Proximate point searching

摘要

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(log d(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(log d(q_(i-1), q_i)) runtime, or any bound that is o(log n).