Efficient computation of minimumarea rectilinear convex hull under rotation and generalizations
Abstract
Let $P$ be a set of $n$ points in the plane. We compute the value of $\theta\in [0,2\pi)$ for which the rectilinear convex hull of $P$, denoted by $\mathcal{RH}_{\theta}(P)$, has minimum (or maximum) area in optimal $O(n\log n)$ time and $O(n)$ space, improving the previous $O(n^2)$ bound. Let $\mathcal{O}$ be a set of $k$ lines through the origin sorted by slope and let $\alpha_i$ be the aperture angles of the $2k$ sectors defined by every pair of two consecutive lines. Let $\Theta_{i}=\pi\alpha_i$ and $\Theta=\min\{\Theta_i:i=1,\ldots,2k\}$. We further obtain: (1) Given a set $\mathcal{O}$ such that $\Theta\ge\frac{\pi}{2}$, we provide an algorithm to compute the $\mathcal{O}$convex hull of $P$ in optimal $O(n\log n)$ time and $O(n)$ space, while if $\Theta<\frac{\pi}{2}$ the complexities are $O(\frac{n}{\Theta}\log n)$ time and $O(\frac{n}{\Theta})$ space. (2) Given a set $\mathcal{O}$ such that $\Theta\ge\frac{\pi}{2}$, we compute and maintain the boundary of the ${\mathcal{O}}_{\theta}$convex hull of $P$ for $\theta\in [0,2\pi)$ in $O(kn\log n)$ time and $O(kn)$ space, or in $O(k\frac{n}{\Theta}\log n)$ time and $O(k\frac{n}{\Theta})$ space if $\Theta<\frac{\pi}{2}$. (3) Finally, given a set $\mathcal{O}$ such that $\Theta\ge\frac{\pi}{2}$, we compute the ${\mathcal{O}}_{\theta}$convex hull of $P$ of minimum (or maximum) area over all $\theta\in [0,2\pi)$ in $O(kn\log n)$ time and $O(kn)$ space.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 DOI:
 10.48550/arXiv.1710.10888
 arXiv:
 arXiv:1710.10888
 Bibcode:
 2017arXiv171010888A
 Keywords:

 Computer Science  Computational Geometry;
 Mathematics  Combinatorics
 EPrint:
 28 pages, 23 figures. Changes in the order in which results are presented, as well as several further improvements and clarifications. This work has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk{\l}odowskaCurie Actions  RISE  grant agreement No 734922  CONNECT