Algorithms Seminar: A Linear-Space Data Structure for Answering Distance Queries for Marked Points on a Grid (draft)

We examine a variant of the clasic nearest neighbor problem, where given a set of k points, we preprocess the points so that we can quickly answer the distance to the nearest neighbor to a query point. We specifically focus on the case where all points—both the k original points as well all query points—are confined to an n×n integer grid in the plane. Furthermore, we only require the distance to the nearest point to be reported (i.e. not the point itself) in the L2 metric. We hope to extend this to higher dimensions and different metrics in the future, but we always require integer coordinates for all points. Our result is a data structure that requires O(N) preprocessing time and O(N) space that can answer queries in O(1) time.