Sixteen space-filling curves and traversals for d-dimensional cubes and simplices

This article describes sixteen different ways to traverse d-dimensional space recursively in a way that is well-defined for any number of dimensions. Each of these traversals has distinct properties that may be beneficial for certain applications. Some of the traversals are novel, some have been known in principle but had not been described adequately for any number of dimensions, some of the traversals have been known. This article is the first to present them all in a consistent notation system. Furthermore, with this article, tools are provided to enumerate points in a regular grid in the order in which they are visited by each traversal. In particular, we cover: five discontinuous traversals based on subdividing cubes into 2^d subcubes: Z-traversal (Morton indexing), U-traversal, Gray-code traversal, Double-Gray-code traversal, and Inside-out traversal; two discontinuous traversals based on subdividing simplices into 2^d subsimplices: the Hill-Z traversal and the Maehara-reflected traversal; five continuous traversals based on subdividing cubes into 2^d subcubes: the Base-camp Hilbert curve, the Harmonious Hilbert curve, the Alfa Hilbert curve, the Beta Hilbert curve, and the Butz-Hilbert curve; four continuous traversals based on subdividing cubes into 3^d subcubes: the Peano curve, the Coil curve, the Half-coil curve, and the Meurthe curve. All of these traversals are self-similar in the sense that the traversal in each of the subcubes or subsimplices of a cube or simplex, on any level of recursive subdivision, can be obtained by scaling, translating, rotating, reflecting and/or reversing the traversal of the complete unit cube or simplex.Comment: 28 pages, 12 figures. v2: fixed a confusing typo on page 12, line

Evaluation of an efficient etack-RLE clustering concept for dynamically adaptive grids

This is the author accepted manuscript. The final version is available from the Society for Industrial and Applied Mathematics via the DOI in this record.Abstract. One approach to tackle the challenge of efficient implementations for parallel PDE simulations on dynamically changing grids is the usage of space-filling curves (SFC). While SFC algorithms possess advantageous properties such as low memory requirements and close-to-optimal partitioning approaches with linear complexity, they require efficient communication strategies for keeping and utilizing the connectivity information, in particular for dynamically changing grids. Our approach is to use a sparse communication graph to store the connectivity information and to transfer data block-wise. This permits efficient generation of multiple partitions per memory context (denoted by clustering) which - in combination with a run-length encoding (RLE) - directly leads to elegant solutions for shared, distributed and hybrid parallelization and allows cluster-based optimizations. While previous work focused on specific aspects, we present in this paper an overall compact summary of the stack-RLE clustering approach completed by aspects on the vertex-based communication that ease up understanding the approach. The central contribution of this work is the proof of suitability of the stack-RLE clustering approach for an efficient realization of different, relevant building blocks of Scientific Computing methodology and real-life CSE applications: We show 95% strong scalability for small-scale scalability benchmarks on 512 cores and weak scalability of over 90% on 8192 cores for finite-volume solvers and changing grid structure in every time step; optimizations of simulation data backends by writer tasks; comparisons of analytical benchmarks to analyze the adaptivity criteria; and a Tsunami simulation as a representative real-world showcase of a wave propagation for our approach which reduces the overall workload by 95% for parallel fully-adaptive mesh refinement and, based on a comparison with SFC-ordered regular grid cells, reduces the computation time by a factor of 7.6 with improved results and a factor of 62.2 with results of similar accuracy of buoy station dataThis work was partly supported by the German Research Foundation (DFG) as part of the Transregional Collaborative Research Centre “Invasive Computing” (SFB/TR 89)