Fast Locality-Sensitive Hashing Frameworks for Approximate Near Neighbor Search

Abstract

The Indyk-Motwani Locality-Sensitive Hashing (LSH) framework (STOC 1998) is a general technique for constructing a data structure to answer approximate near neighbor queries by using a distribution \(\mathcal {H}\) over locality-sensitive hash functions that partition space. For a collection of n points, after preprocessing, the query time is dominated by \(O(n^{\rho } \log n)\) evaluations of hash functions from \(\mathcal {H}\) and \(O(n^{\rho })\) hash table lookups and distance computations where \(\rho \in (0,1)\) is determined by the locality-sensitivity properties of \(\mathcal {H}\). It follows from a recent result by Dahlgaard et al. (FOCS 2017) that the number of locality-sensitive hash functions can be reduced to \(O(\log ^2 n)\), leaving the query time to be dominated by \(O(n^{\rho })\) distance computations and \(O(n^{\rho } \log n)\) additional word-RAM operations. We state this result as a general framework and provide a simpler analysis showing that the number of lookups and distance computations closely match the Indyk-Motwani framework. Using ideas from another locality-sensitive hashing framework by Andoni and Indyk (SODA 2006) we are able to reduce the number of additional word-RAM operations to \(O(n^\rho )\).