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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 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...