Abstract

Indexing schemes for grids based on space-filling curves (e.g., Hilbert indexings) find applications in numerous fields, 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-filling indexing scheme. 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 indexings 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 comp...

Citations