Approximate Furthest Neighbor with Application to Annulus Query

Abstract: Much recent work has been devoted to approximate nearest neighbor queries. Motivated by applications in recommender systems, we consider approximate furthest neighbor (AFN) queries and present a simple, fast, and highly practical data structure for answering AFN queries in high- dimensional Euclidean space. The method builds on the technique of In- dyk (SODA 2003), storing random projections to provide sublinear query time for AFN. However, we introduce a different query algorithm, improving on Indyk's approximation factor and reducing the running time by a logarithmic factor. We also present a variation based on a query- independent ordering of the database points; while this does not have the provable approximation factor of the query-dependent data structure, it offers significant improvement in time and space complexity. We give a theoretical analysis, and experimental results. As an application, the query-dependent approach is used for deriving a data structure for the approximate annulus query problem, which is defined as follows: given an input set S and two parameters r > 0 and w >= 1, construct a data structure that returns for each query point q a point p in S such that the distance between p and q is at least r/w and at most wr.