Approximate Nearest Neighbors in Limited Space

We consider the (1 + ∈)-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 + ∈) min_x∈_X kx − yk for an accuracy parameter ∈ (0, 1). Our main result is a data structure that occupies only O(∈⁻²n log(n) log(1/∈)) bits of space, assuming all point coordinates are integers in the range {−n^O(1) . . . n^O(1)}, i.e., the coordinates have O(log n) bits of precision. This improves over the best previously known space bound of O(∈⁻²n 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.