Approximate Nearest Neighbors in Limited Space

We consider the $(1+ϵ)$-approximate nearest neighbor search problem: given a set $X$ of $n$ points in a $d$-dimensional space, build a data structure that, given any query point $y$, finds a point $x\in X$ whose distance to $y$ is at most $(1+ϵ){\min }_{x\in X}\mathrm{\parallel }x-y\mathrm{\parallel }$ for an accuracy parameter $ϵ\in (0,1)$. Our main result is a data structure that occupies only $O({ϵ}^{-2}n\log (n)\log (1\mathrm{/}ϵ))$ bits of space, assuming all point coordinates are integers in the range ${-n^{O(1)} \ldots n^{O(1)}}$,i\mathrm{.}e\mathrm{.},thecoordinateshave$O(\log n)$ bits of precision. This improves over the best previously known space bound of $O(\epsilon^{-2} n \log(n)^2)$, obtained via the randomized dimensionality reduction method of Johnson and Lindenstrauss (1984). We also consider the more general problem of estimating all distances from a collection of query points to all data points $X$, and provide almost tight upper and lower bounds for the space complexity of this problem.