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.

Indexing schemes for grids based on space-filling 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 multi-dimensional spaces, indexing schemes for them have won considerable interest. Hilbert curves are the most simple and popular space-filling 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 r-dimensional Hilbert indexings for arbitrary r 2. Moreover, we generalize and simplify previous work and clarify the concept of Hilbert curves for multi-dimensional 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...

Indexing schemes for grids based on space-filling 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 space-filling 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.

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 Finite-Element, Finite-Volume and Finite-Difference 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...

this paper a parallelisable and cheap method based on space-filling 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