Efficient computation of minimum-area rectilinear convex hull under rotation and generalizations - Alegría, Carlos; Orden, David; Seara, Carlos; Urrutia, Jorge (Journal of global optimization, 2021)
Let P be a set of n points in the plane. We compute the value of θ ∈ [ 0 , 2 π ) for which the rectilinear convex hull of P , denoted by RH P ( θ ) , has minimum (or maximum) area in optimal O ( n log n ) time and O ( n ) space, improving the previous O ( n 2 ) bound. Let O be a set of k lines through the origin sorted by slope and let α i be the sizes of the 2 k angles defined by pairs of two consecutive lines, i = 1 , … , 2 k . Let Θ i = π - α i and Θ = min { Θ i : i = 1 , … , 2 k } . We obtain: (1) Given a set O such that Θ ≥ π 2 , we provide an algorithm to compute the O -convex hull of P in optimal O ( n log n ) time and O ( n ) space; If Θ < π 2 , the time and space complexities are O ( n Θ log n ) and O ( n Θ ) respectively. (2) Given a set O such that Θ ≥ π 2 , we compute and maintain the boundary of the O θ -convex hull of P for θ ∈ [ 0 , 2 π ) in O ( k n log n ) time and O ( kn ) space, or if Θ < π 2 , in O ( k n Θ log n ) time and O ( k n Θ ) space. (3) Finally, given a set O such that Θ ≥ π 2 , we compute, in O ( k n log n ) time and O ( kn ) space, the angle θ ∈ [ 0 , 2 π ) such that the O θ -convex hull of P has minimum (or maximum) area over all θ ∈ [ 0 , 2 π ) .