On Querz Processing and Optimality Using Spectral Locality-Preserving Mappings

A locality-preserving mapping (LPM) from the multidimensional space into the one-dimensional space is beneficial for many applications (e.g., range queries, ncarestneighbor queries, clustering, and dedustcring) when multidimeIlsional data is placed into one-dimensional storage (e.g., the disk). The idea behind a locality-preserving mapping is to map points that are nearby in the multidimensional space into points that are nearby in the onedimensional space. For the past two decades, fractal::; (e.g., the Hilbert and Penna space-filling curves) have been considered the natural method for providing a locality-preserving mapping to support effieient answer for range queries and similarity search queries. In this paper, we go beyond the Idea of fractals. Instead, we investigate a locality-preserving mapping algorithm (The Spectral LPM) that uses the spectrum of the multi· dimensional space 1. This paper prov~ ably demonstrates how Spectral LPM provide.s a globally optimal mapping from the multi-dimensional .space to the one-dimensional space, and hence outperforms fractals. As an application, in the context of range queries and nearestneighbor queries, empirical results of the performance of Spectral LPM validate our analysis in comparison with Peano, Hilbert, and Gray fractal mappings.