Theocharis Malamatos

Space-time tradeoffs for approximate nearest neighbor searching

Nearest neighbor searching is the problem of preprocessing a set of <i>n</i> point points in <i>d</i>-dimensional space so that, given any query point <i>q</i>, it is possible to report the closest point to <i>q</i> 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 <i>d</i> is a constant and treat <i>n</i> and ϵ as asymptotic quantities. Numerous solutions have been proposed, ranging from low-space solutions having space <i>O</i>(<i>n</i>) and query time <i>O</i>(log <i>n</i> + 1/ϵ^<i>d</i>−1) to high-space solutions having space roughly <i>O</i>((<i>n</i> log <i>n</i>)/ϵ^<i>d</i>) and query time <i>O</i>(log (<i>n</i>/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 <i>O</i>(<i>n</i>γ^<i>d</i>−1 log(1/ϵ)) that can answer queries in time <i>O</i>(log(<i>n</i>γ) + 1/(ϵγ)^{(<i>d</i>−1)/2}. When γ = 2, this yields a data structure of space <i>O</i>(<i>n</i> log (1/ϵ)) that can answer queries in time <i>O</i>(log <i>n</i> + 1/ϵ^{(<i>d</i>−1)/2}). When γ = 1/ϵ, it provides a data structure of space <i>O</i>((<i>n</i>/ϵ^<i>d</i>−1)log(1/ϵ)) that can answer queries in time <i>O</i>(log(<i>n</i>/ϵ)). Our results are based on a data structure called a (<i>t</i>,ϵ)-AVD, which is a hierarchical quadtree-based subdivision of space into cells. Each cell stores up to <i>t</i> representative points of the set, such that for any query point <i>q</i> in the cell at least one of these points is an approximate nearest neighbor of <i>q</i>. 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 <i>O</i>(log (1/ϵ)), our space bounds are asymptotically tight in the two extremes, γ = 2 and γ = 1/ϵ.

Space-efficient approximate Voronoi diagrams

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FINDING PLANAR REGIONS IN A TERRAIN – IN PRACTICE AND WITH A GUARANTEE

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Conic nearest neighbor queries and approximate Voronoi diagrams

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