Approximate Furthest Neighbor with Application to Annulus Query

TL;DR

This paper introduces a practical and efficient data structure for approximate furthest neighbor queries in high-dimensional Euclidean space, improving previous methods and applying it to approximate annulus queries with theoretical and experimental validation.

Contribution

The authors present a novel, faster, and more accurate data structure for AFN queries, including a variation with better time and space complexity, and apply it to approximate annulus queries.

Findings

Improved approximation factor and reduced running time over Indyk's method.

A variation with query-independent ordering offers better time and space efficiency.

The approach is validated through theoretical analysis and experimental results.

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.