Proximate point searching

In the 2D point searching problem, the goal is to preprocess n points P={p ₁,⋯,p _n} in the plane so that, for an online sequence of query points q ₁,⋯,q _m, it can quickly be determined which (if any) of the elements of P are equal to each query point q _i. This problem can be solved in O(logn) time by mapping the problem to one dimension. We present a data structure that is optimized for answering queries quickly when they are geometrically close to the previous successful query. Specifically, our data structure executes queries in time O(logd(q _i-1,q _i)), where d is some distance function between two points, and uses O(nlogn) space. Our structure works with a variety of distance functions. In contrast, it is proved that, for some of the most intuitive distance functions d, it is impossible to obtain an O(logd(q _i-1, q _i)) runtime, or any bound that is o(logn).