On Multidimensional Curves with Hilbert Property

Abstract.

Indexing schemes for grids based on space-filling curves (e.g., Hilbert curves) find applications in numerous fields, ranging from parallel processing over data structures to image processing. Because of an increasing interest in discrete multidimensional spaces, indexing schemes for them have won considerable interest. Hilbert curves are the most simple and popular space-filling indexing schemes. We extend the concept of curves with Hilbert property to arbitrary dimensions and present first results concerning their structural analysis that also simplify their applicability.

We define and analyze in a precise mathematical way r -dimensional Hilbert curves for arbitrary r ≥ 2 . Moreover, we generalize and simplify previous work and clarify the concept of Hilbert curves for multidimensional grids. As we show, curves with Hilbert property can be completely described and analyzed by ``generating elements of order 1,'' thus, in comparison with previous work, reducing their structural complexity decisively. Whereas there is basically one Hilbert curve in the two-dimensional world, our analysis shows that there are 1536 structurally different simple three-dimensional Hilbert curves. Further results include generalizations of locality results for multidimensional indexings and an easy recursive computation scheme for multidimensional curves with Hilbert property. In addition, our formalism lays the groundwork for potential mechanized analysis of locality properties of multidimensional Hilbert curves.