On the Locality Properties of Space-Filling Curves

Abstract

A discrete space-filling curve provides a linear traversal or indexing of a multi-dimensional grid space. We present an analytical study of the locality properties of the m-dimensional k-order discrete Hilbert and z-order curve families, $\{H^m_k | k = 1,2,...\}$ and $\{Z^m_k | k = 1,2,...\}$, respectively, based on the locality measure L _δ that cumulates all index-differences of point-pairs at a common 1-normed distance δ. We derive the exact formulas for L _δ (H _k ^m) and L _δ (Z _k ^m) for m = 2 and arbitrary δ that is an integral power of 2, and m = 3 and δ = 1. The results yield a constant asymptotic ratio lim$_{k\rightarrow\infty}\frac{L_\delta(H^m_k)}{L_\delta(Z^m_k)} > 1$, which suggests that the z-order curve family performs better than the Hilbert curve family over the considered parameter ranges.