On Vertical Ray Shooting in Arrangements

Jiri Matousek Department of Computer Science Charles University, CSFR Report B 92-06 February 1992

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We consider the following problem: Given a collection $H$ of $n$ hyperplanes in $\Ed$, preprocess it so that given a query point $x$, a hyperplane of $H$ lying immediately above $x$ can be detected quickly. We give a relatively simple solution with $O(n^d/\log^{d-1} n)$ space and deterministic preprocessing time and $O(\log n)$ query time. This gives a slightly more efficient and considerably simplified alternative of a previous solution due to Chazelle and Friedman.