Hyperorthogonal well-folded Hilbert curves

TL;DR

This paper introduces a new high-dimensional Hilbert curve generalization that produces significantly tighter bounding boxes for R-tree indexing, improving efficiency by reducing unnecessary volume coverage.

Contribution

The paper presents a novel hyperorthogonal well-folded Hilbert curve that minimizes bounding box volume, outperforming existing generalizations in high-dimensional spaces.

Findings

Bounding boxes are at most 4 times the volume of the curve segment

The new curve generalization is asymptotically optimal with factor 4

Improves R-tree efficiency by reducing bounding box size

Abstract

R-trees can be used to store and query sets of point data in two or more dimensions. An easy way to construct and maintain R-trees for two-dimensional points, due to Kamel and Faloutsos, is to keep the points in the order in which they appear along the Hilbert curve. The R-tree will then store bounding boxes of points along contiguous sections of the curve, and the efficiency of the R-tree depends on the size of the bounding boxes---smaller is better. Since there are many different ways to generalize the Hilbert curve to higher dimensions, this raises the question which generalization results in the smallest bounding boxes. Familiar methods, such as the one by Butz, can result in curve sections whose bounding boxes are a factor $\Omega(2^{d/2})$ larger than the volume traversed by that section of the curve. Most of the volume bounded by such bounding boxes would not contain any data points. In this paper we present a new way of generalizing Hilbert's curve to higher dimensions, which results in much tighter bounding boxes: they have at most 4 times the volume of the part of the curve covered, independent of the number of dimensions. Moreover, we prove that a factor 4 is asymptotically optimal.