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

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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).

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