Dai, Ho-Kwok and Su, Hung-Chi - Approximation and Analytical Studies of Inter-clustering Performances of Space-Filling Curves

dmtcs:3338 - Discrete Mathematics & Theoretical Computer Science, January 1, 2003, DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03)
Approximation and Analytical Studies of Inter-clustering Performances of Space-Filling Curves

Authors: Dai, Ho-Kwok and Su, Hung-Chi

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.


Volume: DMTCS Proceedings vol. AC, Discrete Random Walks (DRW'03)
Section: Proceedings
Published on: January 1, 2003
Submitted on: May 10, 2017
Keywords: space-filling curves,Hilbert curves,z-order curves,clustering,random walk,[INFO.INFO-DS] Computer Science [cs]/Data Structures and Algorithms [cs.DS],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-CG] Computer Science [cs]/Computational Geometry [cs.CG]


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