Abstract
A discrete space-filling curve provides a linear indexing or traversal of a multi-dimensional grid space. We present an analytical study on the locality properties of the 2-dimensional Hilbert curve family. The underlying locality measure, based on the p-normed metric d p , is the maximum ratio of d p (v, u)m to \(d_{p}(\tilde{v},\tilde{u})\) over all corresponding point-pairs (v, u) and \((\tilde{v},\tilde{u})\) in the m-dimensional grid space and (1-dimensional) index space, respectively. Our analytical results close the gaps between the current best lower and upper bounds with exact formulas for p ∈ {1, 2}, and extend to all reals p ≥ 2.
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References
Alber, J., Niedermeier, R.: On multi-dimensional curves with Hilbert property. Theory of Computing Systems 33(4), 295–312 (2000)
Asano, T., Ranjan, D., Roos, T., Welzl, E., Widmayer, P.: Space-filling curves and their use in the design of geometric data structures. Theoretical Computer Science 181(1), 3–15 (1997)
Dai, H.K., Su, H.C.: Approximation and analytical studies of inter-clustering performances of space-filling curves. In: Proceedings of the International Conference on Discrete Random Walks (Discrete Mathematics and Theoretical Computer Science, Volume AC ), September 2003, pp. 53–68 (2003)
Dai, H.K., Su, H.C.: An empirical study of p-norm based locality measures of space-filling curves. In: Proceedings of the 2003 International Conference on Parallel and Distributed Processing Techniques and Applications, Computer Science Research, Education, and Applications Press, June, pp. 1434–1440 (2003)
Dai, H.K., Su, H.C.: On the locality properties of space-filling curves. In: Ibaraki, T., Katoh, N., Ono, H. (eds.) ISAAC 2003. LNCS, vol. 2906, pp. 385–394. Springer, Heidelberg (2003)
Gotsman, C., Lindenbaum, M.: On the metric properties of discrete space-filling curves. IEEE Transactions on Image Processing 5(5), 794–797 (1996)
Mitchison, G., Durbin, R.: Optimal numberings of an N ×N array. SIAM Journal on Algebraic and Discrete Methods 7(4), 571–582 (1986)
Moon, B., Jagadish, H.V., Faloutsos, C., Saltz, J.H.: Analysis of the clustering properties of the Hilbert space-filling curve. IEEE Transactions on Knowledge and Data Engineering 13(1), 124–141 (2001)
Pérez, A., Kamata, S., Kawaguchi, E.: Peano scanning of arbitrary size images. In: Proceedings of the International Conference on Pattern Recognition, pp. 565–568. IEEE Computer Society Press, Los Alamitos (1992)
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Dai, H.K., Su, H.C. (2004). On p-Norm Based Locality Measures of Space-Filling Curves. In: Fleischer, R., Trippen, G. (eds) Algorithms and Computation. ISAAC 2004. Lecture Notes in Computer Science, vol 3341. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30551-4_33
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DOI: https://doi.org/10.1007/978-3-540-30551-4_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-24131-7
Online ISBN: 978-3-540-30551-4
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