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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.
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 ..."
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Cited by 8 (0 self)
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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
On the Quality of SpaceFilling Curve Induced Partitions
 Z. Angew. Math. Mech
, 2000
"... The solution of partial differential equations on a parallel computer is usually done by a domain decomposition approach. The mesh is split into several partitions mapped onto the processors. However, partitioning of unstructured meshes and adaptive refined meshes in general is an NP hard proble ..."
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Cited by 6 (0 self)
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The solution of partial differential equations on a parallel computer is usually done by a domain decomposition approach. The mesh is split into several partitions mapped onto the processors. However, partitioning of unstructured meshes and adaptive refined meshes in general is an NP hard problem and heuristics are used. In this paper spacefilling curve based partition methods are analysed and bounds for the quality of the partitions are given. Furthermore estimates for parallel numerical algorithms such as multigrid and wavelet methods on these partitions are derived. AMS/MSC91 classification: 65Y20, 68Q22, 65N50 1 The partition problem FiniteElement, FiniteVolume and FiniteDifference methods for the solution of partial differential equations are based on meshes. The solution is represented by degrees of freedoms attached to certain locations on the mesh. Numerical algorithms operate on these degrees of freedom during steps like the assembly of a linear equation system or...
On the ManhattanDistance Between Points on SpaceFilling MeshIndexings
, 1996
"... Indexing schemes based on space filling curves like the Hilbert curve are a powerful tool for building efficient parallel algorithms on meshconnected computers. The main reason is that they are localitypreserving, i.e., the Manhattandistance between processors grows only slowly with increasing in ..."
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Cited by 4 (0 self)
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Indexing schemes based on space filling curves like the Hilbert curve are a powerful tool for building efficient parallel algorithms on meshconnected computers. The main reason is that they are localitypreserving, i.e., the Manhattandistance between processors grows only slowly with increasing index differences. We present a simple and easytoverify proof that the Manhattandistance of any indices i and j is bounded by 3 p ji \Gamma jj \Gamma 2 for the 2DHilbert curve. The technique used for the proof is then generalized for a large class of selfsimilar curves. We use this result to show a (quite tight) bound of 4:73458 3 p ji \Gamma jj \Gamma 3 for a 3DHilbert curve. 1 Introduction It has become increasingly clear that meshconnected processor arrays, grids for short, are among the most realistic models of parallel computation [1, 4, 14, 18]. The indexing of the processors is an important aspect in the design of mesh algorithms. Several indexing schemes are wellknown. Mos...
Stereoscopic families of permutations, and their applications (Extended Abstract)
 In 5th IEEE Israel Symposium on the Theory of Computing and Systems
, 1997
"... ) Uriel Feige Robert Krauthgamer y Abstract A stereoscopic family of permutations maps an m dimensional mesh into several 1dimensional lines, in a way that jointly preserves distance information. Specifically, consider any two points and denote their distance on the mdimensional mesh by d. ..."
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Cited by 2 (0 self)
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) Uriel Feige Robert Krauthgamer y Abstract A stereoscopic family of permutations maps an m dimensional mesh into several 1dimensional lines, in a way that jointly preserves distance information. Specifically, consider any two points and denote their distance on the mdimensional mesh by d. Then the distance between their images, on the line on which these images are closest together, is O(d m ). We initiate a systematic study of stereoscopic families of permutations. We show a construction of these families that involves the use of m + 1 images. We also show that under some additional restrictions (namely, adjacent points on the image lines originate at points which are not too far away on the mesh), three images are necessary in order to construct such a family for the 2dimensional mesh. We present two applications for stereoscopic families of permutations. One application is an algorithm for routing on the mesh that guarantees delivery of each packet within a number of s...
The Paderborn University BSP (PUB) Library  Design and Implementation
, 1997
"... The Paderborn University BSP (PUB) library is a C library of message passing functions inspired by the BSP model. The library supports buffered and unbuffered asynchronous communication between processors, and a mechanism for synchronizing the processors in a barrier style. In addition, it provides ..."
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The Paderborn University BSP (PUB) library is a C library of message passing functions inspired by the BSP model. The library supports buffered and unbuffered asynchronous communication between processors, and a mechanism for synchronizing the processors in a barrier style. In addition, it provides routines for collective communication on any arbitrary subset of processors, such as broadcast and parallel prefix, as well as fork and join operations. Compared to other BSP libraries, it provides a richer set of message passing functions, but direct remote memory access is currently not supported. Furthermore, some techniques used in the implementation of the PUB library deviate significantly from the techniques used in other libraries. 1 Introduction Most message passing libraries are based on pairwise sends and receives: for each send operation, a matching receive has to be issued on the destination processor. Widely available message passing libraries like PVM and MPI provide the user w...