On Multi-dimensional Hilbert Indexings

Indexing schemes for grids based on space-lling curves (e.g., Hilbert indexings) nd applications in numerous elds, ranging from parallel processing over data structures to image processing. Because of an increasing interest in discrete multi-dimensional spaces, indexing schemes for them have won considerable interest. Hilbert curves are the most simple and popular space-lling indexing scheme. We extend the concept of curves with Hilbert property to arbitrary dimensions and present rst results concerning their structural analysis that also simplify their applicability. We deene and analyze in a precise mathematical way r-dimensional Hilbert in-dexings for arbitrary r 2. Moreover, we generalize and simplify previous work and clarify the concept of Hilbert curves for multi-dimensional grids. As we show, Hilbert indexings can be completely described and analyzed by \generating elements of order 1", thus, in comparison with previous work, reducing their structural complexity decisively. This structural analysis like e.g. investigations concerning locality can always be reduced to small generating elements of the curves. In addition,whereas there is basically one Hilbert curve in the 2D world, our analysis shows that there are 1536 structurally diierent 3D Hilbert curves. Further results include generalizations of locality results for multi-dimensional indexings and an easy recursive computation scheme for multi-dimensional Hilbert indexings. In addition, our formalism lays the groundwork for potential mechanized analysis of locality properties of multi-dimensional Hilbert indexings.