Conic Nearest Neighbor Queries and Approximate Voronoi Diagrams∗

Given a cone C and a set S of n points in Rd, we want to preprocess S into a data structure so that we can find fast an approximate nearest neighbor to a query point q with respect to the points of S contained in the translation of C with apex at q. We develop an approximate conic Voronoi diagram of Õ(n/εd) size that supports conic nearest neighbor queries in O(log(n/ε)) time. Our preprocessing uses only the well-separated pair decomposition and a compressed quadtree. Previous results were restricted to simplicial cones and achieved polylogarithmic or higher query times. By increasing space to Õ(n/ε2d) our data structure further supports queries for any cone direction and angle.