Synonyms
FASS (space-f illing, self-a voiding, s imple, and self-s imilar)-curves
Definition
A space-filling curve is a continuous and surjective mapping from a 1D parameter interval, say [0,1], onto a higher-dimensional domain, say the unit square in 2D or the unit cube in 3D. Although this, at first glance, seems to be of a purely mathematical interest, space-filling curves and their recursive construction process have obtained a broad impact on scientific computing in general and on the parallelization of numerical algorithms for spatially discretized problems in particular.
Discussion
Introduction
Space-filling curves (SFC) were presented at the end of the nineteenth century – first by Peano (1890) and Hilbert (1891), and later by Moore, Lebesgue, Sierpinski, and others. The idea that some curves (i.e., something actually one-dimensional) may completely cover an area or a volume sounds somewhat strange and formerly caused mathematicians to call them “topological monsters.” The...
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Bibliography
Bader M, Schraufstetter S, Vigh CA, Behrens J (2008) Memory efficient adaptive mesh generation and implementation of multigrid algorithms using Sierpinski curves. Int J Comput Sci Eng 4(1):12–21
Bader M, Zenger Ch (2006) Cache oblivious matrix multiplication using an element ordering based on a Peano curve. Linear Algebra Appl 417(2–3):301–313
Barnes J, Hut P (1986) A hierarchical O(n log n) force-calculation algorithm. Nature 324:446–449
Brázdová V, Bowler DR (2008) Automatic data distribution and load balancing with space-filling curves: implementation in conquest. J Phy Condens Matt 20
Brenk M, Bungartz H-J, Mehl M, Muntean IL, Neckel T, Weinzierl T (2008) Numerical simulation of particle transport in a drift ratchet. SIAM J Sci Comput 30(6):2777–2798
Griebel M, Zumbusch GW (1999) Parallel multigrid in an adaptive PDE solver based on hashing and space-filling curves. Parallel Comput 25:827–843
Günther F, Krahnke A, Langlotz M, Mehl M, Pögl M, Zenger Ch (2004) On the parallelization of a cache-optimal iterative solver for PDES based on hierarchical data structures and space-filling curves. In: Recent Advances in Parallel Virtual Machine and Message Passing Interface: 11th European PVM/MPI Users Group Meeting Budapest, Hungary, September 19–22, 2004. Proceedings, vol 3241 of Lecture Notes in Computer Science. Springer, Heidelberg
Günther F, Mehl M, Pögl M, Zenger C (2006) A cache-aware algorithm for PDEs on hierarchical data structures based on space-filling curves. SIAM J Sci Comput 28(5):1634–1650
Hunter GM, Steiglitz K (1979) Operations on images using quad trees. IEEE Trans Pattern Analy Machine Intell PAMI-1(2):145–154
Jackings C, Tanimoto SL (1980) Octrees and their use in representing three-dimensional objects. Comp Graph Image Process 14(31):249–270
Rokhlin V, Greengard L (1987) A fast algorithms for particle simulations. J Comput Phys 73:325–348
Meagher D (1980) Octree encoding: A new technique for the representation, manipulation and display of arbitrary 3d objects by computer. Technical Report, IPL-TR-80-111
Mehl M, Weinzierl T, Zenger C (2006) A cache-oblivious self-adaptive full multigrid method. Numer Linear Algebr 13(2–3):275–291
Mitchell WF (2007) A refinement-tree based partitioning method for dynamic load balancing with adaptively refined grids. J Parallel Distrib Comput 67(4):417–429
Morton GM (1966) A computer oriented geodetic data base and a new technique in file sequencing. Technical Report, IBM Ltd., Ottawa, Ontario
Mundani R-P, Bungartz H-J, Niggl A, Rank E (2006) Embedding, organisation, and control of simulation processes in an octree-based cscw framework. In: Proceedings of the 11th International Conference on Computing in Civil and Building Engineering, Montreal, pp 3208–3215
Patra A, Oden JT (1995) Problem decomposition for adaptive hp finite element methods. Comput Syst Eng 6(2):97–109
Roberts S, Klyanasundaram S, Cardew-Hall M, Clarke W (1998) A key based parallel adaptive refinement technique for finite element methods. In: Proceedings of the Computational Techniques and Applications: CTAC ’97, Singapore, pp 577–584
Samet H (1980) Region representation: quadtrees from binary arrays. Comput Graph Image Process 13(1):88–93
Saxena M, Finnigan1 PM, Graichen CM, Hathaway AF, Parthasarathy VN (1995) Octree-based automatic mesh generation for non-manifold domains. Eng Comput 11(1):1–14
Schroeder WJ, Shephard MS (1988) A combined octree/delaunay method for fully automatic 3-d mesh generation. Int J Numer Methods Eng 26(1):37–55
Sundar H, Sampath RS, Biros G (2008) Bottom-up construction and 2:1 balance refinement of linear octrees in parallel. SIAM J Sci Comput 30(5):2675–2708
Weinzierl T (2009) A framework for parallel PDE solvers on multiscale adaptive Cartesian grids. Verlag Dr. Hut
Zumbusch GW (2001) On the quality of space-filling curve induced partitions. Z Angew Math Mech 81:25–28
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Bader, M., Bungartz, HJ., Mehl, M. (2011). Space-Filling Curves. In: Padua, D. (eds) Encyclopedia of Parallel Computing. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-09766-4_145
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DOI: https://doi.org/10.1007/978-0-387-09766-4_145
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