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 ∈ X whose distance to y is at most (1+ε) minx ∈ X |x-y| for an accuracy parameter ε ∈ (0,1). Our main result is a data structure that occupies only O(ε-2 n log(n) log(1/ε)) bits of space, assuming all point coordinates are integers in the range -nO(1) ldots nO(1), i.e., the coordinates have O(log n) bits of precision. This improves over the best previously known space bound of O(ε-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.