Discrete Random Walks, DRW'03
Cyril Banderier and Christian Krattenthaler (eds.)
DMTCS Conference Volume AC (2003), pp. 5368
author:  HoKwok Dai and HungChi Su 

title:  Approximation and Analytical Studies of Interclustering Performances of SpaceFilling Curves 
keywords:  spacefilling curves, Hilbert curves, zorder curves, clustering, random walk 
abstract: 
A discrete spacefilling curve provides a linear
traversal/indexing of a multidimensional grid space. This
paper presents an application of random walk to the study
of interclustering of spacefilling curves and an
analytical study on the interclustering performances of
2
dimensional Hilbert and zorder curve families. Two
underlying measures are employed: the mean intercluster
distance over all intercluster gaps and the mean total
intercluster distance over all subgrids. We show how
approximating the mean intercluster distance statistics of
continuous multidimensional spacefilling 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
intercluster distance statistics suggests that the random
walk may furnish an effective model to develop
approximations to clustering and locality statistics for
spacefilling curves. Based upon the analytical results,
the asymptotic comparisons indicate that zorder curve
family performs better than Hilbert curve family with
respect to both statistics.

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reference:  HoKwok Dai and HungChi Su (2003), Approximation and Analytical Studies of Interclustering Performances of SpaceFilling Curves, in Discrete Random Walks, DRW'03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC, pp. 5368 
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