Efficient computation of minimum-area rectilinear convex hull under rotation and generalizations

TL;DR

This paper presents an efficient algorithm to compute the rotation angle that minimizes or maximizes the area of the rectilinear convex hull of a point set, improving previous bounds to optimal $O(n ext{log} n)$ time.

Contribution

It introduces a novel $O(n ext{log} n)$ algorithm for optimal rotation of rectilinear convex hulls, extending to generalizations involving $ heta$-convex hulls and multiple line sets.

Findings

Achieved $O(n ext{log} n)$ time complexity for minimum-area rectilinear convex hull rotation.

Extended algorithms to handle $ heta$-convex hulls with multiple line orientations.

Provided methods to compute and maintain convex hull boundaries under rotation.

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 sizes of the $2k$ angles defined by pairs of two consecutive lines, $i=1, \ldots , 2k$. Let $\Theta_{i}=\pi-\alpha_i$ and $\Theta=\min\{\Theta_i \colon i=1,\ldots,2k\}$. We 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; If $\Theta < \frac{\pi}{2}$, the time and space complexities are $O(\frac{n}{\Theta}\log n)$ and $O(\frac{n}{\Theta})$ respectively. (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 if $\Theta < \frac{\pi}{2}$, in $O(k\frac{n}{\Theta}\log n)$ time and $O(k\frac{n}{\Theta})$ space. (3) Finally, given a set $\mathcal{O}$ such that $\Theta\ge\frac{\pi}{2}$, we compute, in $O(kn\log n)$ time and $O(kn)$ space, the angle $\theta\in [0,2\pi)$ such that the $\mathcal{O}_{\theta}$-convex hull of $P$ has minimum (or maximum) area over all $\theta\in [0,2\pi)$.