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- W4302375763 abstract "$renewcommand{Re}{{rm I!hspace{-0.025em} R}} newcommand{SetX}{mathsf{X}} newcommand{eps}{varepsilon} newcommand{VorX}[1]{mathcal{V} pth{#1}} newcommand{Polygon}{mathsf{P}} newcommand{IntRange}[1]{[ #1 ]} newcommand{Space}{ovebarline{mathsf{m}}} newcommand{pth}[2][!]{#1left({#2}right)} newcommand{Arr}{{cal A}}$ 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 $(1pmeps)n/r$. 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, 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." @default.
- W4302375763 created "2022-10-06" @default.
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- W4302375763 date "2017-01-01" @default.
- W4302375763 modified "2023-09-26" @default.
- W4302375763 title "Approximating the k-Level in Three-Dimensional Plane Arrangements" @default.
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- W4302375763 doi "https://doi.org/10.1007/978-3-319-44479-6_19" @default.
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