Norm-Based Locality Measures of Two-Dimensional Hilbert Curves

Abstract

A discrete space-filling curve provides a 1-dimensional indexing or traversal of a multi-dimensional grid space. Applications of space-filling curves include multi-dimensional indexing methods, parallel computing, and image compression. Common goodness-measures for the applicability of space-filling curve families are locality and clustering. Locality reflects proximity preservation that close-by grid points are mapped to close-by indices or vice versa. We present an analytical study on the locality property of the 2-dimensional Hilbert curve family. The underlying locality measure, based on the p-normed metric $d_{p}$, is the maximum ratio of $d_{p}(u, v)^{m}$ to $d_{p}(\tilde{u}, \tilde{v})$ over all corresponding point-pairs (u, v) and $(\tilde{u}, \tilde{v})$ in the m-dimensional grid space and 1-dimensional index space, respectively. Our analytical results identify all candidate representative grid-point pairs (realizing the locality-measure values) for all real norm-parameters in the unit interval [1, 2] and grid-orders. Together with the known results for other norm-parameter values, we have almost complete knowledge of the locality measure of 2-dimensional Hilbert curves over the entire spectrum of possible norm-parameter values.