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Mapping with Space Filling Surfaces
September 2007 (vol. 18 no. 9)
pp. 1258-1269
ASCII Text
x 
 
Masood Ahmed, Shahid Bokhari, "Mapping with Space Filling Surfaces," IEEE Transactions on Parallel and Distributed Systems, vol. 18, no. 9, pp. 1258-1269, September, 2007.
 
 
BibTex
x
 
@article{ 10.1109/TPDS.2007.1049,
author = {Masood Ahmed and Shahid Bokhari},
title = {Mapping with Space Filling Surfaces},
journal ={IEEE Transactions on Parallel and Distributed Systems},
volume = {18},
number = {9},
issn = {1045-9219},
year = {2007},
pages = {1258-1269},
doi = {http://doi.ieeecomputersociety.org/10.1109/TPDS.2007.1049},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
}
 
 
RefWorks Procite/RefMan/Endnote
x 
 
TY - JOUR
JO - IEEE Transactions on Parallel and Distributed Systems
TI - Mapping with Space Filling Surfaces
IS - 9
SN - 1045-9219
SP1258
EP1269
EPD - 1258-1269
A1 - Masood Ahmed,
A1 - Shahid Bokhari,
PY - 2007
KW - Fractals
KW - Hilbert curve
KW - parallel computing
KW - Peano curve
KW - Sierpinski carpet
KW - space filling curves
KW - space filling surfaces
VL - 18
JA - IEEE Transactions on Parallel and Distributed Systems
ER -
 
The use of space filling curves for proximity-improving mappings is well known and has found many useful applications in parallel computing. Such curves permit a linear array to be mapped onto a 2(respectively, 3)D structure such that points distance d apart in the linear array are distance O(d^1/2) (O(d^1/3)) apart in the 2(3)D array and vice-versa. We extend the concept of space filling curves to space filling surfaces and show how these surfaces lead to mappings from 2D to 3D so that points at distance d^1/2 on the 2D surface are mapped to points at distance O(d^1/3) in the 3D volume. Three classes of surfaces, associated respectively with the Peano curve, Sierpinski carpet, and the Hilbert curve, are presented. A methodology for using these surfaces to map from 2D to 3D is developed. These results permit efficient execution of 2D computations on processors interconnected in a 3D grid. The space filling surfaces proposed by us are the first such fractal objects to be formally defined and are thus also of intrinsic interest in the context of fractal geometry.

[1] G. Cantor, “Über Unendliche, Lineare Punktmannigfaltigkeiten V,” Mathematische Annalen, vol. 21, pp. 545-591, 1883.
[2] G. Peano, “Sur une Courbe, Qui Remplit Toute une Aire Plane,” Mathematische Annalen, vol. 36, pp. 157-160, 1890.
[3] D. Hilbert, “Über die Stetige Abbildung Einer Linie auf Ein Flächenstück,” Mathematische Annalen, vol. 38, pp. 459-460, 1891.
[4] W. Sierpiński, “Sur une Courbe Cantorienne Qui Contient une Image Biunivoque et Continue de Toute Courbe Donnée,” Comptes Rendus de l'Academie des Sciences, Paris, vol. 162, pp. 629-632, 1916.
[5] B.B. Mandelbrot, The Fractal Geometry of Nature. W.H. Freeman and Co., 1977.
[6] M. Schroeder, Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise. W.H. Freeman and Co., 1991.
[7] S.H. Bokhari, T.W. Crockett, and D.M. Nicol, “Parametric Binary Dissection,” Technical Report 93-39, Inst. for Computer Applications in Science and Eng., 1993.
[8] C.-W. Ou, M. Gunwani, and S. Ranka, “Architecture-Independent Locality-Improving Transformations of Computational Graphs Embedded in k-Dimensions,” Proc. Ninth ACM Int'l Conf. Supercomputing, pp. 289-297, July 1995.
[9] J.R. Pilkington and S.B. Baden, “Dynamic Partitioning of Non-Uniform Structured Workloads with Spacefilling Curves,” IEEE Trans. Parallel and Distributed Systems, vol. 7, no. 3, pp. 288-300, Mar. 1996.
[10] S. Aluru and F. Sevilgen, “Parallel Domain Decomposition and Load Balancing Using Space-Filling Curves,” Proc. Fourth IEEE Int'l Conf. High Performance Computing, pp. 230-235, 1997.
[11] S. Chatterjee, A. Lebeck, P. Patnala, and M. Thottethodi, “Recursive Array Layouts and Fast Parallel Matrix Multiplication,” Proc. Ann. ACM Symp. Parallel Algorithms and Architectures (SPAA), pp. 222-231, 1999.
[12] Y. Zhu and Y. Hu, “Efficient, Proximity-Aware Load Balancing for DHT-Based P2P Systems,” IEEE Trans. Parallel and Distributed Systems, vol. 16, no. 4, pp. 349-361, Apr. 2005.
[13] J.K. Lawder, “Calculation of Mappings between One and n-Dimensional Values Using the Hilbert Space-Filling Curve,” www.dcs.bbk.ac.uk/TriStarp/pubsJL1_00.ps.Z , Aug. 2000.
[14] J.J. Bartholdi III and P. Goldsman, “Vertex-Labeling Algorithms for the Hilbert Spacefilling Curve,” Software—Practice and Experience, vol. 31, no. 5, pp. 395-408, 2001.
[15] M.F. Mokbel, W.G. Aref, and I. Kamel, “Performance of Multi-Dimensional Space-Filling Curves,” Proc. 10th ACM Int'l Symp. Advances in Geographical Infomation Systems, pp. 149-154, 2002.
[16] R. Niedermeier, K. Reinhardt, and P. Sanders, “Towards Optimal Locality in Mesh-Indexings,” Discrete Applied Mathematics, vol. 117, no. 1-3, pp. 211-237, 2002.
[17] H.V. Jagadish, “Analysis of the Hilbert Curve for Representing Two-Dimensional Space,” Information Processing Letters, vol. 62, pp. 17-22, 1997.
[18] B. Moon, H.V. Jagadish, C. Faloutsos, and J.H. Saltz, “Analysis ofthe Clustering Properties of the Hilbert Space-Filling Curve,” IEEE Trans. Knowledge and Data Eng., vol. 13, no. 1, pp. 124-141, Jan.-Feb. 2001.
[19] T. Kurç, Ü. Çatalyürek, C. Chang, A. Sussman, and J. Saltz, “Visualization of Large Data Sets with the Active Data Repository,” IEEE Computer Graphics and Applications, vol. 21, pp. 24-33, July/Aug. 2001.
[20] A. Alhosni and S.Y. Berkovich, “Application of Lebesgue Space Filling Curve in Progressive Image Transmission,” Proc. Third IASTED Int'l Conf. Visualization, Imaging, and Image Processing, Sept. 2003.
[21] M. Quweider and E. Salari, “Use of Space Filling Curves in Fast Encoding of VQ Images,” Proc. Int'l Conf. Image Processing, vol. 3, p.3101, 1995.
[22] S. Patil, S.R. Das, and A. Nasipuri, “Serial Data Fusion Using Space-Filling Curves in Wireless Sensor Networks,” Proc. IEEE Comm. Soc. Conf. Sensors and Ad Hoc Comm. and Networks, vol. 1, pp. 182-190, Oct. 2004.
[23] M.J. Aftosmis, M.J. Berger, and S. Murman, “Applications of Space Filling Curves to Cartesian Methods for CFD,” Proc. 42nd AIAA Aerospace Sciences Meeting, 2004.
[24] D.J. Mavriplis, M.J. Aftosmis, and M. Berger, “High Resolution Aerospace Applications Using the NASA Columbia Supercomputer,” Proc. Conf. High-Performance Networking and Computing (SC '05), Nov. 2005.
[25] J.M. Dennis, “Partitioning with Space-Filling Curves on the Cubed-Sphere,” Proc. 17th Int'l Parallel and Distributed Processing Symp. (IPDPS '03), p. 269, Apr.22-26, 2003.
[26] G. Jin and J. Mellor-Crummey, “Sfcgen: A Framework for Efficient Generation of Multi-Dimensional Space-Filling Curves by Recursion,” ACM Trans. Mathematical Software, vol. 31, pp. 120-148, Mar. 2005.
[27] E. Skubalska-Rafajlowicz and A. Krzyzak, “Fast k-NN Classification Rule Using Metric on Space-Filling Curves,” Proc. 13th Int'l Conf. Pattern Recognition, vol. 2, pp. 121-125, 1996.
[28] H.-L. Chen and Y.-I. Chang, “Neighbor-Finding Based on Space-Filling Curves,” Infomation Systems, vol. 30, pp. 205-226, May 2005.
[29] S.H. Bokhari, “On the Mapping Problem,” IEEE Trans. Computers, vol. C-30, pp. 207-214, Mar. 1981.
[30] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, 1990.
[31] H. Sagan, Space-Filling Curves. Springer-Verlag, 1994.
[32] “Chaotic Dynamics and Fractals,” Notes and Reports in Mathematics in Science and Engineering, M.F. Barnsley and S.G. Demko, eds., vol. 2, 1986.
[33] M.F. Barnsley, Fractals Everywhere, second ed. Academic Press, 1993.
[34] H. von Koch, “Sur Une Courbe Continue Sans Tangente, Obtenue Par une Construction Géométrique Élémentaire,” Arkiv för Matematik, vol. 1, pp. 681-704, 1904.
[35] K. Menger, “Allgemeine Räume und Cartesische Räume,” Proc. Koniklijke Akademie van Wetenschappen te Amsterdam, vol. 29, pp.1125-1128, 1926.
[36] S.H. Bokhari, T.W. Crockett, and D.M. Nicol, “Binary Dissection: Variants and Applications,” Technical Report 97-29, Inst. for Computer Applications in Science and Eng., www.compsci.wm. edu/~tom97-29.pdf, 1997.
[37] C. Gotsman and M. Lindenbaum, “On the Metric Properties ofDiscrete Space-Filling Curves,” www.cs.technion.ac.il/~gotsman/AmendedPubl/ OnTheMetricOnTheMetric.pdf, July 1996.
[38] G. Chochia and M. Cole, “Recursive 3D Mesh Indexing with Improved Locality,” Proc. Int'l Conf. and Exhibition High-Performance Computing and Networking (HPCN Europe), pp. 1014-1015, 1997.
[39] A.J. Cole, “Compaction Techniques for Raster Scan Graphics Using Space-Filling Curves,” Computer J., vol. 30, no. 1, pp.87-92, 1987.

Index Terms:
Fractals, Hilbert curve, parallel computing, Peano curve, Sierpinski carpet, space filling curves, space filling surfaces
Citation:
Masood Ahmed, Shahid Bokhari, "Mapping with Space Filling Surfaces," IEEE Transactions on Parallel and Distributed Systems, vol. 18, no. 9, pp. 1258-1269, Sept. 2007, doi:10.1109/TPDS.2007.1049
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