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On Spatial Orders and Location Codes
March 2009 (vol. 58 no. 3)
pp. 424-432
L. J. Stocco, The University of British Columbia, Vancouver
G. Schrack, The University of British Columbia, Vancouver
Spatial orders such as the Morton (Z) order, Uorder, or X-order have applications in matrix manipulation, graphic rendering and data encryption. It is shown that these spatial orders are single examples of entire classes of spatial orders which can be defined in arbitrary numbers of dimensions and base values. Secondly, an algorithm is proposed which can be used to transform between these spatial orders and cartesian coordinates. It is shown that the efficiency of the algorithm improves with a larger base value. By choosing a base value that corresponds to the available memory page size, the computational effort required to perform operations such as matrix multiplication can be optimized.

[1] V. Valsalam and A. Skjellum, “A Framework for High-Performance Matrix Multiplication Base on Hierarchical Abstractions, Algorithms and Optimized Low-Level Kernels,” Concurrency and Computation: Practice and Experience, vol. 14, John Wiley & Sons, pp. 805-839, 2002.
[2] G. Schrack, “Finding Neighbors of Equal Size in Linear Quadtrees and Octrees in Constant Time,” CVGIP: Image Understanding, vol. 55, no. 3, pp.221-230, 1992.
[3] G. Peano, “Sur une Courbe, Qui Remplit Toute une Aire Plaine,” Math. Ann., vol. 36, pp. 157-160, 1890.
[4] D. Hilbert, “Über die Stetige Abbildung einer Linie auf ein Flächenstück,” Math. Ann., vol. 38, pp. 459-460, 1891.
[5] G.M. Morton, A Computer Oriented Geodetic Data Base and a New Technique in File Sequencing, technical report, IBM, Mar. 1966.
[6] G. Schrack and X. Liu, “The Spatial U-Order and Some of Its Mathematical Characteristics,” Proc. IEEE Pacific Rim Conf. Comm., Computers and Signal Processing (PACRIM '95), pp. 416-419, 1995.
[7] X. Liu and G.F. Schrack, “A New Ordering Strategy Applied to Spatial Data Processing,” Int'l J. Geographical Information Science, vol. 12, no. 1, pp. 3-22, 1988.
[8] F.C. Holroyd and D.C. Mason, “Efficient Linear Quadtree Construction Algorithm,” Image and Vision Computing, vol. 8, no. 3, pp. 218-224, 1990.
[9] L. Stocco and G. Schrack, “Integer Dilation and Contraction for Quadtrees and Octrees,” Proc. IEEE Pacific Rim Conf. Comm., Computers and Signal Processing (PACRIM '95), pp. 426-428, 1995.

Index Terms:
Numerical algorithms, Algorithm design and analysis, Constructive solid geometry, Volumetric
L. J. Stocco, G. Schrack, "On Spatial Orders and Location Codes," IEEE Transactions on Computers, vol. 58, no. 3, pp. 424-432, March 2009, doi:10.1109/TC.2008.171
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