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

Abstract

Let P be a set of n points in the plane. We compute the value of ¿¿[0,2p) for which the rectilinear convex hull of P, denoted by RHP(¿), has minimum (or maximum) area in optimal O(nlogn) 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 ai be the sizes of the 2k angles defined by pairs of two consecutive lines, i=1,…,2k. Let Ti=p-ai and T=min{Ti:i=1,…,2k}. We obtain: (1) Given a set O such that T=p2, we provide an algorithm to compute the O-convex hull of P in optimal O(nlogn) time and O(n) space; If T<p2, the time and space complexities are O(nTlogn) and O(nT) respectively. (2) Given a set O such that T=p2, we compute and maintain the boundary of the O¿-convex hull of P for ¿¿[0,2p) in O(knlogn) time and O(kn) space, or if T<p2, in O(knTlogn) time and O(knT) space. (3) Finally, given a set O such that T=p2, we compute, in O(knlogn) time and O(kn) space, the angle ¿¿[0,2p) such that the O¿-convex hull of P has minimum (or maximum) area over all ¿¿[0,2p).