Rectilinear Convex Hull with Minimum Area

Let P be a set of n points in the plane. We solve the problem of computing the orientations for which the rectilinear convex hull of P has minimum area in optimal ⇥(n log n) time and O(n) space. Introduction The interest in the rectilinear convex hull of planar point sets arises from the study of ortho-convexity [10], a relaxation of traditional convexity. Unlike convex regions, an ortho-convex region might be disconnected, which makes the study of the ortho-convex closure for a point set [5, 8] harder. Several definitions have been presented by different authors. We will use a definition stated by Ottman et al. [8] as the mr-convex hull, see also Matousěk et al. [5, 7]. The study of rectilinear convex hulls has gained attention partly because of some applications in digital image processing [3] and VLSI circuit layout design [11]. The rectilinear convex hull of point sets is an orientation-dependent region, i.e., it changes as the orientation of the plane changes. In this paper we are interested in computing an orientation for which the rectilinear convex hull of P has minimum area. We show that the set of orientations ✓ 2 [0, 2⇡) can be divided into a set of linear intervals such that, within each interval I, the angle ✓ 2 I which minimizes the area of the rectilinear convex hull of a point set (save the first one we process) can be calculated in constant time. These intervals can be computed in O(n log n) time and O(n) space. Using this result and based on techniques from Avis et al. [1], Bae et al. [2], and DíazBáñez et al. [4], we present an optimal ⇥(n log n) time and O(n) space algorithm for this problem. Our result improves the O(n2) time complexity presented by Bae et al. [2]. 1 Terminology and notation An orthogonal wedge is the intersection of two open half-planes whose supporting lines are orthogonal. The apex of the wedge is the intersection point of these supporting lines. An orthogonal wedge is P -free if it does not contain points of P in its interior. An orthogonal wedge is called a ✓-wedge if its supporting lines can be obtained by first rotating the X1Coauthors from UNAM are partially supported by projects MTM2006-03909 of Spain and SEPCONACYT 80268 of México. The coauthor from UPC is partially supported by projects MTM2009-07242 and Gen. Cat. DGR 2009GR1040. CRM Documents, vol. 8, Centre de Recerca Matemàtica, Bellaterra (Barcelona), 2011 185 186 Rectilinear convex hull with minimum area and Y -axis ✓ degrees, and then translating the origin to the apex of our wedge. The rectilinear convex hull of P with orientation ✓ is the region RH✓(P ) = R [ w2W✓ w, where W✓ is the set of all P -free orthogonal ✓-wedges [2, 4, 8]. As ✓ changes, the set of orthogonal P -free ✓-wedges change, and, thus, RH✓(P ) changes (see Figure 1). A ✓-orientation of the plane, ✓ 2 [0, 2⇡), is the coordinate system obtained by rotating the axes of R by ✓ degrees with respect to the origin. For a fixed ✓, RH✓(P ) has a close relation to the maxima problem [6, 9]. A vertex of RH✓(P ) is a point in P that lies on the boundary of RH✓(P ). Let X✓(P ) be the set of maximal points of P with respect to vector dominance in the ✓-orientation of the plane. The set of vertices of RH✓(P ) is equal to the set X✓(P )[X✓+⇡ 2 (P )[X✓+⇡(P )[X✓+ 3 2 ⇡(P ) [2, 8]. Given a fixed ✓, RH✓(P ) can be computed in optimal ⇥(n log n) time and O(n) space [6, 9]. We say that a point p 2 P is ✓-maximal with respect to P if there is an orthogonal P -free wedge with apex at p in a ✓-orientation of the plane. The set of orientations for which p is ✓-maximal forms at most three intervals. The endpoints of each interval mark the inand an outevents of p, i.e., the ✓-orientations when p becomes and stops being ✓-maximal. The set of intervals corresponding to the elements of P and the set of angles at which these points of P start and stop being ✓-maximal can be computed in optimal ⇥(n log n) time and O(n) space [4].