AbstractÐSeveral schemes for the linear mapping of a multidimensional space have been proposed for various applications, such as access methods for spatio-temporal 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 space-filling curve achieves the best clustering [1], [14]. In this paper, we analyze the clustering property of the Hilbert space-filling curve by deriving closed-form 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.

In this work, we show that the standard graph-partitioning based decomposition of sparse matrices does not reflect the actual communication volume requirement for parallel matrix-vector multiplication. We propose two computational hypergraph models which avoid this crucial deficiency of the graph model. The proposed models reduce the decomposition problem to the well-known hypergraph partitioning problem. The recently proposed successful multilevel framework is exploited to develop a multilevel hypergraph partitioning tool PaToH for the experimental verification of our proposed hypergraph models. Experimental results on a wide range of realistic sparse test matrices confirm the validity of the proposed hypergraph models. In the decomposition of the test matrices, the hypergraph models using PaToH and hMeTiS result in up to 63% less communication volume (30%--38% less on the average) than the graph model using MeTiS, while PaToH is only 1.3--2.3 times slower than MeTiS on the average. ...

The efficiency of many data structures and algorithms relies on "locality-preserving" indexing schemes for meshes. We concentrate on the case in which the maximal distance between two mesh nodes indexed i and j shall be a slow-growing function of ji jj. We present a new 2-D indexing scheme we call H-indexing , which has superior (possibly optimal) locality in comparison with the well-known Hilbert indexings. H-indexings form a Hamiltonian cycle and we prove that they are optimally locality-preserving among all cyclic indexings. We provide fairly tight lower bounds for indexings without any restriction. Finally, illustrated by investigations concerning 2-D and 3-D Hilbert indexings, we present a framework for mechanizing upper bound proofs for locality.

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Space-filling curves are continuous self-similar functions which map compact multi-dimensional sets into one-dimensional ones. Since their invention they have found applications in a wide variety of fields [12, 21]. In the context of scientific computing and database systems, spacefilling curves can significantly improve data reuse and request times because of their locality properties [9, 13, 15]. In particular, the Hilbert curve has been shown to be the best choice for these applications [21]. However, in database systems it is often the case that not all dimensions of the data have the same cardinality, leading to an inefficiency in the use of space-filling curves due to their being naturally constrained to spaces where all dimensions are of equal size. We explore the Hilbert curve, reproducing classical algorithms for their generation and manipulation through an intuitive and rigorous geometric approach. We then extend these basic results to construct compact Hilbert indices which are able to capture the ordering properties of the regular Hilbert curve but without the associated inefficiency in representation for spaces with mismatched dimensions.

In this paper we define a new compact Hilbert index which, while maintaining all of the advantages of the standard Hilbert curve, permits spaces with unequal dimension cardinalities. The compact Hilbert index can be used in any application that would have previously relied on Hilbert curves but, in the case of unequal side lengths, provides a more memory efficient representation. This advantage is particularly important in distributed applications (Parallel, P2P and Grid), in which not only is memory space saved but communication volume is significantly reduced.