On p-Norm Based Locality Measures of Space-Filling Curves

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 dp , is the maximum ratio of dp(v, u)m to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$d_{p}(\tilde{v},\tilde{u})$\end{document} over all corresponding point-pairs (v, u) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(\tilde{v},\tilde{u})$\end{document} 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.