An optimal algorithm for approximate nearest neighbor searching in fixed dimensions

Consider a set S of n data points in real d-dimensional space, R^d, where distances are measured using any Minkowski metric. In nearest neighbor searching, we preprocess S into a data structure, so that given any query point q ∈ R^d, is the closest point of 5 to q can be reported quickly. Given any positive real ∈, a data point p is a (1 + ∈)-approximate nearest neighbor of q if its distance from q is within a factor of (1 + ∈) of the distance to the true nearest neighbor. We show that it is possible to preprocess a set_of n points in R^d in O(dn log n) time and O(dn) space, so that given a query point q ∈ R^d, and ∈ > 0, a (1 + ∈)-approximate nearest neighbor of q can be computed in O(c_d,∈ log n) time, where c_d,∈ ≤ d ⌈1 + 6d/∈⌉^d is a factor depending only on dimension and ∈. In general, we show that given an integer k ≥ 1, (1 + ∈)-approximations to the k nearest neighbors of q can be computed in additional O(kd log n) time.