ANN queries: covering Voronoi diagram with hyperboxes

Given a set S of n points in d-dimensional Euclidean metric space X and a small positive real number ǫ, we present an algorithm to preprocess S and answer queries that require finding a set S′ ⊆ S of ǫ-approximate nearest neighbors (ANNs) to a given query point q ∈ X . The following are the characteristics of points belonging to set S′: ∀s ∈ S′, ∃ a point p ∈ X such that |pq| ≤ ǫ and the nearest neighbor of p is s, and ∃ a s′ ∈ S′ such that s′ is a nearest neighbor of q. During the preprocessing phase, from the Voronoi diagram of S we construct a set of box trees of size O(4 V δ ( ǫ )d−1) which facilitate in querying ANNs of any input query point in O( 1 d lg V δ + ( ǫ )d−1) time. Here δ equals to ( ǫ 2 √ d ), and V is the volume of a large bounding box that contains all the points of set S. The average case cardinality of S′ is shown to rely on S and ǫ.