| author: | Ho-Kwok Dai and Hung-Chi Su |
|---|---|
| title: | Approximation and Analytical Studies of Inter-clustering Performances of Space-Filling Curves |
| keywords: | space-filling curves, Hilbert curves, z-order curves, clustering, random walk |
| abstract: |
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.
|
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| reference: | 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 |
| bibtex: | For a corresponding BibTeX entry, please consider our BibTeX-file. |
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| pdf-source: | dmAC0106.pdf (196 K) |
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