Efficient computation of minimum-area rectilinear convex hull under rotation and generalizations
Let P beasetofn points in the plane. We compute the value of θ ∈[0,2π) for which the rectilinear convex hull of P, denoted by RHP(θ), has minimum (or maximum) area in optimal O(n logn) time and O(n) space, improving the previous O(n2) bound. Let O be a set of k lines through the origin sorted by slo...
| Autores: | , , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2021 |
| País: | España |
| Institución: | Universidad de Alcalá (UAH) |
| Repositorio: | e_Buah Biblioteca Digital Universidad de Alcalá |
| Idioma: | inglés |
| OAI Identifier: | oai:ebuah.uah.es:10017/64217 |
| Acceso en línea: | http://hdl.handle.net/10017/64217 https://dx.doi.org/10.1007/s10898-020-00953-5 |
| Access Level: | acceso abierto |
| Palabra clave: | Rectilinear convex hull Restricted orientation convex hull Minimum area Matemáticas Mathematics |
| Sumario: | Let P beasetofn points in the plane. We compute the value of θ ∈[0,2π) for which the rectilinear convex hull of P, denoted by RHP(θ), has minimum (or maximum) area in optimal O(n logn) time and O(n) space, improving the previous O(n2) bound. Let O be a set of k lines through the origin sorted by slope and let αi be the sizes of the 2k angles defined by pairs of two consecutive lines, i = 1,...,2k.Let i = π − αi and =min{ i : i =1,...,2k}.Weobtain:(1)Givenaset O suchthat ≥ π 2 ,weprovidean algorithm to compute the O-convex hull of P in optimal O(n logn) time and O(n) space; If <π 2 , the time and space complexities are O( n logn) and O( n) respectively. (2) Given a set O such that ≥ π 2 , we compute and maintain the boundary of the Oθ-convex hull of P for θ ∈[0,2π)in O(knlogn)time and O(kn)space, or if <π 2 ,inO(k n logn)time and O(k n) space. (3) Finally, given a set O such that ≥ π 2 , we compute, in O(knlogn) time and O(kn) space, the angle θ ∈[0,2π)such that the Oθ-convex hull of P has minimum (or maximum) area over all θ ∈[0,2π). |
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