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

• Published in: Journal of global optimization Vol. 79; no. 3; pp. 687 - 714
• Main Authors: Alegría, Carlos, Orden, David, Seara, Carlos, Urrutia, Jorge
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2021; Springer; Springer Nature B.V
• Subjects: Algorithms; Computational geometry; Computer Science; Convexity; K lines; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Real Functions; Restricted orientation convex hull; Minimum area; Rectilinear convex hull
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-020-00953-5

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 ) .