Space-time tradeoffs for approximate nearest neighbor searching

Abstract

Nearest neighbor searching is the problem of preprocessing a set of n point points in d -dimensional space so that, given any query point q , it is possible to report the closest point to q rapidly. In approximate nearest neighbor searching, a parameter ε > 0 is given, and a multiplicative error of (1 + ε) is allowed. We assume that the dimension d is a constant and treat n and ε as asymptotic quantities. Numerous solutions have been proposed, ranging from low-space solutions having space O ( n ) and query time O (log n + 1/ε d −1 ) to high-space solutions having space roughly O (( n log n )/ε d ) and query time O (log ( n /ε)). We show that there is a single approach to this fundamental problem, which both improves upon existing results and spans the spectrum of space-time tradeoffs. Given a tradeoff parameter γ, where 2 ≤ γ ≤ 1/ε, we show that there exists a data structure of space O ( n γ d −1 log(1/ε)) that can answer queries in time O (log( n γ) + 1/(εγ) ( d −1)/2 . When γ = 2, this yields a data structure of space O ( n log (1/ε)) that can answer queries in time O (log n + 1/ε ( d −1)/2 ). When γ = 1/ε, it provides a data structure of space O (( n /ε d −1 )log(1/ε)) that can answer queries in time O (log( n /ε)). Our results are based on a data structure called a ( t ,ε)-AVD, which is a hierarchical quadtree-based subdivision of space into cells. Each cell stores up to t representative points of the set, such that for any query point q in the cell at least one of these points is an approximate nearest neighbor of q . We provide new algorithms for constructing AVDs and tools for analyzing their total space requirements. We also establish lower bounds on the space complexity of AVDs, and show that, up to a factor of O (log (1/ε)), our space bounds are asymptotically tight in the two extremes, γ = 2 and γ = 1/ε.