An Optimal Algorithm for Approximate Nearest Neighbor Searching in Fixed Dimensions

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An Optimal Algorithm for Approximate Nearest Neighbor Searching in Fixed Dimensions (1994)

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by Sunil Arya , David M. Mount , Nathan S. Netanyahu , Ruth Silverman , Angela Y. Wu

Venue:	ACM-SIAM SYMPOSIUM ON DISCRETE ALGORITHMS
Citations:	984 - 32 self

BibTeX

@INPROCEEDINGS{Arya94anoptimal,
    author = {Sunil Arya and David M. Mount and Nathan S. Netanyahu and Ruth Silverman and Angela Y. Wu},
    title = {An Optimal Algorithm for Approximate Nearest Neighbor Searching in Fixed Dimensions},
    booktitle = {ACM-SIAM SYMPOSIUM ON DISCRETE ALGORITHMS},
    year = {1994},
    pages = {573--582},
    publisher = {}
}

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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 structure, so that given any query point q 2 R d , the closest point of S to q can be reported quickly. Given any positive real ffl, a data point p is a (1 + ffl)-approximate nearest neighbor of q if its distance from q is within a factor of (1 + ffl) 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 2 R d , and ffl ? 0, a (1 + ffl)-approximate nearest neighbor of q can be computed in O(c d;ffl log n) time, where c d;ffl d d1 + 6d=ffle d is a factor depending only on dimension and ffl. In general, we show that given an integer k 1, (1 + ffl)-approximations to the k nearest neighbors of q can be computed in additional O(kd log n) time.

Keyphrases

fixed dimension optimal algorithm approximate nearest neighbor searching data point query point data structure real d-dimensional space positive real ffl dn log

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