## DMTCS Proceedings, Discrete Random Walks, DRW'03

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DMTCS Conference vol AC (2003), pp. 53-68

## Discrete Random Walks, DRW'03

### DMTCS Conference Volume AC (2003), pp. 53-68

author: Ho-Kwok Dai and Hung-Chi Su Approximation and Analytical Studies of Inter-clustering Performances of Space-Filling Curves space-filling curves, Hilbert curves, z-order curves, clustering, random walk A discrete space-filling curve provides a linear traversal/indexing of a multi-dimensional grid space. This paper presents an application of random walk to the study of inter-clustering of space-filling curves and an analytical study on the inter-clustering performances of 2 -dimensional Hilbert and z-order curve families. Two underlying measures are employed: the mean inter-cluster distance over all inter-cluster gaps and the mean total inter-cluster distance over all subgrids. We show how approximating the mean inter-cluster distance statistics of continuous multi-dimensional space-filling curves fits into the formalism of random walk, and derive the exact formulas for the two statistics for both curve families. The excellent agreement in the approximate and true mean inter-cluster distance statistics suggests that the random walk may furnish an effective model to develop approximations to clustering and locality statistics for space-filling curves. Based upon the analytical results, the asymptotic comparisons indicate that z-order curve family performs better than Hilbert curve family with respect to both statistics. If your browser does not display the abstract correctly (because of the different mathematical symbols) you may look it up in the PostScript or PDF files. Ho-Kwok Dai and Hung-Chi Su (2003), Approximation and Analytical Studies of Inter-clustering Performances of Space-Filling Curves, in Discrete Random Walks, DRW'03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC, pp. 53-68 For a corresponding BibTeX entry, please consider our BibTeX-file. dmAC0106.ps.gz (84 K) dmAC0106.ps (244 K) dmAC0106.pdf (196 K)