An optimal algorithm for approximate nearest neighbor searching fixed dimensions

  • Authors:
  • Sunil Arya;David M. Mount;Nathan S. Netanyahu;Ruth Silverman;Angela Y. Wu

  • Affiliations:
  • Hong Kong Univ. of Science of Technology, Kowloon, Hong Kong;Univ. of Maryland, College Park;Univ. of Maryland, College Park/ and NASA, Greenbelt, MD;Univ. of the District of Columbia, Washington, D.C./ and Univ. of Maryland, College Park;American Univ., Washington, D.C.

  • Venue:
  • Journal of the ACM (JACM)
  • Year:
  • 1998

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Abstract

Consider a set of S of n data points in real d-dimensional space, Rd, where distances are measured using any Minkowski metric. In nearest neighbor searching, we preprocess S into a data structure, so that given any query point q ∈ Rd, is the closest point of S to q can be reported quickly. Given any positive real &egr;, data point p is a (1 +&egr;)-approximate nearest neighbor of q if its distance from q is within a factor of (1 + &egr;) of the distance to the true nearest neighbor. We show that it is possible to preprocess a set of n points in Rd in O(dn log n) time and O(dn) space, so that given a query point q ∈ Rd, and &egr; 0, a (1 + &egr;)-approximate nearest neighbor of q can be computed in O(cd, &egr; log n) time, where cd,&egr;≤d 1 + 6d/e;d is a factor depending only on dimension and &egr;. In general, we show that given an integer k ≥ 1, (1 + &egr;)-approximations to the k nearest neighbors of q can be computed in additional O(kd log n) time.