半代数范围报告和空搜索及其应用
Semi-algebraic Range Reporting and Emptiness Searching with Applications
摘要：
在典型的范围空搜索中（分别地，报告）问题时，我们被给出$\reals^d$中的$n$点的集合$P$，并且希望将其预处理成支持有效范围空性的数据结构（分别，报告）查询，其中我们指定范围$σ$，其通常是恒定描述复杂度的$\reals^d$中的半代数集合，并且希望确定是否为$P\capσ=\emptyset$，或者报告$P\capσ$中的所有点。范围空搜索和报告出现在许多应用中，并且已经由Matoušek \cite{Ma:rph}在范围是由超平面限定的半空间的特殊情况下处理。如\cite{Ma:rph}中所示，这两个问题密切相关，并且具有具有类似性能界限的解决方案（对于半空间的情况）。在本文中，我们扩展到一般的半代数范围的分析，并显示如何适应Matoušek的技术，而不需要{ \em线性化}的范围到一个高维空间。这将产生更有效的解决方案，几个有用的问题，我们证明了新技术在四个应用程序。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
