Locality-Sensitive Hashing Without False Negatives for \(l_p\)

  • Andrzej Pacuk
  • Piotr Sankowski
  • Karol Wegrzycki
  • Piotr WygockiEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9797)


In this paper, we show a construction of locality-sensitive hash functions without false negatives, i.e., which ensure collision for every pair of points within a given radius R in d dimensional space equipped with \(l_p\) norm when \(p \in [1,\infty ]\). Furthermore, we show how to use these hash functions to solve the c-approximate nearest neighbor search problem without false negatives. Namely, if there is a point at distance R, we will certainly report it and points at distance greater than cR will not be reported for \(c=\varOmega (\sqrt{d},d^{1-\frac{1}{p}})\). The constructed algorithms work:
  • with preprocessing time \(\mathcal {O}(n \log (n))\) and sublinear expected query time,

  • with preprocessing time \(\mathcal {O}(\mathrm {poly}(n))\) and expected query time \(\mathcal {O}(\log (n))\).

Our paper reports progress on answering the open problem presented by Pagh [8], who considered the nearest neighbor search without false negatives for the Hamming distance.



This work was supported by ERC PoC project PAAl-POC 680912 and FET project MULTIPLEX 317532. We would also like to thank Rafał Latała for meaningful discussions.


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Andrzej Pacuk
    • 1
  • Piotr Sankowski
    • 1
  • Karol Wegrzycki
    • 1
  • Piotr Wygocki
    • 1
    Email author
  1. 1.Institute of InformaticsUniversity of WarsawWarsawPoland

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