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Approximating the $k$-Level in Three-Dimensional Plane Arrangements $\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\IntRange}[1]{[ #1 ]} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)} \newcommand{\Arr}{{\cal A}}$

Sariel Har-Peled, Haim Kaplan, and Micha Sharir.
Let $H$ be a set of $n$ planes in three dimensions, and let $r \leq n$ be a parameter. We give a simple alternative proof of the existence of a $(1/r)$-cutting of the first $n/r$ levels of $\Arr(H)$, which consists of $O(r)$ semi-unbounded vertical triangular prisms. The same construction yields an approximation of the $(n/r)$-level by a terrain consisting of $O(r/\eps^3)$ triangular faces, which lies entirely between the levels $(1\pm\eps)n/r$. The proof does not use sampling, and exploits techniques based on planar separators and various structural properties of levels in three-dimensional arrangements and of planar maps. The proof is constructive, and leads to a simple randomized algorithm, with expected near-linear running time. An application of this technique allows us to mimic Matousek's construction of cuttings in the plane~\cite{m-cen-90}, to obtain a similar construction of ``layered'' $(1/r)$-cutting of the entire arrangement $\Arr(H)$, of optimal size $O(r^3)$. Another application is a simplified optimal approximate range counting algorithm in three dimensions, competing with that of Afshani and Chan.
slides of SODA talk.
Last modified: Fri Sep 25 15:19:22 CDT 2015