Faster deterministic and Las vegas algorithms for offline approximate nearest neighbors in high dimensions

Josh Alman, Timothy M. Chan, Ryan Williams

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We present a deterministic, truly subquadratic algorithm for offline (1 + ε)-approximate nearest or farthest neighbor search (in particular, the closest pair or diameter problem) in Hamming space in any dimension d ≤ nδ, for a sufficiently small constant δ > 0. The running time of the algorithm is roughly n2ε1/2+O(δ) for nearest neighbors, or n2−Ω(√ε/log(1/ε)) for farthest. The algorithm follows from a simple combination of expander walks, Chebyshev polynomials, and rectangular matrix multiplication. We also show how to eliminate errors in the previous Monte Carlo randomized algorithm of Alman, Chan, and Williams [FOCS'16] for offline approximate nearest or farthest neighbors, and obtain a Las Vegas randomized algorithm with expected running time n2−Ω(ε1/3/log(1/ε)) . Finally, we note a simplification of Alman, Chan, and Williams' method and obtain a slightly improved Monte Carlo randomized algorithm with running time n2−Ω(ε1/3/log2/3(1/ε)) . As one application, we obtain improved deterministic and randomized (1+ε)-approximation algorithms for MAX-SAT.

Original languageEnglish (US)
Title of host publication31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020
EditorsShuchi Chawla
PublisherAssociation for Computing Machinery
Pages637-649
Number of pages13
ISBN (Electronic)9781611975994
StatePublished - 2020
Event31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020 - Salt Lake City, United States
Duration: Jan 5 2020Jan 8 2020

Publication series

NameProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
Volume2020-January

Conference

Conference31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020
Country/TerritoryUnited States
CitySalt Lake City
Period1/5/201/8/20

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

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