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Hardness of approximate nearest neighbor search

Published:20 June 2018Publication History

ABSTRACT

We prove conditional near-quadratic running time lower bounds for approximate Bichromatic Closest Pair with Euclidean, Manhattan, Hamming, or edit distance. Specifically, unless the Strong Exponential Time Hypothesis (SETH) is false, for every δ>0 there exists a constant ε>0 such that computing a (1+ε)-approximation to the Bichromatic Closest Pair requires Ω(n2−δ) time. In particular, this implies a near-linear query time for Approximate Nearest Neighbor search with polynomial preprocessing time.

Our reduction uses the recently introduced Distributed PCP framework, but obtains improved efficiency using Algebraic Geometry (AG) codes. Efficient PCPs from AG codes have been constructed in other settings before, but our construction is the first to yield new hardness results.

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      cover image ACM Conferences
      STOC 2018: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing
      June 2018
      1332 pages
      ISBN:9781450355599
      DOI:10.1145/3188745

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