A lower bound on proximity preservation by space filling curves

A space filling curve (SFC) is a proximity preserving mapping from a high dimensional space to a single dimensional space. SFCs have been used extensively in dealing with multi-dimensional data in parallel computing, scientific computing, and databases. The general goal of an SFC is that points that are close to each other in high-dimensional space are also close to each other in the single dimensional space. While SFCs have been used widely, the extent to which proximity can be preserved by an SFC is not precisely understood yet. We consider natural metrics, including the "nearest-neighbor stretch" of an SFC, which measure the extent to which an SFC preserves proximity. We first show a powerful negative result, that there is an inherent lower bound on the stretch of any SFC. We then show that the stretch of the commonly used Z curve is within a factor of 1.5 from the optimal, irrespective of the number of dimensions. Further we show that a very simple SFC also achieves the same stretch as the Z curve. Our results apply to SFCs in any dimension d such that d is a constant.