Conic nearest neighbor queries and approximate Voronoi diagrams

Abstract

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 O˜(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 O˜(n/ε2d) our data structure further supports queries for any cone direction and angle.