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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 ..."
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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 10 (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.
Parallel Sorting Pattern
"... A large number of parallel applications contain a computationally intensive phase in which a large list of elements must be ordered based on some common attribute of the elements. How do we sort a sequence of elements on multiple processing units so as to minimize redistribution of keys while allo ..."
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A large number of parallel applications contain a computationally intensive phase in which a large list of elements must be ordered based on some common attribute of the elements. How do we sort a sequence of elements on multiple processing units so as to minimize redistribution of keys while allowing processing units to do independent sorting work? 2. CONTEXT Sorting is the process of reordering a sequence taken as input and producing one that is ordered according to an attribute. Parallel sorting is the process of using multiple processing units to collectively sort an unordered sequence. The unsorted sequence is composed of disjoint subsequences, each of which is associated with a unique processing unit. Parallel sorting produces a fully sorted sequence composed of ordered subsequences, each of which is associated with a unique processing unit. The produced sequences are typically ordered according to the given processor ordering and are of roughly equal length. It is important to realize that parallelization of sorting algorithms (particularly under the distributed memory model which we focus on) is a fundamentally different problem from acceleration or hardware performance tuning of a sorting algorithm. Indeed, such acceleration and tuning is often implemented using techniques similar to parallel sorting algorithms implemented within a distributed memory model [1] [2]. However, the semantics and goals of such acceleration and tuning of sorting is often different than that of parallel sorting under the distributed memory model and deserves separate analysis that will not be provided here. In particular, we focus on sorting a sequence composed of a set of sequences, each associated with a logical memory. Parallel sorting under the distributed memory model is a fundamentally different problem from sorting under a sequential or shared memory programming model. Sequential