Approximate nearest neighbor queries in fixed dimensions

Given a set of n points in d-dimensional Euclidean space, S ⊂ E, and a query point q ∈ E, we wish to determine the nearest neighbor of q, that is, the point of S whose Euclidean distance to q is minimum. The goal is to preprocess the point set S, such that queries can be answered as efficiently as possible. We assume that the dimension d is a constant independent of n. Although reasonably good solutions to this problem exist when d is small, as d increases the performance of these algorithms degrades rapidly. We present a randomized algorithm for approximate nearest neighbor searching. Given any set of n points S ⊂ E, and a constant ǫ > 0, we produce a data structure, such that given any query point, a point of S will be reported whose distance from the query point is at most a factor of (1 + ǫ) from that of the true nearest neighbor. Our algorithm runs in O(log n) expected time and requires O(n log n) space. The data structure can be built in O(n) expected time. The constant factors depend on d and ǫ. Because of the practical importance of nearest neighbor searching in higher dimensions, we have implemented a practical variant of this algorithm, and show empirically that for many point distributions this variant of the algorithm finds the nearest neighbor in moderately large dimension significantly faster than existing practical approaches.