p-Adic scaled space filling curve indices for high dimensional data

Abstract: Space filling curves are widely used in Computer Science. In particular Hilbert curves and their generalisations to higher dimension are used as an indexing method because of their nice locality properties. This article generalises this concept to the systematic construction of p-adic versions of Hilbert curves based on affine transformations of the p-adic Gray code, and develops an efficient scaled indexing method for data taken from high-dimensional spaces based on these new curves, which with increasing dimension is shown to be less space consuming than the optimal standard static Hilbert curve index. A measure is derived which allows to assess the local sparsity of a data set, and is tested on some data.