  $\newcommand{\ball}{\mathbb{B}}\newcommand{\dsQ}{{\mathcal{Q}}}\newcommand{\dsS}{{\mathcal{S}}}$In
this work we study a fair variant of the near neighbor problem. Namely, given a
set of $n$ points $P$ and a parameter $r$, the goal is to preprocess the
points, such that given a query point $q$, any point in the $r$-neighborhood of
the query, i.e., $\ball(q,r)$, have the same probability of being reported as
the near neighbor.
  We show that LSH based algorithms can be made fair, without a significant
loss in efficiency. Specifically, we show an algorithm that reports a point in
the $r$-neighborhood of a query $q$ with almost uniform probability. The query
time is proportional to $O\bigl( \mathrm{dns}(q.r) \dsQ(n,c) \bigr)$, and its
space is $O(\dsS(n,c))$, where $\dsQ(n,c)$ and $\dsS(n,c)$ are the query time
and space of an LSH algorithm for $c$-approximate near neighbor, and
$\mathrm{dns}(q,r)$ is a function of the local density around $q$.
  Our approach works more generally for sampling uniformly from a
sub-collection of sets of a given collection and can be used in a few other
applications. Finally, we run experiments to show performance of our approach
on real data.
