Near Neighbor: Who is the Fairest of Them All?

Abstract

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., $B(q,r)$, 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 $p$ in the $r$-neighborhood of a query $q$ with almost uniform probability. The time to report such a point is proportional to $O(\dns(q.r) Q(n,c))$, and its space is $O(S(n,c))$, where $Q(n,c)$ and $S(n,c)$ are the query time and space of an LSH algorithm for $c$-approximate near neighbor, and $\dns(q,r)$ 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.