近似凸交检测及其在宽度和Minkowski和中的应用
Approximate Convex Intersection Detection with Applications to Width and Minkowski Sums
摘要：
$d$维空间中涉及单个凸体的逼近问题在计算几何界受到了极大的关注。相比之下，涉及多个凸体的工作通常限于尺寸$d \leq 3$和/或不考虑近似。在本文中，我们考虑两个自然的问题，涉及多个凸体的近似：检测是否两个多面体相交和计算他们的Minkowski和。给定一个近似参数$\varepsilon > 0$，我们展示了如何独立地将两个多面体$A,B$预处理成大小为$O(1/\varepsilon^{(d-1)/2})$的数据结构，这样我们就可以在多对数时间内回答$A$和$B$是否近似相交。更一般地说，我们可以对$A$和$B$在仿射变换下的图像回答这个问题。接下来，我们展示了如何$\varepsilon$ -近似Minkowski和两个给定的多面体定义为交叉的$n$半空间在$O(n \log(1/\varepsilon) + 1/\varepsilon^{(d-1)/2 + α})$时间，任何常数$α> 0$。最后，我们提出了一个令人惊讶的影响，这些结果的一个很好的研究问题，认为一个单一的凸体。我们展示了如何$\varepsilon$ -近似的宽度的一组$n$点在$O(n \log(1/\varepsilon) + 1/\varepsilon^{(d-1)/2 + α})$时间，对于任何恒定的$α> 0$，一个重大的改进，在以前的边界大致$O(n + 1/\varepsilon^{d-1})$时间。
英文摘要：
Approximation problems involving a single convex body in $d$-dimensional space have received a great deal of attention in the computational geometry community. In contrast, works involving multiple convex bodies are generally limited to dimensions $d \leq 3$ and/or do not consider approximation. In this paper, we consider approximations to two natural problems involving multiple convex bodies: detecting whether two polytopes intersect and computing their Minkowski sum. Given an approximation parameter $\varepsilon > 0$, we show how to independently preprocess two polytopes $A,B$ into data structures of size $O(1/\varepsilon^{(d-1)/2})$ such that we can answer in polylogarithmic time whether $A$ and $B$ intersect approximately. More generally, we can answer this for the images of $A$ and $B$ under affine transformations. Next, we show how to $\varepsilon$-approximate the Minkowski sum of two given polytopes defined as the intersection of $n$ halfspaces in $O(n \log(1/\varepsilon) + 1/\varepsilon^{(d-1)/2 + α})$ time, for any constant $α> 0$. Finally, we present a surprising impact of these results to a well studied problem that considers a single convex body. We show how to $\varepsilon$-approximate the width of a set of $n$ points in $O(n \log(1/\varepsilon) + 1/\varepsilon^{(d-1)/2 + α})$ time, for any constant $α> 0$, a major improvement over the previous bound of roughly $O(n + 1/\varepsilon^{d-1})$ time.
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