Locality and Bounding-Box Quality of Two-Dimensional Space-Filling Curves

TL;DR

This paper introduces measures for evaluating the effectiveness of space-filling curves in organizing planar points into bounding-box hierarchies, providing bounds, algorithms, and empirical analysis of various curves.

Contribution

It develops general bounds and an approximation algorithm for bounding-box quality and locality of space-filling curves, including new and known curves, revealing surprising trade-offs.

Findings

Some curves with poor locality have good bounding-box quality.

The best locality curve has relatively poor bounding-box quality.

The paper provides a generic algorithm to evaluate space-filling curves.

Abstract

Space-filling curves can be used to organise points in the plane into bounding-box hierarchies (such as R-trees). We develop measures of the bounding-box quality of space-filling curves that express how effective different space-filling curves are for this purpose. We give general lower bounds on the bounding-box quality measures and on locality according to Gotsman and Lindenbaum for a large class of space-filling curves. We describe a generic algorithm to approximate these and similar quality measures for any given curve. Using our algorithm we find good approximations of the locality and the bounding-box quality of several known and new space-filling curves. Surprisingly, some curves with relatively bad locality by Gotsman and Lindenbaum's measure, have good bounding-box quality, while the curve with the best-known locality has relatively bad bounding-box quality.