Conic nearest neighbor queries and approximate Voronoi diagrams

Abstract

Given a cone C and a set S of n points in ${\mathbb{R}}^d$, 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 $\tilde{O}(n/{\epsilon }^d)$ size that supports conic nearest neighbor queries in $O(\log (n/\epsilon ))$ 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 $\tilde{O}(n/{\epsilon }^{2d})$ our data structure further supports queries for any cone direction and angle.