  We present a framework for similarity search based on Locality-Sensitive
Filtering (LSF), generalizing the Indyk-Motwani (STOC 1998) Locality-Sensitive
Hashing (LSH) framework to support space-time tradeoffs. Given a family of
filters, defined as a distribution over pairs of subsets of space with certain
locality-sensitivity properties, we can solve the approximate near neighbor
problem in $d$-dimensional space for an $n$-point data set with query time
$dn^{\rho_q+o(1)}$, update time $dn^{\rho_u+o(1)}$, and space usage $dn + n^{1
+ \rho_u + o(1)}$. The space-time tradeoff is tied to the tradeoff between
query time and update time, controlled by the exponents $\rho_q, \rho_u$ that
are determined by the filter family. Locality-sensitive filtering was
introduced by Becker et al. (SODA 2016) together with a framework yielding a
single, balanced, tradeoff between query time and space, further relying on the
assumption of an efficient oracle for the filter evaluation algorithm. We
extend the LSF framework to support space-time tradeoffs and through a
combination of existing techniques we remove the oracle assumption.
  Building on a filter family for the unit sphere by Laarhoven (arXiv 2015) we
use a kernel embedding technique by Rahimi & Recht (NIPS 2007) to show a
solution to the $(r,cr)$-near neighbor problem in $\ell_s^d$-space for $0 < s
\leq 2$ with query and update exponents
$\rho_q=\frac{c^s(1+\lambda)^2}{(c^s+\lambda)^2}$ and
$\rho_u=\frac{c^s(1-\lambda)^2}{(c^s+\lambda)^2}$ where $\lambda\in[-1,1]$ is a
tradeoff parameter. This result improves upon the space-time tradeoff of
Kapralov (PODS 2015) and is shown to be optimal in the case of a balanced
tradeoff. Finally, we show a lower bound for the space-time tradeoff on the
unit sphere that matches Laarhoven's and our own upper bound in the case of
random data.
