Abstract. Consider a set S of n data points in real d-dimensional space, R d , where distances are measured using any Minkowski metric. In nearest neighbor searching, we preprocess S into a data Permission to make digital / hard copy of part or all of this work for personal or classroom use is granted without fee provided that the copies are not made or distributed for profit or commercial advantage, the copyright notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery (ACM), Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and / or a fee. © 1999 ACM 0004-5411/99/1100-0891 $5.00 Journal of the ACM, Vol. 45, No. 6, November 1998, pp. 891-923. structure, so that given any query point q ʦ R d , is the closest point of S to q can be reported quickly. Given any positive real ⑀, a data point p is a (1 ϩ ⑀)-approximate nearest neighbor of q if its distance from q is within a factor of (1 ϩ ⑀) of the distance to the true nearest neighbor. We show that it is possible to preprocess a set of n points in R d in O(dn log n) time and O(dn) space, so that given a query point q ʦ R d , and ⑀ Ͼ 0, a (1 ϩ ⑀)-approximate nearest neighbor of q can be computed in O(c d,⑀ log n) time, where c d,⑀ Յ d 1 ϩ 6d/⑀ d is a factor depending only on dimension and ⑀. In general, we show that given an integer k Ն 1, (1 ϩ ⑀)-approximations to the k nearest neighbors of q can be computed in additional O(kd log n) time.
One formulation is (1+ ) -approximate k-NN search REF , in which the distance of the points returned by the algorithm is at most (1 + ) times the distance of the true kth nearest neighbor of the query point.
An optimal algorithm for approximate nearest neighbor searching fixed dimensions
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Mathematics Computer Science