Fast Locality-sensitive Hashing Frameworks For Approximate Near Neighbor Search

Christiani Tobias. Arxiv 2017

[Paper]    
ARXIV FOCS Independent LSH

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 $O(n^{\rho }logn)$ 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 $O(lo{g}^{2}n)$, leaving the query time to be dominated by $O(n^{\rho })$ distance computations and $O(n^{\rho }logn)$ 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, making it a viable replacement in practice. 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 $O(n^{\rho })$.

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