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Towards Optimal Locality in MeshIndexings
, 1997
"... The efficiency of many data structures and algorithms relies on "localitypreserving" 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 slowgrowing function of ji jj. We present a new 2D indexing scheme we call H ..."
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Cited by 31 (4 self)
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The efficiency of many data structures and algorithms relies on "localitypreserving" 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 slowgrowing function of ji jj. We present a new 2D indexing scheme we call Hindexing , which has superior (possibly optimal) locality in comparison with the wellknown Hilbert indexings. Hindexings form a Hamiltonian cycle and we prove that they are optimally localitypreserving among all cyclic indexings. We provide fairly tight lower bounds for indexings without any restriction. Finally, illustrated by investigations concerning 2D and 3D Hilbert indexings, we present a framework for mechanizing upper bound proofs for locality.
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 Multidimensional Curves with Hilbert Property
, 2000
"... Indexing schemes for grids based on spacefilling curves (e.g., Hilbert curves) 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 c ..."
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Cited by 9 (0 self)
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Indexing schemes for grids based on spacefilling curves (e.g., Hilbert curves) 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 schemes. 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.
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 ..."
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Cited by 1 (1 self)
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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.