Conic nearest neighbor queries and approximate Voronoi diagrams

Given a cone C and a set S of n points in 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 O ? ( n / e d ) size that supports conic nearest neighbor queries in O ( log ? ( n / e ) ) 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 / e 2 d ) our data structure further supports queries for any cone direction and angle.

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