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An optimal algorithm for approximate nearest neighbor searching fixed dimensions

Authors:

David M. Mount,

Nathan S. Netanyahu,

Ruth Silverman,

Angela Y. WuAuthors Info & Claims

Journal of the ACM (JACM), Volume 45, Issue 6

Pages 891 - 923

https://doi.org/10.1145/293347.293348

Published: 01 November 1998 Publication History

Abstract

Consider a set of 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 structure, so that given any query point q∈ R^d, is the closest point of S to q can be reported quickly. Given any positive real ϵ, data point p is a (1 +ϵ)-approximate nearest neighbor of q if its distance from q is within a factor of (1 + ϵ) 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 ∈ R^d, and ϵ > 0, a (1 + ϵ)-approximate nearest neighbor of q can be computed in O(c_{d, ϵ} log n) time, where c_d,ϵ≤d ⌈1 + 6d/ϵ⌉^d is a factor depending only on dimension and ϵ. In general, we show that given an integer k ≥ 1, (1 + ϵ)-approximations to the k nearest neighbors of q can be computed in additional O(kd log n) time.

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Published In

cover image Journal of the ACM

Journal of the ACM Volume 45, Issue 6

Nov. 1998

185 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/293347

Editor:
F. Thomson Leighton
Massachusetts Institute of Technology, Cambridge

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Copyright © 1998 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Publication History

Published: 01 November 1998

Published in JACM Volume 45, Issue 6

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