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

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## 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 $$\mathcal {RH}_{P}({\theta })$$, 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 $$\Theta =\min \{\Theta _i :i=1,\ldots ,2k\}$$. 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 $$\theta \in [0,2\pi )$$.

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Correspondence to Carlos Seara.

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Extended abstracts related to this work were presented at the XIV Spanish Meeting on Computational Geometry [2] and the 30th European Workshop on Computational Geometry (EuroCG) [4]. Carlos Alegría: Research supported in part by UNAM Project PAEP, and by MIUR Proj. “AHeAD” No. 20174LF3T8. David Orden: Research supported by Projects MTM2017-83750-P of the Spanish Ministry of Science (AEI/FEDER, UE), and Project PID2019-104129GB-I00 of the Spanish Ministry of Science and Innovation. Carlos Seara: Research supported by Projects Gen. Cat. DGR 2017SGR1640, MINECO MTM2015-63791-R, and Project PID2019-104129GB-I00 of the Spanish Ministry of Science and Innovation. Jorge Urrutia: Research supported in part by SEP-CONACYT 80268, PAPPIIT IN102117 Programa de Apoyo a la Investigación e Innovación Tecnológica UNAM. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 734922.

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Alegría, C., Orden, D., Seara, C. et al. Efficient computation of minimum-area rectilinear convex hull under rotation and generalizations. J Glob Optim 79, 687–714 (2021). https://doi.org/10.1007/s10898-020-00953-5