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Duality-based approximation algorithms for depth queries and maximum depth
arXiv - CS - Computational Geometry Pub Date : 2020-06-22 , DOI: arxiv-2006.12318
Dror Aiger, Haim Kaplan, Micha Sharir

We design an efficient data structure for computing a suitably defined approximate depth of any query point in the arrangement $\mathcal{A}(S)$ of a collection $S$ of $n$ halfplanes or triangles in the plane or of halfspaces or simplices in higher dimensions. We then use this structure to find a point of an approximate maximum depth in $\mathcal{A}(S)$. Specifically, given an error parameter $\epsilon>0$, we compute, for any query point $q$, an underestimate $d^-(q)$ of the depth of $q$, that counts only objects containing $q$, but is allowed to exclude objects when $q$ is $\epsilon$-close to their boundary. Similarly, we compute an overestimate $d^+(q)$ that counts all objects containing $q$ but may also count objects that do not contain $q$ but $q$ is $\epsilon$-close to their boundary. Our algorithms for halfplanes and halfspaces are linear in the number of input objects and in the number of queries, and the dependence of their running time on $\epsilon$ is considerably better than that of earlier techniques. Our improvements are particularly substantial for triangles and in higher dimensions.

中文翻译:

用于深度查询和最大深度的基于对偶的近似算法

我们设计了一种有效的数据结构,用于计算平面或半空间中的 $n$ 个半平面或三角形的集合 $S$ 的排列 $\mathcal{A}(S)$ 中任何查询点的适当定义的近似深度更高维度的简单化。然后我们使用这个结构在 $\mathcal{A}(S)$ 中找到一个近似最大深度的点。具体来说,给定一个错误参数 $\epsilon>0$,我们计算,对于任何查询点 $q$,$q$ 深度的低估 $d^-(q)$,它只计算包含 $q$ 的对象,但允许在 $q$ 接近 $\epsilon$ 边界时排除对象。类似地,我们计算高估 $d^+(q)$,它计算包含 $q$ 的所有对象,但也可能计算不包含 $q$ 但 $q$ 接近其边界的对象。我们的半平面和半空间算法在输入对象的数量和查询的数量上是线性的,并且它们的运行时间对 $\epsilon$ 的依赖性比早期的技术要好得多。对于三角形和更高维度,我们的改进尤其显着。
更新日期:2020-06-23
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