Approximate Furthest Neighbor in High Dimensions

Much recent work has been devoted to approximate nearest neighbor queries. Motivated by applications in recommender systems, we consider approximate furthest neighbor (AFN) queries. We present a simple, fast, and highly practical data structure for answering AFN queries in high-dimensional Euclidean space. We build on the technique of Indyk (SODA 2003), storing random projections to provide sublinear query time for AFN. However, we introduce a di↵erent 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 queryindependent ordering of the database points; while this does not have the provable approximation factor of the query-dependent data structure, it o↵ers significant improvement in time and space complexity. We give a theoretical analysis, and experimental results.