effectiveness of the resulting visualizations. We show this by evaluating the optimization functions and comparing it the results to the visualization obtained in a number of different application. Index TermsÐInformation visualization, visualizing large data sets, visualizing multidimensional and multivariate data, visual data exploration, visual data mining. 1
Visualization Techniques for Mining Large Databases: A Comparison
 IEEE Transactions on Knowledge and Data Engineering
, 1996
"... Visual data mining techniques have proven to be of high value in exploratory data analysis and they also have a high potential for mining large databases. In this article, we describe and evaluate a new visualizationbased approach to mining large databases. The basic idea of our visual data mining ..."
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Cited by 75 (1 self)
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Visual data mining techniques have proven to be of high value in exploratory data analysis and they also have a high potential for mining large databases. In this article, we describe and evaluate a new visualizationbased approach to mining large databases. The basic idea of our visual data mining techniques is to represent as many data items as possible on the screen at the same time by mapping each data value to a pixel of the screen and arranging the pixels adequately. The major goal of this article is to evaluate our visual data mining techniques and to compare them to other wellknown visualization techniques for multidimensional data: the parallel coordinate and stick figure visualization techniques. For the evaluation of visual data mining techniques, in the first place the perception of properties of the data counts, and only in the second place the CPU time and the number of secondary storage accesses are important. In addition to testing the visualization techniques using re...
On Partitioning Dynamic Adaptive Grid Hierarchies
 Proceedings of the 29th Annual Hawaii International Conference on System Sciences
, 1996
"... This paper presents a computationally efficient runtime partitioning and loadbalancing scheme for the Distributed Adaptive Grid Hierarchies that underlie adaptive meshrefinement methods. The partitioning scheme yields an efficient parallel computational structure that maintains locality to reduce ..."
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Cited by 72 (22 self)
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This paper presents a computationally efficient runtime partitioning and loadbalancing scheme for the Distributed Adaptive Grid Hierarchies that underlie adaptive meshrefinement methods. The partitioning scheme yields an efficient parallel computational structure that maintains locality to reduce communications. Further, it enables dynamic repartitioning and loadbalancing of the adaptive grid hierarchy to be performed costeffectively. The runtime partitioning support presented has been implemented within the framework of a datamanagement infrastructure supporting dynamic distributed datastructures for parallel adaptive numerical techniques. This infrastructure is the foundational layer of a computational toolkit for the Binary BlackHole NSF Grand Challenge project. 1 Introduction Dynamically adaptive methods for the solution of partial differential equations that employ locally optimal approximations can yield highly advantageous ratios for cost/accuracy when compared to metho...
Nonlinear Array Layouts for Hierarchical Memory Systems
, 1999
"... Programming languages that provide multidimensional arrays and a flat linear model of memory must implement a mapping between these two domains to order array elements in memory. This layout function is fixed at language definition time and constitutes an invisible, nonprogrammable array attribute. ..."
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Cited by 72 (5 self)
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Programming languages that provide multidimensional arrays and a flat linear model of memory must implement a mapping between these two domains to order array elements in memory. This layout function is fixed at language definition time and constitutes an invisible, nonprogrammable array attribute. In reality, modern memory systems are architecturally hierarchical rather than flat, with substantial differences in performance among different levels of the hierarchy. This mismatch between the model and the true architecture of memory systems can result in low locality of reference and poor performance. Some of this loss in performance can be recovered by reordering computations using transformations such as loop tiling. We explore nonlinear array layout functions as an additional means of improving locality of reference. For a benchmark suite composed of dense matrix kernels, we show by timing and simulation that two specific layouts (4D and Morton) have low implementation costs (25% of total running time) and high performance benefits (reducing execution time by factors of 1.12.5); that they have smooth performance curves, both across a wide range of problem sizes and over representative cache architectures; and that recursionbased control structures may be needed to fully exploit their potential.
On the construction of some capacityapproaching coding schemes
, 2000
"... This thesis proposes two constructive methods of approaching the Shannon limit very closely. Interestingly, these two methods operate in opposite regions, one has a block length of one and the other has a block length approaching infinity. The first approach is based on novel memoryless joint source ..."
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Cited by 56 (2 self)
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This thesis proposes two constructive methods of approaching the Shannon limit very closely. Interestingly, these two methods operate in opposite regions, one has a block length of one and the other has a block length approaching infinity. The first approach is based on novel memoryless joint sourcechannel coding schemes. We first show some examples of sources and channels where no coding is optimal for all values of the signaltonoise ratio (SNR). When the source bandwidth is greater than the channel bandwidth, joint coding schemes based on spacefilling curves and other families of curves are proposed. For uniform sources and modulo channels, our coding scheme based on spacefilling curves operates within 1.1 dB of Shannon’s ratedistortion bound. For Gaussian sources and additive white Gaussian noise (AWGN) channels, we can achieve within 0.9 dB of the ratedistortion bound. The second scheme is based on lowdensity paritycheck (LDPC) codes. We first demonstrate that we can translate threshold values of an LDPC code between channels accurately using a simple mapping. We develop some models for density evolution
Recursive Array Layouts and Fast Parallel Matrix Multiplication
 In Proceedings of Eleventh Annual ACM Symposium on Parallel Algorithms and Architectures
, 1999
"... Matrix multiplication is an important kernel in linear algebra algorithms, and the performance of both serial and parallel implementations is highly dependent on the memory system behavior. Unfortunately, due to false sharing and cache conflicts, traditional columnmajor or rowmajor array layouts i ..."
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Cited by 48 (4 self)
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Matrix multiplication is an important kernel in linear algebra algorithms, and the performance of both serial and parallel implementations is highly dependent on the memory system behavior. Unfortunately, due to false sharing and cache conflicts, traditional columnmajor or rowmajor array layouts incur high variability in memory system performance as matrix size varies. This paper investigates the use of recursive array layouts for improving the performance of parallel recursive matrix multiplication algorithms. We extend previous work by Frens and Wise on recursive matrix multiplication to examine several recursive array layouts and three recursive algorithms: standard matrix multiplication, and the more complex algorithms of Strassen and Winograd. We show that while recursive array layouts significantly outperform traditional layouts (reducing execution times by a factor of 1.22.5) for the standard algorithm, they offer little improvement for Strassen's and Winograd's algorithms;...
Distributed Dynamic DataStructures for Parallel Adaptive MeshRefinement
 PROCEEDINGS OF THE INTERNATIONAL CONFERENCE FOR HIGH PERFORMANCE COMPUTING
, 1995
"... This paper presents the design and implementation of dynamic distributed datastructures to support parallel adaptive (multigrid) finite difference codes based on hierarchical adaptive meshrefinement (AMR) techniques for the solution of partial differential equations. The abstraction provided by th ..."
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Cited by 35 (11 self)
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This paper presents the design and implementation of dynamic distributed datastructures to support parallel adaptive (multigrid) finite difference codes based on hierarchical adaptive meshrefinement (AMR) techniques for the solution of partial differential equations. The abstraction provided by the datastructures is a dynamic hierarchical grid where operations on the grid are independent of its distribution across processors in a parallel execution environment, and of the number of levels in the grid hierarchy. The distributed dynamic datastructures have been implemented aspart of a computational toolkit for the Binary BlackHole NSF Grand Challenge project.
DOT: A spatial access method using fractals
 Proc. 7th IEEE Internat. Conf. on Data Engineering
, 1991
"... Existing Database Management Systems (DBMSs) do not handle efficiently multidimensional data such as boxes, polygons, or even points in a multidimensional space. We examine access methods for these data with two design goals in mind: (a) efficiency in terms of search speed and space overhead and ( ..."
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Cited by 34 (1 self)
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Existing Database Management Systems (DBMSs) do not handle efficiently multidimensional data such as boxes, polygons, or even points in a multidimensional space. We examine access methods for these data with two design goals in mind: (a) efficiency in terms of search speed and space overhead and (b) ability to be integrated in a DBMS easily. We propose a method to map multidimensional objects into points in a 1dimensional space; thus, traditional primarykey access methods can be applied, with very few extensions on the part of the DBMS. We propose such mappings based on fractals; we implemented the whole method on top of a B +tree, along with several mappings. Simulation experiments on several distributions of the input data show
Recursive Array Layouts and Fast Matrix Multiplication
, 1999
"... The performance of both serial and parallel implementations of matrix multiplication is highly sensitive to memory system behavior. False sharing and cache conflicts cause traditional columnmajor or rowmajor array layouts to incur high variability in memory system performance as matrix size var ..."
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Cited by 31 (0 self)
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The performance of both serial and parallel implementations of matrix multiplication is highly sensitive to memory system behavior. False sharing and cache conflicts cause traditional columnmajor or rowmajor array layouts to incur high variability in memory system performance as matrix size varies. This paper investigates the use of recursive array layouts to improve performance and reduce variability. Previous work on recursive matrix multiplication is extended to examine several recursive array layouts and three recursive algorithms: standard matrix multiplication, and the more complex algorithms of Strassen and Winograd. While recursive layouts significantly outperform traditional layouts (reducing execution times by a factor of 1.22.5) for the standard algorithm, they offer little improvement for Strassen's and Winograd's algorithms. For a purely sequential implementation, it is possible to reorder computation to conserve memory space and improve performance between ...