Advertisement

Case Study: Cache Efficient Algorithms for Matrix Operations

  • Michael Bader
Chapter
First Online:
Part of the Texts in Computational Science and Engineering book series (TCSE, volume 9)

Abstract

In Chaps. 10 and 11, we discussed applications of space-filling curves for parallelisation, which were motivated by their locality properties. In the following two chapters, we will discuss further applications, which again exploit the intrinsic locality properties of space-filling curves. As both applications will focus on inherently cache-efficient algorithms, we will start with a short introduction to cache-memory architectures, and discuss the resulting requirements on cache-efficient algorithms.

Keywords

Matrix Block Access Pattern Cache Line Element Operation Cache Memory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

References

  1. 1.
    D. J. Abel and D. M. Mark. A comparative analysis of some two-dimensional orderings. International Journal of Geographical Information Systems, 4(1):21–31, 1990.Google Scholar
  2. 2.
    D. J. Abel and J. L. Smith. A data structure and algorithm based on a linear key for a rectangle retrieval problem. Computer Vision, Graphics, and Image Processing, 24:1–13, 1983.Google Scholar
  3. 3.
    K. Abend, T. J. Harley, and L. N. Kanal. Classification of binary random patterns. IEEE Transactions on Information Theory, IT-11(4):538–544, 1965.Google Scholar
  4. 4.
    M. Aftosmis, M. Berger, and J. Melton. Robust and efficient Cartesian mesh generation for component-based geometry. In 35 th Aerospace Sciences Meeting and Exhibit, 1997. AIAA 1997-0196.Google Scholar
  5. 5.
    M. J. Aftosmis, M. J. Berger, and S. M. Murman. Applications of space-filling curves to Cartesian methods for CFD. AIAA Paper 2004-1232, 2004.Google Scholar
  6. 6.
    J. Akiyama, H. Fukuda, H. Ito, and G. Nakamura. Infinite series of generalized Gosper space filling curves. In CJCDGCGT 2005, volume 4381 of Lecture Notes in Computer Science, pages 1–9, 2007.Google Scholar
  7. 7.
    I. Al-Furaih and S. Ranka. Memory hierarchy management for iterative graph structures. In Parallel Processing Symposium, IPPS/SPDP 1998, pages 298–302. IEEE Computer Society, 1998.Google Scholar
  8. 8.
    F. Alauzet and A. Loseille. On the use of space filling curves for parallel anisotropic mesh adaptation. In Proceedings of the 18th International Meshing Roundtable, pages 337–357. Springer, 2009.Google Scholar
  9. 9.
    J. Alber and R. Niedermeier. On multidimensional curves with Hilbert property. Theory of Computing Systems, 33:295–312, 2000.Google Scholar
  10. 10.
    S. Aluru and F. E. Sevilgen. Parallel domain decomposition and load balancing using space-filling curves. In Proceedings of the Fourth International Conference on High-Performance Computing, pages 230–235. IEEE Computer Society, 1997.Google Scholar
  11. 11.
    N. Amenta, S. Choi, and G. Rote. Incremental constructions con BRIO. In Proceedings of the Nineteenth ACM Symposium on Computational Geometry, pages 211–219, 2003.Google Scholar
  12. 12.
    M. Amor, F. Arguello, J. López, O. Plata, and E. L. Zapata. A data-parallel formulation for divide and conquer algorithms. The Computer Journal, 44(4):303–320, 2001.Google Scholar
  13. 13.
    E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. DuCroz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen. LAPACK: a portable linear algebra library for high-performance computers. Technical Report CS-90-105, LAPACK Working Note #20, University of Tennessee, Knoxville, TN, 1990.Google Scholar
  14. 14.
    J. A. Anderson, C. D. Lorenz, and A. Travesset. General purpose molecular dynamics simulations fully implemented on graphics processing units. Journal of Computational Physics, 227(10):5342–5359, 2008.Google Scholar
  15. 15.
    A. Ansari and A. Fineberg. Image data ordering and compression using Peano scan and LOT. IEEE Transactions on Consumer Electronics, 38(3):436–445, 1992.Google Scholar
  16. 16.
    L. Arge, M. De Berg, H. Haverkort, and K. Yi. The priority R-tree: A practically efficient and worst-case optimal R-tree. ACM Transactions on Algorithms, 4(1):9:1–9:29, 2008.Google Scholar
  17. 17.
    D. N. Arnold, A. Mukherjee, and L. Pouly. Locally adapted tetrahedral meshes using bisection. SIAM Journal on Scientific Computing, 22(2):431–448, 2000.Google Scholar
  18. 18.
    T. Asano, D. Ranjan, T. Roos, E. Welzl, and P. Widmayer. Space-filling curves and their use in the design of geometric data structures. Theoretical Computer Science, 181(1):3–15, 1997.Google Scholar
  19. 19.
    M. Bader. Exploiting the locality properties of Peano curves for parallel matrix multiplication. In Proceedings of the Euro-Par 2008, volume 5168 of Lecture Notes in Computer Science, pages 801–810, 2008.Google Scholar
  20. 20.
    M. Bader, C. Böck, J. Schwaiger, and C. Vigh. Dynamically adaptive simulations with minimal memory requirement – solving the shallow water equations using Sierpinksi curves. SIAM Journal on Scientific Computing, 32(1):212–228, 2010.Google Scholar
  21. 21.
    M. Bader, R. Franz, S. Guenther, and A. Heinecke. Hardware-oriented implementation of cache oblivious matrix operations based on space-filling curves. In Parallel Processing and Applied Mathematics, 7th International Conference, PPAM 2007, volume 4967 of Lecture Notes in Computer Science, pages 628–638, 2008.Google Scholar
  22. 22.
    M. Bader and A. Heinecke. Cache oblivious dense and sparse matrix multiplication based on Peano curves. In Proceedings of the PARA ’08, 9th International Workshop on State-of-the-Art in Scientific and Parallel Computing, volume 6126/6127 of Lecture Notes in Computer Science, 2010. In print.Google Scholar
  23. 23.
    M. Bader, S. Schraufstetter, C. A Vigh, and J. Behrens. Memory efficient adaptive mesh generation and implementation of multigrid algorithms using Sierpinski curves. International Journal of Computational Science and Engineering, 4(1):12–21, 2008.Google Scholar
  24. 24.
    M. Bader and C. Zenger. Cache oblivious matrix multiplication using an element ordering based on a Peano curve. Linear Algebra and Its Applications, 417(2–3):301–313, 2006.Google Scholar
  25. 25.
    M. Bader and C. Zenger. Efficient storage and processing of adaptive triangular grids using Sierpinski curves. In Computational Science – ICCS 2006, volume 3991 of Lecture Notes in Computer Science, pages 673–680, 2006.Google Scholar
  26. 26.
    Y. Bandou and S.-I. Kamata. An address generator for an n-dimensional pseudo-Hilbert scan in a hyper-rectangular parallelepiped region. In International Conference on Image Processing, ICIP 2000, pages 707–714, 2000.Google Scholar
  27. 27.
    R. Bar-Yehuda and C. Gotsman. Time/space tradeoffs for polygon mesh rendering. ACM Transactions on Graphics, 15(2):141–152, 1996.Google Scholar
  28. 28.
    J. Barnes and P. Hut. A hierarchical O(NlogN) force-calculation algorithm. Nature, 324:446–449, 1986.Google Scholar
  29. 29.
    J.J. Bartholdi III and P. Goldsman. Vertex-labeling algorithms for the Hilbert spacefilling curve. Software: Practice and Experience, 31(5):395–408, 2001.Google Scholar
  30. 30.
    J. J. Bartholdi III and P. Goldsman. Multiresolution indexing of triangulated irregular networks. IEEE Transactions on Visualization and Computer Graphics, 10(3):1–12, 2004.Google Scholar
  31. 31.
    J. J. Bartholdi III and L. K. Platzman. An O(NlogN) planar travelling salesman heuristic based on spacefilling curves. Operations Research Letters, 1(4):121–125, 1982.Google Scholar
  32. 32.
    J. J. Bartholdi III and L. K. Platzman. Heuristics based on spacefilling curves for combinatorial problems in the Euclidian space. Management Science, 34(3):291–305, 1988.Google Scholar
  33. 33.
    A. C. Bauer and A. K. Patra. Robust and efficient domain decomposition preconditioners for adaptive hp finite element approximations of linear elasticity with and without discontinuous coefficients. International Journal for Numerical Methods in Engineering, 59:337–364, 2004.Google Scholar
  34. 34.
    K.E. Bauman. The dilation factor of the Peano–Hilbert curve. Mathematical Notes, 80(5):609–620, 2006. Translated from Matematicheskie Zametki, vol. 80, no. 5, 2006, pp. 643–656. Original Russian Text Copyright 2006 by K. E. Bauman.Google Scholar
  35. 35.
    R. Bayer. The universal B-tree for multidimensional indexing. Technical Report TUM-I9637, Institut für Informatik, Technische Universität München, 1996.Google Scholar
  36. 36.
    J. Behrens. Multilevel optimization by space-filling curves in adaptive atmospheric modeling. In F. Hülsemann, M. Kowarschik, and U. Rüde, editors, Frontiers in Simulation - 18th Symposium on Simulation Techniques, pages 186–196. SCS Publishing House, Erlangen, 2005.Google Scholar
  37. 37.
    J. Behrens. Adaptive atmospheric modeling: key techniques in grid generation, data structures, and numerical operations with applications, volume 54 of Lecture Notes in Computational Science and Engineering. Springer, 2006.Google Scholar
  38. 38.
    J. Behrens and M. Bader. Efficiency considerations in triangular adaptive mesh refinement. Philosophical Transactions of the Royal Society A, 367:4577–4589, 2009. Theme Issue ’Mesh generation and mesh adaptation for large-scale Earth-system modelling’.Google Scholar
  39. 39.
    J. Behrens, N. Rakowsky, W. Hiller, D. Handorf, M. Läuter, J. Päpke, and K. Dethloff. Parallel adaptive mesh generator for atmospheric and oceanic simulation. Ocean Modelling, 10:171–183, 2005.Google Scholar
  40. 40.
    J. Behrens and J. Zimmermann. Parallelizing an unstructured grid generator with a space-filling curve approach. In Euro-Par 2000 Parallel Processing, volume 1900 of Lecture Notes in Computer Science, pages 815–823, 2000.Google Scholar
  41. 41.
    D. Bertsimas and M. Grigni. Worst-case example for the spacefilling curve heuristics for the Euclidian traveling salesman problem. Operations Research Letters, 8:241–244, 1989.Google Scholar
  42. 42.
    T. Bially. Space-filling curves: Their generation and their application to bandwidth reduction. IEEE Transactions on Information Theory, IT-15(6):658–664, 1969.Google Scholar
  43. 43.
    E. Bänsch. Local mesh refinement in 2 and 3 dimensions. IMPACT of Computing in Science and Engineering, 3(3):181–191, 1991.Google Scholar
  44. 44.
    A. Bogomjakov and C. Gotsman. Universal rendering sequences for transparent vertex caching of progressive meshes. Computer Graphics forum, 21(2):137–148, 2002.Google Scholar
  45. 45.
    E. Borel. Elements de la Theorie des Ensembles. Editions Albin Michel, Paris, 1949. Note IV: La courbe de Péano.Google Scholar
  46. 46.
    V. Brázdová and D. R. Bowler. Automatic data distribution and load balancing with space-filling curves: implementation in CONQUEST. Journal of Physics: Condensed Matter, 20, 2008.Google Scholar
  47. 47.
    G. Breinholt and Ch. Schierz. Algorithm 781: Generating Hilbert’s space-filling curve by recursion. ACM Transactions on Mathematical Software, 24(2):184–189, 1998.Google Scholar
  48. 48.
    M. Brenk, H.-J. Bungartz, M. Mehl, I. L. Muntean, T. Neckel, and T. Weinzierl. Numerical simulation of particle transport in a drift ratchet. SIAM Journal on Scientific Computing, 30(6):2777–2798, 2008.Google Scholar
  49. 49.
    K. Brix, S. Sorana Melian, S. Müller, and G. Schieffer. Parallelisation of multiscale-based grid adaption using space-filling curves. ESAIM: Proceedings, 29:108–129, 2009.Google Scholar
  50. 50.
    K. Buchin. Constructing Delauney triangulations along space-filling curves. In ESA 2009, volume 5757, pages 119–130, 2009.Google Scholar
  51. 51.
    E. Bugnion, T. Roos, R. Wattenhofer, and P. Widmayer. Space filling curves versus random walks. In Algorithmic Foundations of Geographic Information Systems, volume 1340, pages 199–211, 1997.Google Scholar
  52. 52.
    H.-J. Bungartz, M. Mehl, T. Neckel, and T. Weinzierl. The PDE framework Peano applied to fluid dynamics: an efficient implementation of a parallel multiscale fluid dynamics solver on octree-like adaptive Cartesian grids. Computational Mechanics, 46(1):103–114, 2010.Google Scholar
  53. 53.
    H.-J. Bungartz, M. Mehl, and T. Weinzierl. A parallel adaptive Cartesian PDE solver using space-filling curves. In E.W. Nagel, V.W. Walter, and W. Lehner, editors, Euro-Par 2006, Parallel Processing, 12th International Euro-Par Conference, volume 4128 of Lecture Notes in Computer Science, pages 1064–1074, 2006.Google Scholar
  54. 54.
    C. Burstedde, O. Ghattas, M. Gurnis, G. Stadler, E. Tan, T. Tu, L. C. Wilcox, and S. Zhong. Scalable adaptive mantle convection simulation on petascale supercomputers. In SC ’08: Proceedings of the 2008 ACM/IEEE Conference on Supercomputing, pages 1–15. IEEE Press, 2008.Google Scholar
  55. 55.
    C. Burstedde, L. C. Wilcox, and O. Ghattas. p4est: Scalable algorithms for parallel adaptive mesh refinement on forests of octrees. SIAM Journal on Scientific Computing, 33(3):1103–1133, 2011.Google Scholar
  56. 56.
    A. Buttari, J. Dongarra, J. Kurzak, J. Langou, P. Luszczek, and S. Tomov. The impact of multicore on math software. In Applied Parallel Computing, State of the Art in Scientific Computing, volume 4699 of Lecture Notes in Computer Science, pages 1–10, 2007.Google Scholar
  57. 57.
    A. Buttari, J. Langou, J. Kurzak, and J. Dongarra. A class of parallel tiled linear algebra algorithms for multicore architectures. Technical Report UT-CS-07-600, LAPACK Working Note #191, ICL, University Tennessee, 2007.Google Scholar
  58. 58.
    A. R. Butz. Convergence with Hilbert’s space filling curve. Journal of Computer and System Sciences, 3:128–146, 1969.Google Scholar
  59. 59.
    A. R. Butz. Alternative algorithm for Hilbert’s space-filling curve. IEEE Transactions on Computers, C-20(4):424–426, 1971.Google Scholar
  60. 60.
    A. C. Calder, B. C. Curtis, L. J. Dursi, B. Fryxell, P. MacNeice, K. Olson, P. Ricker, R. Rosner, F. X. Timmes, H. M. Tufo, J. W. Turan, M. Zingale, and G. Henry. High performance reactive fluid flow simulations using adaptive mesh refinement on thousands of processors. In Supercomputing ’00: Proceedings of the 2000 ACM/IEEE Conference on Supercomputing, page 56. IEEE Computer Society, 2000.Google Scholar
  61. 61.
    P. M. Campbell, K. D. Devine, J. E. Flaherty, L. G. Gervasio, and J. D. Terescoy. Dynamic octree load balancing using space-filling curves. Technical Report CS-03-01, Williams College Department of Computer Science, 2003.Google Scholar
  62. 62.
    X. Cao and Z. Mo. A new scalable parallel method for molecular dynamics based on cell-block data structure. In Parallel and Distributed Processing and Applications, 3358, pages 757–764, 2005.Google Scholar
  63. 63.
    J. Castro, M. Georgiopoulos, R. Demara, and A. Gonzalez. Data-partitioning using the hilbert space filling curves: Effect on the speed of convergence of Fuzzy ARTMAP for large database problems. Neural Networks, 18:967–984, 2005.Google Scholar
  64. 64.
    E. Cesaro. Remarques sur la courbe de von Koch. Atti della R. Accad. della Scienze fisiche e matem. Napoli, 12(15):1–12, 1905. Reprinted in: Opere scelte, a cura dell’Unione matematica italiana e col contributo del Consiglio nazionale delle ricerche, Vol. 2: Geometria, analisi, fisica matematica. Rome: Edizioni Cremonese, pp. 464-479, 1964.Google Scholar
  65. 65.
    S. Chatterjee, V. V. Jain, A. R. Lebeck, S. Mundhra, and M. Thottethodi. Nonlinear array layouts for hierarchical memory systems. In International Conference on Supercomputing (ICS’99). ACM, New York, 1999.Google Scholar
  66. 66.
    H.-L. Chen and Y.-I. Chang. Neighbor-finding based on space-filling curves. Information Systems, 30(3):205–226, 2005.Google Scholar
  67. 67.
    G. Chochia, M. Cole, and T. Heywood. Implementing the hierarchical PRAM on the 2d mesh: analyses and experiments. IEEE Symposium on Parallel and Distributed Processing, 0:587–595, 1995. Preprint on http://homepages.inf.ed.ac.uk/mic/Pubs/ECS-CSG-10-95.ps.gz.
  68. 68.
    W. J. Coirier and K. G. Powell. Solution-adaptive Cartesian cell approach for viscous and inviscid fluid flows. AIAA Journal, 34(5):938–945, 1996.Google Scholar
  69. 69.
    A. J. Cole. A note on space-filling curves. Software: Practice and Experience, 13:1181–1189, 1983.Google Scholar
  70. 70.
    S. Craver, B.-L. Yeo, and M. Yeung. Multilinearization data structure for image browsing. In Storage and Retrieval for Image and Video Databases VII, volume 3656, pages 155–166, 1998.Google Scholar
  71. 71.
    R. Dafner, D. Cohen-Or, and Y. Matias. Context-based space filling curves. Computer Graphics Forum, 19(3):209–218, 2000.Google Scholar
  72. 72.
    K. Daubner. Geometrische Modellierung mittels Oktalbäumen und Visualisierung von Simulationsdaten aus der Strömungsmechanik. Institut für Informatik, Technische Universität München, 2005.Google Scholar
  73. 73.
    L. De Floriani and E. Puppo. Hierarchical triangulation for multiresolution surface description. ACM Transactions on Graphics, 14(4):363–411, 1995.Google Scholar
  74. 74.
    J. M. Dennis. Partitioning with space-filling curves on the cubed-sphere. In Proceedings of the 17th International Symposium on Parallel and Distributed Processing, pages 269–275. IEEE Computer Society, 2003.Google Scholar
  75. 75.
    J. M. Dennis. Inverse space-filling curve partitioning of a global ocean model. In Parallel and Distributed Processing Symposium, IPDPS 2007, pages 1–10. IEEE International, 2007.Google Scholar
  76. 76.
    K. D. Devine, E. G. Boman, R. T. Heaphy, B. A. Hendrickson, J. D. Terescoy, J. Falk, J. E. Flaherty, and L. G. Gervasio. New challenges in dynamic load balancing. Applied Numerical Mathematics, 52:133–152, 2005.Google Scholar
  77. 77.
    R. Diekmann, J. Hungershöfer, M. Lux, M. Taenzer, and J.-M. Wierum. Using space filling curves for efficient contact searching. In 16th IMACS World Congress, 2000.Google Scholar
  78. 78.
    J. J. Dongarra, J. Du Croz, S. Hammarling, and I. Duff. A set of level 3 basic linear algebra subprograms. ACM Transactions on Mathematical Software, 16(1):1–28, 1990.Google Scholar
  79. 79.
    J. Dreher and R. Grauer. Racoon: A parallel mesh-adaptive framework for hyperbolic conservation laws. Parallel Computing, 31(8–9):913–932, 2005.Google Scholar
  80. 80.
    M. Duchaineau, M. Wolinsky, D. E. Sigeti, M. C. Miller, C. Aldrich, and M. B. Mineev-Weinstein. ROAMing terrain: real-time optimally adapting meshes. In VIS ’97: Proceedings of the 8th Conference on Visualization ’97, pages 81–88. IEEE Computer Society Press, 1997.Google Scholar
  81. 81.
    E. Elmroth, F. Gustavson, I. Jonsson, and B. Kågström. Recursive blocked algorithms and hybrid data structures for dense matrix library software. SIAM Review, 46(1):3–45, 2004.Google Scholar
  82. 82.
    M. Elshafei and M. S. Ahmed. Fuzzification using space-filling curves. Intelligent Automation and Soft Computing, 7(2):145–157, 2001.Google Scholar
  83. 83.
    M. Elshafei-Ahmed. Fast methods for split codebooks. Signal Processing, 80:2553–2565, 2000.Google Scholar
  84. 84.
    W. Evans, D. Kirkpatrick, and G. Townsend. Right-triangulated irregular networks. Algorithmica, 30(2):264–286, 2001.Google Scholar
  85. 85.
    C. Faloutsos. Analytical results on the quadtree decomposition of arbitrary rectangles. Pattern Recognition Letters, 13:31–40, 1992.Google Scholar
  86. 86.
    C. Faloutsos and S. Roseman. Fractals for secondary key retrieval. In Proceedings of the Eighth ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, pages 247–252, 1989.Google Scholar
  87. 87.
    R. Finkel and Bentley J. L. Quad trees: a data structure for retrieval on composite keys. Acta Informatica, 4(1):1–9, 1974.Google Scholar
  88. 88.
    J. E. Flaherty, R. M. Loy, M. S. Shephard, B. K. Szymanski, J. D. Terescoy, and L. H. Ziantz. Adaptive local refinement with octree load balancing for the parallel solution of three-dimensional conservation laws. Journal of Parallel and Distributed Computing, 47:139–152, 1997.Google Scholar
  89. 89.
    A. C. Frank. Organisationsprinzipien zur Integration von geometrischer Modellierung, numerischer Simulation und Visualisierung. Herbert Utz Verlag, Dissertation, Institut für Informatik, Technische Universität München, 2000.Google Scholar
  90. 90.
    J. Frens and D. S. Wise. Auto-blocking matrix-multiplication or tracking BLAS3 performance from source code. In Proceedings of the 6th ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming, pages 206–216, 1997.Google Scholar
  91. 91.
    J. Frens and D. S. Wise. QR factorization with Morton-ordered quadtree matrices for memory re-use and parallelism. In Proceedings of the 9th ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming, pages 144–154, 2003.Google Scholar
  92. 92.
    M. Frigo, C. E. Leiserson, H. Prokop, and S. Ramachandran. Cache-oblivious algorithms. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science, pages 285–297, 1999.Google Scholar
  93. 93.
    H. Fukuda, M. Shimizu, and G. Nakamura. New Gosper space filling curves. In Proceedings of the International Conference on Computer Graphics and Imaging (CGIM2001), pages 34–38, 2001.Google Scholar
  94. 94.
    J. Gao and J. M. Steele. General spacefilling curve heuristics and limit theory for the traveling salesman problem. Journal of Complexity, 10:230–245, 1994.Google Scholar
  95. 95.
    M. Gardner. Mathematical games – in which “monster” curves force redefinition of the word “curve”. Scientific American, 235:124–133, Dec. 1976.Google Scholar
  96. 96.
    I. Gargantini. An effective way to represent quadtrees. Communications of the ACM, 25(12):905–910, 1982.Google Scholar
  97. 97.
    T. Gerstner. Multiresolution Compression and Visualization of Global Topographic Data. GeoInformatica, 7(1):7–32, 2003. (shortened version in Proc. Spatial Data Handling 2000, P. Forer, A.G.O. Yeh, J. He (eds.), pp. 14–27, IGU/GISc, 2000, also as SFB 256 report 29, Univ. Bonn, 1999).Google Scholar
  98. 98.
    P. Gibbon, W. Frings, S. Dominiczak, and B. Mohr. Performance analysis and visualization of the n-body tree code PEPC on massively parallel computers. In Parallel Computing: Current & Future Issues of High-End Computing, Proceedings of the International Conference ParCo 2005, volume 33 of NIC Series, pages 367–374, 2006.Google Scholar
  99. 99.
    W. Gilbert. A cube-filling Hilbert curve. The Mathematical Intelligencer, 6(3):78–79, 1984.Google Scholar
  100. 100.
    J. Gips. Shape Grammars and their Uses. Interdisciplinary Systems Research. Birkhäuser Verlag, 1975.Google Scholar
  101. 101.
    L. M. Goldschlager. Short algorithms for space-filling curves. Software: Practice and Experience, 11:99–100, 1981.Google Scholar
  102. 102.
    M. F. Goodchild and D. M. Mark. The fractal nature of geographic phenomena. Annals of the Association of American Geographers, 77(2):265–278, 1987.Google Scholar
  103. 103.
    K. Goto and R. van de Geijn. On reducing TLB misses in matrix multiplication. FLAME working note #9. Technical Report TR-2002-55, The University of Texas at Austin, Department of Computer Sciences, 2002.Google Scholar
  104. 104.
    K. Goto and R. A. van de Geijn. Anatomy of a high-performance matrix multiplication. ACM Transactions on Mathematical Software, 34(3):12:1–12:25, 2008.Google Scholar
  105. 105.
    C. Gotsman and M. Lindenbaum. On the metric properties of space-filling curves. IEEE Transactions on Image Processing, 5(5):794–797, 1996.Google Scholar
  106. 106.
    P. Gottschling, D. S. Wise, and A. Joshi. Generic support of algorithmic and structural recursion for scientific computing. International Journal of Parallel, Emergent and Distributed Systems, 24(6):479–503, 2009.Google Scholar
  107. 107.
    M. Griebel and M. A. Schweitzer. A particle-partition of unity method—part IV: Parallelization. In Meshfree Methods for Partial Differential Equations, volume 26 of Lecture Notes in Computational Science and Engineering, pages 161–192, 2002.Google Scholar
  108. 108.
    M. Griebel and G. Zumbusch. Parallel multigrid in an adaptive PDE solver based on hashing and space-filling curves. Parallel Computing, 27(7):827–843, 1999.Google Scholar
  109. 109.
    M. Griebel and G. Zumbusch. Hash based adaptive parallel multilevel methods with space-filling curves. In H. Rollnik and D. Wolf, editors, NIC Symposium 2001, volume 9 of NIC Series, pages 479–492. Forschungszentrum Jülich, 2002.Google Scholar
  110. 110.
    J. G. Griffiths. Table-driven algorithm for generating space-filling curves. Computer Aided Design, 17(1):37–41, 1985.Google Scholar
  111. 111.
    J. G. Griffiths. An algorithm for displaying a class of space-filling curves. Software: Practice and Experience, 16(5):403–411, 1986.Google Scholar
  112. 112.
    J. Gunnels, F. Gustavson, K. Pingali, and K. Yotov. Is cache-oblivious DGEMM viable? In Applied Parallel Computing. State of the Art in Scientific Computing, volume 4699 of Lecture Notes in Computer Science, pages 919–928, 2007.Google Scholar
  113. 113.
    J. A. Gunnels, F. G. Gustavson, G. M. Henry, and R. A. van de Geijn. FLAME: formal linear algebra methods environment. ACM Transactions on Mathematical Software, 27(4):422–455, 2001.Google Scholar
  114. 114.
    F. Günther, M. Mehl, M. Pögl, and C. Zenger. A cache-aware algorithm for PDEs on hierarchical data structures based on space-filling curves. SIAM Journal on Scientific Computing, 28(5):1634–1650, 2006.Google Scholar
  115. 115.
    F. Gustavson, L. Karlsson, and B. Kågström. Parallel and cache-efficient in-place matrix storage format conversion. Transactions on Mathematical Software, 38(3):17:1–17:32, 2012.Google Scholar
  116. 116.
    F. G. Gustavson. Recursion leads to automatic variable blocking for dense linear-algebra algorithms. IBM Journal of Research and Development, 41(6), 1999.Google Scholar
  117. 117.
    G. Haase, M. Liebmann, and G. Plank. A Hilbert-order multiplication scheme for unstructured sparse matrices. International Journal of Parallel, Emergent and Distributed Systems, 22(4):213–220, 2007.Google Scholar
  118. 118.
    C. H. Hamilton and A. Rau-Chaplin. Compact Hilbert indices: Space-filling curves for domains with unequal side lengths. Information Processing Letters, 105:155–163, 2008.Google Scholar
  119. 119.
    H. Han and C.-W. Tseng. Exploiting locality for irregular scientific codes. IEEE Transactions on Parallel and Distributed Systems, 17(7):606–618, 2006.Google Scholar
  120. 120.
    J. Hartmann, A. Krahnke, and C. Zenger. Cache efficient data structures and algorithms for adaptive multidimensional multilevel finite element solvers. Applied Numerical Mathematics, 58(4):435–448, 2008.Google Scholar
  121. 121.
    A. Haug. Sierpinski-Kurven zur speichereffizienten numerischen Simulation auf adaptiven Tetraedergittern. Diplomarbeit, Fakultät für Informatik, Technische Universität München, 2006.Google Scholar
  122. 122.
    H. Haverkort and F. van Walderveen. Four-dimensional Hilbert curves for R-trees. In Proceedings of the Workshop on Algorithm Engineering and Experiments (ALENEX), pages 63–73, 2009.Google Scholar
  123. 123.
    H. Haverkort and F. van Walderveen. Locality and bounding-box quality of two-dimensional space-filling curves. Computational Geometry: Theory and Applications, 43(2):131–147, 2010.Google Scholar
  124. 124.
    G. Heber, R. Biswas, and G. R. Gao. Self-avoiding walks over adaptive unstructured grids. Concurrency: Practice and Experience, 12:85–109, 2000.Google Scholar
  125. 125.
    D. J. Hebert. Symbolic local refinement of tetrahedral grids. Journal of Symbolic Computation, 17(5):457–472, 1994.Google Scholar
  126. 126.
    D. J. Hebert. Cyclic interlaced quadtree algorithms for quincunx multiresolution. Journal of Algorithms, 27:97–128, 1998.Google Scholar
  127. 127.
    A. Heinecke and M. Bader. Parallel matrix multiplication based on space-filling curves on shared memory multicore platforms. In Proceedings of the 2008 Computing Frontiers Conference and co-located workshops: MAW’08 and WRFT’08, pages 385–392. ACM, 2008.Google Scholar
  128. 128.
    B. Hendrickson. Load balancing fictions, falsehoods and fallacies. Applied Mathematical Modelling, 25:99–108, 2000.Google Scholar
  129. 129.
    B. Hendrickson and K. Devine. Dynamic load balancing in computational mechanics. Computer Methods in Applied Mechanical Engineering, 184:485–500, 2000.Google Scholar
  130. 130.
    J. R. Herrero and J. J. Navarro. Analysis of a sparse hypermatrix Cholesky with fixed-sized blocking. Applicable Algebra in Engineering, Communication and Computing, 18(3):279–295, 2007.Google Scholar
  131. 131.
    D. Hilbert. Über die stetige Abbildung einer Linie auf ein Flächenstück. Mathematische Annalen, 38:459–460, 1891. Available online on the Göttinger Digitalisierungszentrum.Google Scholar
  132. 132.
    J.-W. Hong and H. T. Kung. I/O complexity: the red-blue pebble game. In Proceedings of ACM Symposium on Theory of Computing, pages 326–333, 1981.Google Scholar
  133. 133.
    H. Hoppe. Optimization of mesh locality for transparent vertex caching. In SIGGRAPH ’99: Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques, pages 269–276. ACM Press/Addison-Wesley Publishing Co., 1999.Google Scholar
  134. 134.
    Y. C. Hu, A. Cox, and W. Zwaenepoel. Improving fine-grained irregular shared-memory benchmarks by data reordering. In Proceedings of the 2000 ACM/IEEE Conference on Supercomputing, pages # 33. IEEE Computer Society, 2000.Google Scholar
  135. 135.
    J. Hungershöfer and J.-M. Wierum. On the quality of partitions based on space-filling curves. In ICCS 2002, volume 2331 of Lecture Notes in Computer Science, pages 36–45, 2002.Google Scholar
  136. 136.
    G. M. Hunter and K. Steiglitz. Operations on images using quad trees. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-1(2):145–154, 1979.Google Scholar
  137. 137.
    L. M. Hwa, M. A. Duchaineau, and K. i. Joy. Adaptive 4-8 texture hierarchies. In VIS ’04: Proceedings of the Conference on Visualization ’04, pages 219–226. IEEE Computer Society, 2004.Google Scholar
  138. 138.
    C. Jackings and S. L. Tanimoto. Octrees and their use in representing three-dimensional objects. Computer Graphics and Image Processing, 14(31):249–270, 1980.Google Scholar
  139. 139.
    H. V. Jagadish. Linear clustering of objects with multiple attributes. ACM SIGMOD Record, 19(2):332–342, 1990.Google Scholar
  140. 140.
    H. V. Jagadish. Analysis of the Hilbert curve for representing two-dimensional space. Information Processing Letters, 62(1):17–22, 1997.Google Scholar
  141. 141.
    G. Jin and J. Mellor-Crummey. SFCGen: a framework for efficient generation of multi-dimensional space-filling curves by recursion. ACM Transactions on Mathematical Software, 31(1):120–148, 2005.Google Scholar
  142. 142.
    Bentley J. L. and D. F. Stanat. Analysis of range searches in quad trees. Information Processing Letters, 3(6):170–173, 1975.Google Scholar
  143. 143.
    M. Kaddoura, C.-W. Ou, and S. Ranka. Partitioning unstructured computational graphs for nonuniform and adaptive environments. IEEE Concurrency, 3(3):63–69, 1995.Google Scholar
  144. 144.
    C. Kaklamanis and G. Persiano. Branch-and-bound and backtrack search on mesh-connected arrays of processors. Mathematical Systems Theory, 27:471–489, 1994.Google Scholar
  145. 145.
    S. Kamata, R. O. Eason, and Y. Bandou. A new algorithm for n-dimensional Hilbert scanning. IEEE Transactions on Image Processing, 8(7):964–973, 1999.Google Scholar
  146. 146.
    I. Kamel and C. Faloutsos. On packing R-trees. In Proceedings of the Second International ACM Conference on Information and Knowledge Management, pages 490–499. ACM New York, 1993.Google Scholar
  147. 147.
    E. Kawaguchi and T. Endo. On a method of binary-picture representation and its application to data compression. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-2(1):27–35, 1980.Google Scholar
  148. 148.
    A. Klinger. Data structures and pattern recognition. In Proceedings of the First International Joint Conference on Pattern Recognition, pages 497–498. IEEE, 1973.Google Scholar
  149. 149.
    A. Klinger and C. R. Dyer. Experiments on picture representation using regular decomposition. Computer Graphics and Image Processing, 5:68–106, 1976.Google Scholar
  150. 150.
    K. Knopp. Einheitliche Erzeugung und Darstellung der Kurven von Peano, Osgood und von Koch. Archiv der Mathematik und Physik, 26:103–115, 1917.Google Scholar
  151. 151.
    I. Kossaczky. A recursive approach to local mesh refinement in two and three dimensions. Journal of Computational and Applied Mathematics, 55:275–288, 1994.Google Scholar
  152. 152.
    R. Kriemann. Parallel ℋ-matrix arithmetics on shared memory systems. Computing, 74:273–297, 2005.Google Scholar
  153. 153.
    J. P. Lauzon, D. M. Mark, L. Kikuchi, and J. A. Guevara. Two-dimensional run-encoding for quadtree representation. Computer Vision, Graphics, and Image Processing, 30(1):56–69, 1985.Google Scholar
  154. 154.
    J. K. Lawder and P. J. H. King. Querying multi-dimensional data indexed using the Hilbert space-filling curve. ACM SIGMOD Record, 30(1):19–24, 2001.Google Scholar
  155. 155.
    J. K. Lawder and P. J. H. King. Using state diagrams for Hilbert curve mappings. International Journal of Computer Mathematics, 78:327–342, 2001.Google Scholar
  156. 156.
    D. Lea. Digital and Hilbert k-d-trees. Information Processing Letters, 27:35–41, 1988.Google Scholar
  157. 157.
    H. L. Lebesgue. Leçons sur l’intégration et la recherche des fonctions primitives. Gauthier-Villars, Paris, 1904. Available online on the University of Michigan Historical Math Collection.Google Scholar
  158. 158.
    J.-H. Lee and Y.-C. Hsueh. Texture classification method using multiple space filling curves. Pattern Recognition Letters, 15:1241–1244, 1994.Google Scholar
  159. 159.
    M. Lee and H. Samet. Navigating through triangle meshes implemented as linear quadtrees. ACM Transactions on Graphics, 19(2):79–121, 2000.Google Scholar
  160. 160.
    A. Lempel and J. Ziv. Compression of two-dimensional data. IEEE Transactions on Information Theory, IT-32(1):2–8, 1986.Google Scholar
  161. 161.
    S. Liao, M. A. Lopez, and S. T. Leutenegger. High dimensional similarity search with space filling curves. In Proceedings of the 17th International Conference on Data Engineering, pages 615–622. IEEE Computer Society, 2000.Google Scholar
  162. 162.
    A. Lindenmayer. Mathematical models for cellular interactions in development. Journal of Theoretical Biology, 18:280–299, 1968.Google Scholar
  163. 163.
    P. Lindstrom, D. Koller, W. Ribarsky, L. F. Hodges, N. Faust, and G. A. Turner. Real-time, continuous level of detail rendering of height fields. In SIGGRAPH ’96: Proceedings of the 23rd Annual Conference on Computer Graphics and Interactive Techniques, pages 109–118. ACM, 1996.Google Scholar
  164. 164.
    P. Lindstrom and V. Pascucci. Terrain simplification simplified: A general framework for view-dependent out-of-core visualization. Technical Report UCRL-JC-147847, 2002.Google Scholar
  165. 165.
    A Liu and B. Joe. On the shape of tetrahedra from bisection. Mathematics of Computation, 63:141–154, 1994.Google Scholar
  166. 166.
    A Liu and B. Joe. Quality local refinement of tetrahedral meshes based on bisection. SIAM Journal on Scientific Computing, 16:1269–1291, 1995.Google Scholar
  167. 167.
    P. Liu and S. N. Bhatt. Experiences with parallel n-body simulation. IEEE Transactions on Parallel and Distributed Systems, 11(12):1306–1323, 2000.Google Scholar
  168. 168.
    X. Liu. Four alternative patterns of the Hilbert curve. Applied Mathematics and Communication, 147:741–752, 2004.Google Scholar
  169. 169.
    X. Liu and G. Schrack. Encoding and decoding the Hilbert order. Software: Practice and Experience, 26(12):1335–1346, 1996.Google Scholar
  170. 170.
    X. Liu and G. F. Schrack. A new ordering strategy applied to spatial data processing. International Journal of Geographical Information Science, 12(1):3–22, 1998.Google Scholar
  171. 171.
    Y. Liu and J. Snoeyink. A comparison of five implementations of 3d Delaunay tessellation. Combinatorial and Computational Geometry, 52:439–458, 2005.Google Scholar
  172. 172.
    J. Luitjens, M. Berzins, and T. Henderson. Parallel space-filling curve generation through sorting. Concurrency and Computation: Practice and Experience, 19:1387–1402, 2007.Google Scholar
  173. 173.
    G. Mainar-Ruiz and J.-C. Perez-Cortes. Approximate nearest neighbor search using a single space-filling curve and multiple representations of the data points. In 18th International Conference on Pattern Recognition, 2006 – ICPR 2006, pages 502–505, 2006.Google Scholar
  174. 174.
    B. Mandelbrot. The Fractal Geometry of Nature. Freeman and Company, 1977, 1982, 1983.Google Scholar
  175. 175.
    Y. Matias and A. Shamir. A video scrambling technique based on space filling curves. In Advances in Cryptology – CRYPTO ’87, volume 293, pages 398–417, 1987.Google Scholar
  176. 176.
    J. M. Maubach. Local bisection refinement for n-simplicial grids generated by reflection. SIAM Journal on Scientific Computing, 16(1):210–227, 1995.Google Scholar
  177. 177.
    J. M. Maubach. Space-filling curves for 2-simplicial meshes created with bisections and reflections. Applications of Mathematics, 3:309–321, 2005.Google Scholar
  178. 178.
    D. Meagher. Geometric modelling using octree encoding. Computer Graphics and Image Processing, 19:129–147, 1980.Google Scholar
  179. 179.
    D. Meagher. Octree encoding: A new technique for the representation, manipulation and display of arbitrary 3d objects by computer. Technical report, Image Processing Laboratory, Rensselaer Polytechnic Institute, 1980.Google Scholar
  180. 180.
    M. Mehl, M. Brenk, H.-J. Bungartz, K. Daubner, I. L. Muntean, and T. Neckel. An Eulerian approach for partitioned fluid-structure simulations on Cartesian grids. Computational Mechanics, 43(1):115–124, 2008.Google Scholar
  181. 181.
    M. Mehl, T. Neckel, and P. Neumann. Navier-stokes and lattice-boltzmann on octree-like grids in the Peano framework. International Journal for Numerical Methods in Fluids, 65(1):67–86, 2010.Google Scholar
  182. 182.
    M. Mehl, T. Neckel, and T. Weinzierl. Concepts for the efficient implementation of domain decomposition approaches for fluid-structure interactions. In U. Langer, M. Discacciati, D.E. Keyes, O.B. Widlund, and W. Zulehner, editors, Domain Decomposition Methods in Science and Engineering XVII, volume 60 of Lecture Notes in Science an Enginnering, 2008.Google Scholar
  183. 183.
    M. Mehl, T. Weinzierl, and C. Zenger. A cache-oblivious self-adaptive full multigrid method. Numerical Linear Algebra with Applications, 13(2–3):275–291, 2006.Google Scholar
  184. 184.
    J. Mellor-Crummey, D. Whalley, and K. Kennedy. Improving memory hierarchy performance for irregular applications using data and computation reorderings. International Journal of Parallel Programming, 29(3):217–247, 2001.Google Scholar
  185. 185.
    N. Memon, D. L. Neuhoff, and S. Shende. An analysis of some common scanning techniques for lossless image coding. IEEE Transactions on Image Processing, 9(11):1837–1848, 2000.Google Scholar
  186. 186.
    R. Miller and Q. F. Stout. Mesh computer algorithms for computational geometry. IEEE Transactions on Computers, 38(3):321–340, 1989.Google Scholar
  187. 187.
    W. F. Mitchell. Adaptive refinement for arbitrary finite-element spaces with hierarchical bases. Journal of Computational and Applied Mathematics, 36:65–78, 1991.Google Scholar
  188. 188.
    W. F. Mitchell. Hamiltonian paths through two- and three-dimensional grids. Journal of Research of the National Institute of Standards and Technology, 110(2):127–136, 2005.Google Scholar
  189. 189.
    W. F. Mitchell. A refinement-tree based partitioning method for dynamic load balancing with adaptively refined grids. Journal of Parallel and Distributed Computing, 67(4):417–429, 2007.Google Scholar
  190. 190.
    G. Mitchison and R. Durbin. Optimal numberings of an N ×N-array. SIAM Journal on Algebraic and Discrete Methods, 7(4):571–582, 1986.Google Scholar
  191. 191.
    B. Moon, H. V. Jagadish, C. Faloutsos, and J. H. Saltz. Analysis of the clustering properties of the Hilbert space-filling curve. IEEE Transactions on Knowledge and Data Engineering, 13(1):124–141, 2001.Google Scholar
  192. 192.
    A. Mooney, J. G. Keating, and D. M. Heffernan. A detailed study of the generation of optically detectable watermarks using the logistic map. Chaos, Solitons and Fractals, 30(5):1088–1097, 2006.Google Scholar
  193. 193.
    E. H. Moore. On certain crinkly curves. Transactions of the American Mathematical Society, 1(1):72–90, 1900. Available online on JSTOR.Google Scholar
  194. 194.
    G. M. Morton. A computer oriented geodetic data base and a new technique in file sequencing. Technical report, IBM Ltd., Ottawa, Ontario, 1966.Google Scholar
  195. 195.
    R. D. Nair, H.-W. Choi, and H. M. Tufo. Computational aspects of a scalable high-order discontinuous Galerkin atmospheric dynamical core. Computers & Fluids, 38(2):309–319, 2009.Google Scholar
  196. 196.
    R. Niedermeier, K. Reinhardt, and P. Sanders. Towards optimal locality in mesh-indexings. Discrete Applied Mathematics, 117:211–237, 2002.Google Scholar
  197. 197.
    M. G. Norman and P. Moscato. The Euclidian traveling salesman problem and a space-filling curve. Chaos, Solitons & Fractals, 6:389–397, 1995.Google Scholar
  198. 198.
    A. Null. Space-filling curves, or how to waste time with a plotter. Software: Practice and Experience, 1:403–410, 1971.Google Scholar
  199. 199.
    Y. Ohno and K. Ohyama. A catalog of non-symmetric self-similar space-filling curves. Journal of Recreational Mathematics, 23(4):247–254, 1991.Google Scholar
  200. 200.
    Y. Ohno and K. Ohyama. A catalog of symmetric self-similar space-filling curves. Journal of Recreational Mathematics, 23(3):161–174, 1991.Google Scholar
  201. 201.
    M. A. Oliver and N. E. Wiseman. Operations on quadtree encoded images. The Computer Journal, 26(1):83–91, 1983.Google Scholar
  202. 202.
    J. A. Orenstein. Spatial query processing in an object-oriented database system. ACM SIGMOD Record, 15(2):326–336, 1986.Google Scholar
  203. 203.
    J. A. Orenstein and F. A. Manola. PROBE spatial data modeling in an image database and query processing application. IEEE Transactions on Software Engineering, 14(5):611–629, 1988.Google Scholar
  204. 204.
    J. A. Orenstein and T. H. Merrett. A class of data structures for associative searching. In Proceedings of the 3rd ACM SIGACT-SIGMOD Symposium on Principles of Database Systems, pages 181–190. ACM, 1984.Google Scholar
  205. 205.
    C.-W. Ou, M. Gunwani, and S. Ranka. Architecture-independent locality-improving transformations of computational graphs embedded in k-dimensions. In ICS ’95: Proceedings of the 9th International Conference on Supercomputing, pages 289–298, 1995.Google Scholar
  206. 206.
    C.-W. Ou, S. Ranka, and G. Fox. Fast and parallel mapping algorithms for irregular problems. Journal of Supercomputing, 10:119–140, 1996.Google Scholar
  207. 207.
    R. Pajarola. Large scale terrain visualization using the restricted quadtree triangulation. In VIS ’98: Proceedings of the Conference on Visualization ’98, pages 19–26. IEEE Computer Society Press, 1998.Google Scholar
  208. 208.
    S. Papadomanolakis, A. Ailamaki, J. C. Lopez, T. Tu, D. R. O’Hallaron, and G. Heber. Efficient query processing on unstructured tetrahedral meshes. In SIGMOD ’06: Proceedings of the 2006 ACM SIGMOD International Conference on Management of Data, pages 551–562, 2006.Google Scholar
  209. 209.
    M. Parashar, J. C. Browne, C. Edwards, and K. Klimkowski. A common data management infrastructure for parallel adaptive algorithms for PDE solutions. In Proceedings of the 1997 ACM/IEEE Conference on Supercomputing, pages 1–22. ACM Press, 1997.Google Scholar
  210. 210.
    M. Parashar and J. C. Browne. On partitioning dynamic adaptive grid hierarchies. In Proceedings of the 29th Annual Hawaii International Conference on System Sciences, pages 604–613, 1996.Google Scholar
  211. 211.
    V. Pascucci. Isosurface computation made simple: Hardware acceleration, adaptive refinement and tetrahedral stripping. In Joint Eurographics-IEEE TVCG Symposium on Visualization (VisSym), pages 293–300, 2004.Google Scholar
  212. 212.
    A. Patra and J. T. Oden. Problem decomposition for adaptive hp finite element methods. Computing Systems in Engineering, 6(2):97–109, 1995.Google Scholar
  213. 213.
    A. K. Patra, A. Laszloffy, and J. Long. Data structures and load balancing for parallel adaptive hp finite-element methods. Computers & Mathematics with Applications, 46(1):105–123, 2003.Google Scholar
  214. 214.
    G. Peano. Sur une courbe, qui remplit toute une aire plane. Mathematische Annalen, 36:157–160, 1890. Available online on the Göttinger Digitalisierungszentrum.Google Scholar
  215. 215.
    A. Pérez, S. Kamata, and E. Kawagutchi. Peano scanning of arbitrary size images. In 11th IAPR International Conference on Pattern Recognition, 1992. Vol.III. Conference C: Image, Speech and Signal Analysis, pages 565–568. IEEE, 1992.Google Scholar
  216. 216.
    E. Perlman, R. Burns, Y. Li, and C. Meneveau. Data exploration of turbulence simulations using a database cluster. In SC ’07: Proceedings of the 2007 ACM/IEEE Conference on Supercomputing, pages 1–11, 2007.Google Scholar
  217. 217.
    J. R. Pilkington and S. B. Baden. Partitioning with spacefilling curves. CSE Technical Report Number CS94–349, University of California, San Diego, 1994.Google Scholar
  218. 218.
    J. R. Pilkington and S. B. Baden. Dynamic partitioning of non-uniform structured workloads with spacefilling curves. IEEE Transactions on Parallel and Distributed Systems, 7(3):288–300, 1996.Google Scholar
  219. 219.
    L. K. Platzman and J. J. Bartholdi III. Spacefilling curves and the planar travelling salesman problem. Journal of the ACM, 36(4):719–737, 1989.Google Scholar
  220. 220.
    G. Polya. Über eine Peanosche Kurve. Bulletin de l’Académie des Sciences de Cracovie, Série A, pages 1–9, 1913.Google Scholar
  221. 221.
    P. Prusinkiewicz. Graphical applications of L-systems. In Proceedings of Graphics Interface ’86/Vision Interface ’86, pages 247–253, 1986.Google Scholar
  222. 222.
    P. Prusinkiewicz and A. Lindenmayer. The Algorithmic Beauty of Plants. Springer, 1990.Google Scholar
  223. 223.
    P. Prusinkiewicz, A. Lindenmayer, and F. D. Fracchia. Synthesis of space-filling curves on the square grid. In Fractals in the Fundamental and Applied Sciences, pages 341–366. North Holland, Elsevier Science Publisher B.V., 1991.Google Scholar
  224. 224.
    J. Quinqueton and M. Berthod. A locally adaptive Peano scanning algorithm. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-3(4):403–412, 1981.Google Scholar
  225. 225.
    A. Rahimian, I. Lashuk, S. K. Veerapaneni, A. Chandramowlishwaran, D. Malhotra, L. Moon, R. Sampath, A. Shringarpure, J. Vetter, R. Vuduc, D. Zorin, and G. Biros. Petascale direct numerical simulation of blood flow on 200k cores and heterogeneous architectures. In ACM/IEEE Conference on Supercomputing, 2010, pages 1–11, 2010.Google Scholar
  226. 226.
    F. Ramsak, V. Markl, R. Fenk, Elhardt K., and R. Bayer. Integrating the UB-tree into a database system kernel. In Proceedings of the 26th International Conference on Very Large Databases, pages 263–272, 2000.Google Scholar
  227. 227.
    M.-C. Rivara and Ch. Levin. A 3-d refinement algorithm suitable for adaptive and multi-grid techniques. Communications in Applied Numerical Methods, 8:281–290, 1992.Google Scholar
  228. 228.
    S. Roberts, S. Kalyanasundaram, M. Cardew-Hall, and W. Clarke. A key based parallel adaptive refinement technique for finite element methods. In Computational Techniques and Applications: CTAC 97, pages 577–584. World Scientific Press, 1998.Google Scholar
  229. 229.
    B. Roychoudhury and J. F. Muth. The solution of travelling salesman problems based on industrial data. Journal of Complexity, 46(3):347–353, 1995.Google Scholar
  230. 230.
    H. Sagan. Some reflections on the emergence of space-filling curves. Journal of the Franklin Institute, 328:419–430, 1991.Google Scholar
  231. 231.
    H. Sagan. On the geometrization of the Peano curve and the arithmetization of the Hilbert curve. International Journal of Mathematical Education in Science and Technology, 23(3):403–411, 1992.Google Scholar
  232. 232.
    H. Sagan. A three-dimensional Hilbert curve. International Journal of Mathematical Education in Science and Technology, 24(4):541–545, 1993.Google Scholar
  233. 233.
    H. Sagan. Space-Filling Curves. Universitext. Springer, 1994.Google Scholar
  234. 234.
    J. K. Salmon, M. S. Warren, and G. S. Winckelmans. Fast parallel tree codes for gravitational and fluid dynamical n-body problems. International Journal of Supercomputer Applications, 8(2):129–142, 1994.Google Scholar
  235. 235.
    H. Samet. The quadtree and related hierarchical data structures. ACM Computing Surveys, 16(2):187–260, 1984.Google Scholar
  236. 236.
    P. Sanders and T. Hansch. Efficient massively parallel quicksort. In Proceedings of the 4th International Symposium on Solving Irregularly Structured Problems in Parallel, volume 1253 of Lecture Notes in Computer Science, pages 13–24, 1997.Google Scholar
  237. 237.
    M. Saxena, P. M. Finnigan, C. M. Graichen, A. F. Hathaway, and V. N. Parthasarathy. Octree-based automatic mesh generation for non-manifold domains. Engineering with Computers, 11(1):1–14, 1995.Google Scholar
  238. 238.
    S. Schamberger and J.-M. Wierum. A locality preserving graph ordering approach for implicit partitioning: Graph-filling curves. In Proceedings of the 17th International Conference on Parallel and Distributed Computing Systems, (PDCS’04), pages 51–57. ISCA, 2004.Google Scholar
  239. 239.
    S. Schamberger and J.-M. Wierum. Partitioning finite element meshes using space-filling curves. Future Generation Computer Systems, 21:759–766, 2005.Google Scholar
  240. 240.
    K. Schloegel, G. Karypis, and V. Kumar. Graph partitioning for high-performance scientific simulations, pages 491–541. Morgan Kaufmann Publishers Inc., 2003.Google Scholar
  241. 241.
    W. J. Schroeder and M. S. Shephard. A combined octree/Delaunay method for fully automatic 3-d mesh generation. International Journal for Numerical Methods in Engineering, 26(1):37–55, 1988.Google Scholar
  242. 242.
    E. G. Sewell. Automatic generation of triangulation for piecewise polynomial approximation. Technical Report CSD-TR83, Purdue University, 1972. PhD Thesis.Google Scholar
  243. 243.
    M. S. Shephard and M. K. Georges. Automatic three-dimensional mesh generation by the finite octree technique. International Journal for Numerical Methods in Engineering, 32(4):709–749, 1991.Google Scholar
  244. 244.
    W. Sierpinski. Sur une nouvelle courbe continue qui remplit toute une aire plane. Bulletin de l’Académie des Sciences de Cracovie, Série A, pages 462–478, 1912.Google Scholar
  245. 245.
    J. P. Singh, C. Holt, T. Totsuka, A. Gupta, and J. Hennessy. Load balancing and data locality in adaptive hierarchical n-body methods: Barnes–Hut, fast multipole, and radiosity. Journal of Parallel and Distributed Computing, 27:118–141, 1995.Google Scholar
  246. 246.
    R. Siromoney and K. G. Subramanian. Space-filling curves and infinite graphs. In Graph-Grammars and Their Application to Computer Science, volume 153 of Lecture Notes in Computer Science, pages 380–391, 1983.Google Scholar
  247. 247.
    B. Smith, P. Bjørstad, and W. Gropp. Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge University Press, 1996.Google Scholar
  248. 248.
    V. Springel. The cosmological simulation code GADGET-2. Monthly Notices of the Royal Astronomical Society, 364:1105–1134, 2005.Google Scholar
  249. 249.
    J. Steensland, S. Chandra, and M. Parashar. An application-centric characterization of domain-based SFC partitioners for parallel SAMR. IEEE Transactions on Parallel and Distributed Systems, 13(12):1275–1289, 2002.Google Scholar
  250. 250.
    R. J. Stevens, A. F. Lehar, and F. H. Preston. Manipulation and presentation of multidimensional image data using the Peano scan. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-5(5):520–526, 1983.Google Scholar
  251. 251.
    Q. F. Stout. Topological matching. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pages 24–31, 1983.Google Scholar
  252. 252.
    H. Sundar, R. S. Sampath, S. S. Adavani, C. Davatzikos, and G. Biros. Low-constant parallel algorithms for finite element simulations using linear octrees. In SC ’07: Proceedings of the 2007 ACM/IEEE Conference on Supercomputing, pages 1–12. ACM, 2007.Google Scholar
  253. 253.
    H. Sundar, R. S. Sampath, and G. Biros. Bottom-up construction and 2:1 balance refinement of linear octrees in parallel. SIAM Journal on Scientific Computing, 30(5):2675–2708, 2008.Google Scholar
  254. 254.
    S. Tanimoto and T. Pavlidis. A hierarchical data structure for picture processing. Computer Graphics and Image Processing, 4:104–119, 1975.Google Scholar
  255. 255.
    J. Thiyagalingam, O. Beckmann, and P. H. J. Kelly. Is Morton layout competitive for large two-dimensional arrays yet? Concurrency and Computation: Practice and Experience, 18:1509–1539, 2006.Google Scholar
  256. 256.
    S. Tirthapura, S. Seal, and W. Aluru. A formal analysis of space filling curves for parallel domain decomposition. In Proceedings of the 2006 International Conference on Parallel Processing (ICPP’06), pages 505–512. IEEE Computer Society, 2006.Google Scholar
  257. 257.
    H. Tropf and H. Herzog. Multidimensional range search in dynamically balanced trees. Angewandte Informatik (Applied Informatics), 2:71–77, 1981.Google Scholar
  258. 258.
    T. Tu, D. R. O’Hallaron, and O. Ghattas. Scalable parallel octree meshing for terascale applications. In SC ’05: Proceedings of the 2005 ACM/IEEE Conference on Supercomputing, page 4. IEEE Computer Society, 2005.Google Scholar
  259. 259.
    L. Velho and J. de Miranda Gomes. Digital halftoning with space filling curves. ACM SIGGRAPH Computer Graphics, 25(4):81–90, 1991.Google Scholar
  260. 260.
    L. Velho and J. Gomes. Variable resolution 4-k meshes: Concepts and applications. Computer Graphics forum, 19(4):195–212, 2000.Google Scholar
  261. 261.
    B. Von Herzen and A. H. Barr. Accurate triangulations of deformed, intersecting surfaces. ACM SIGGRAPH Computer Graphics, 21(4):103–110, 1987.Google Scholar
  262. 262.
    J. Wang and J. Shan. Space filling curve based point clouds index. In Proceedings of the 8th International Conference on GeoComputation, pages 551–562, 2005.Google Scholar
  263. 263.
    J. Warnock. A hidden surface algorithm for computer generated halftone pictures. Technical Report TR 4-15,, Computer Science Department, University of Utah, 1969.Google Scholar
  264. 264.
    M. S. Warren and J. K. Salmon. A parallel hashed oct-tree n-body algorithm. In Conference on High Performance Networking and Computing, Proceedings of the 1993 ACM/IEEE Conference on Supercomputing, pages 12–21. ACM, 1993.Google Scholar
  265. 265.
    T. Weinzierl. A Framework for Parallel PDE Solvers on Multiscale Adaptive Cartesian Grids. Dissertation, Institut für Informatik, Technische Universität München, 2009.Google Scholar
  266. 266.
    T. Weinzierl and M. Mehl. Peano – a traversal and storage scheme for octree-like adaptive Cartesian multiscale grids. SIAM Journal on Scientific Computing, 33(5):2732–2760, 2011.Google Scholar
  267. 267.
    R. C. Whaley, A. Petitet, and J. J. Dongarra. Automated empirical optimization of software and the ATLAS project. Parallel Computing, 27(1–2):3–35, 2001.Google Scholar
  268. 268.
    J.-M. Wierum. Definition of a new circular space-filling curve – βΩ-indexing. Technical Report TR-001-02, Paderborn Center for Parallel Computing, PC2, 2002.Google Scholar
  269. 269.
    N. Wirth. Algorithmen und Datenstrukturen. Teubner, 1975.Google Scholar
  270. 270.
    N. Wirth. Algorithms + Data Structures = Programs. Prentice Hall, 1976.Google Scholar
  271. 271.
    D. S. Wise. Representing matrices as quadtrees for parallel processors. Information Processing Letters, 20(4):195–199, 1985.Google Scholar
  272. 272.
    D. S. Wise and S. Franco. Costs of quadtree representation of nondense matrices. Journal of Parallel and Distributed Computing, 9(3):282–296, 1990.Google Scholar
  273. 273.
    I. H. Witten and R. M. Neal. Using Peano curves for bilevel display of continuous tone images. IEEE Computer Graphics and Applications, pages 47–52, May 1982.Google Scholar
  274. 274.
    I. H. Witten and B. Wyvill. On the generation and use of space-filling curves. Software: Practice and Experience, 13:519–525, 1983.Google Scholar
  275. 275.
    Dawes W. N., S. A. Harvey, S. Fellows, N. Eccles, D. Jaeggi, and W. P Kellar. A practical demonstration of scalable, parallel mesh generation. In 47th AIAA Aerospace Sciences Meeting & Exhibit, 2009. AIAA-2009-0981.Google Scholar
  276. 276.
    W. Wunderlich. Irregular curves and functional equations. Ganita, 5:215–230, 1954.Google Scholar
  277. 277.
    W. Wunderlich. Über Peano-Kurven. Elemente der Mathematik, 28(1):1–24, 1973.Google Scholar
  278. 278.
    K. Yang and M. Mills. Fractal based image coding scheme using Peano scan. In Proceedings of ISCAS ’88, volume 1470, pages 2301–2304, 1988.Google Scholar
  279. 279.
    M.-M. Yau and S. N. Srihari. A hierarchical data structure for multidimensional digital images. Communications of the ACM, 26(7):504–515, 1983.Google Scholar
  280. 280.
    L. Ying, G. Biros, D. Zorin, and H. Langston. A new parallel kernel-independent fast multipole method. In SC ’03: Proceedings of the 2003 ACM/IEEE Conference on Supercomputing. IEEE Computer Society, 2003.Google Scholar
  281. 281.
    K. Yotov, T. Roeder, K. Pingali, J. Gunnels, and F. Gustavson. An experimental comparison of cache-oblivious and cache-conscious programs. In Proceedings of the 19th annual ACM symposium on Parallel algorithms and architectures, pages 93–104, 2007.Google Scholar
  282. 282.
    Y. Zhang and R. E. Webber. Space diffusion: an improved parallel halftoning technique using space-filling curves. In Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques, pages 305–312. ACM New York, 1993.Google Scholar
  283. 283.
    S. Zhou and C. B. Jones. HCPO: an efficient insertion order for incremental delaunay triangulation. Information Processing Letters, 93:37–42, 2005.Google Scholar
  284. 284.
    U. Ziegler. The NIRVANA code: Parallel computational MHD with adaptive mesh refinement. Computer Physics Communications, 179(4):227–244, 2008.Google Scholar
  285. 285.
    G. Zumbusch. On the quality of space-filling curve induced partitions. Zeitschrift für Angewandte Mathematik und Mechanik, 81, Suppl. 1:25–28, 2001.Google Scholar
  286. 286.
    G. Zumbusch. Load balancing for adaptively refined grids. Proceedings in Applied Mathematics and Mechanics, 1:534–537, 2002.Google Scholar
  287. 287.
    G. Zumbusch. Parallel Multilevel Methods: Adaptive Mesh Refinement and Loadbalancing. Vieweg+Teubner, 2003.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Michael Bader
    • 1
  1. 1.Department of InformaticsTechnische Universität MünchenMunichGermany

Personalised recommendations

Cite chapter

Buy options