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arXiv · 0908.4061

Semi-algebraic Range Reporting and Emptiness Searching with Applications

Micha Sharir·Hayim Shaul
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Abstract

In a typical range emptiness searching (resp., reporting) problem, we are given a set $P$ of $n$ points in $\reals^d$, and wish to preprocess it into a data structure that supports efficient range emptiness (resp., reporting) queries, in which we specify a range $σ$, which, in general, is a semi-algebraic set in $\reals^d$ of constant description complexity, and wish to determine whether $P\capσ=\emptyset$, or to report all the points in $P\capσ$. Range emptiness searching and reporting arise in many applications, and have been treated by Matoušek \cite{Ma:rph} in the special case where the ranges are halfspaces bounded by hyperplanes. As shown in \cite{Ma:rph}, the two problems are closely related, and have solutions (for the case of halfspaces) with similar performance bounds. In this paper we extend the analysis to general semi-algebraic ranges, and show how to adapt Matoušek's technique, without the need to {\em linearize} the ranges into a higher-dimensional space. This yields more efficient solutions to several useful problems, and we demonstrate the new technique in four applications.

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Micha Sharir, Hayim Shaul. 2009-08-31. Semi-algebraic Range Reporting and Emptiness Searching with Applications. https://arxiv.org/abs/0908.4061

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