Four-dimensional hilbert curves for R-trees

Two-dimensional R-trees are a class of spatial index structures in which objects are arranged to enable fast window queries: report all objects that intersect a given query window. One of the most successful methods of arranging the objects in the index structure is based on sorting the objects according to the positions of their centers along a two-dimensional Hilbert space-filling curve. Alternatively, one may use the coordinates of the objects' bounding boxes to represent each object by a four-dimensional point, and sort these points along a four-dimensional Hilbert-type curve. In experiments by Kamel and Faloutsos and by Arge et al., the first solution consistently outperformed the latter when applied to point data, while the latter solution clearly outperformed the first on certain artificial rectangle data. These authors did not specify which four-dimensional Hilbert-type curve was used; many exist.

In this article, we show that the results of the previous articles can be explained by the choice of the four-dimensional Hilbert-type curve that was used and by the way it was rotated in four-dimensional space. By selecting a curve that has certain properties and choosing the right rotation, one can combine the strengths of the two-dimensional and the four-dimensional approach into one, while avoiding their apparent weaknesses. The effectiveness of our approach is demonstrated with experiments on various datasets. For real data taken from VLSI design, our new curve yields R-trees with query times that are better than those of R-trees that were obtained with previously used curves.