New techniques for exact and approximate dynamic closest-point problems
Sanjiv Kapoor, Michiel Smid · 1994
Let S be a set of n points in RD. It is shown that a range tree can be used to find an L∞ -nearest neighbor in S of any query point, in O((logn)D-1 loglogn) time. This data structure has size O(n(logn)D-1) and an amortized update time of O((logn)D-1 loglogn). This result is used to solve the (1+ ϵ)-approximate L2-nearest neighbor problem within the same bounds. In this problem, for any query point p, a point ∈ is computed such that the euclidean distance between p and q is at most (1+ϵ) times the euclidean distance between p and its true nearest neighbor. This is the first dynamic data structure for this problem having close to linear size and polylogarithmic query and update times.