CCCG 2009, Vancouver, BC, August 17–19, 2009 Optimal Empty Pseudo-Triangles in a Point Set * Hee-Kap Ahn † Sang Won Bae † Iris Reinbacher † Abstract Given n points in the plane, we study three optimiza- tion problems of computing an empty pseudo-triangle: we consider minimizing the perimeter, maximizing the area, and minimizing the longest maximal concave chain. We consider two versions of the problem: First, we assume that the three convex vertices of the pseudo- triangle are given. Let n denote the number of points that lie inside the convex hull of the three given ver- tices, we can compute the minimum perimeter or maxi- mum area pseudo-triangle in O(n 3 ) time. We can com- pute the pseudo-triangle with minimum longest con- cave chain in O(n 2 log n) time. If the convex vertices are not given, we achieve running times of O(n log n) for minimum perimeter, O(n 6 ) for maximum area, and O(n 2 log n) for minimum longest concave chain. In any case, we use only linear space. 1 Introduction A pseudo-triangle is a simply connected region of R 2 with exactly three convex vertices such that the boundary curves connecting pairs of these convex ver- tices must be concave. When all three boundary curves are polygonal, a pseudo-triangle is a simple polygon with exactly three convex vertices, that is, all the other vertices are concave (we consider a vertex with inter- nal angle π to be concave). By definition, any triangle is a pseudo-triangle. Moreover, the convex hull of any pseudo-triangle is a triangle. Pseudo-triangles were introduced in the context of computing visibility relations among convex obstacles in R 2 [8, 7]. Later, a number of different optimiza- tion problems of pseudo-triangulations, partitionings of a region into polygonal pseudo-triangles, have been studied [1, 2, 5, 6, 9]. For an overview of pseudo- triangulations, we refer to the survey by Rote et al. [10]. Here, we are interested in an “empty” pseudo-triangle in the sense that, given a finite set of points in R 2 , it contains no points from the set in its interior. In par- ticular, we consider the following problems: (1) either minimizing the perimeter or maximizing the area of the * This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Re- search Promotion Fund) (KRF-2007-331-D00372). † Department of Computer Science and Engineering, POSTECH, {heekap, swbae, irisrein}@postech.ac.kr empty pseudo-triangle, and (2) minimizing the longest maximal concave curve of the empty pseudo-triangle. We first study these problems when the three convex vertices are given. We will later consider optimizations over all possible pseudo-triangles in a point set. 2 Empty Pseudo-Triangles with Given Corners In this section, we consider finding empty pseudo- triangles in a given triangle that contains a set P of n points in its interior. 2.1 Preliminary Observations Observation 1 Every empty pseudo-triangle parti- tions P into three subsets. It follows that the convex hull of each subset is enclosed by the convex hull of each concave curve. An empty pseudo-triangle that either minimizes the perimeter or maximizes the area, is a simple polygon with concave vertices taken from P such that there are no points of P in its interior. The longest maximal concave curve is minimized by a concave polygonal chain on vertices of P . Therefore, it suffices to find an empty polygo- nal pseudo-triangle for each of the three optimization problems, and in the remainder of the paper we only consider such polygonal pseudo-triangles. Let T,L, and R be the three corners of the input tri- angle. W.l.o.g. we assume that the segment connecting L and R is horizontal with L to the left of R, and that T lies above the segment. Let P be a set of n points in- side the triangle. We let L = t 0 ,t 1 ,...,t n ,t n+1 = R be the sequence of points in P ∪{L, R} sorted in counter-clockwise order around T . Similarly, let R = l 0 ,l 1 ,...,l n ,l n+1 = T be the sorted sequence around L, and let T = r 0 ,r 1 ,...,r n ,r n+1 = L be the sorted sequence around R in counter-clockwise order. Assume that we are given an empty pseudo-triangle. At each corner, there are two concave chains, left and right. Let l i ,r j , and t k be the first vertices encountered when we follow the left concave chain from L, R, and T , respectively. Figure 1 shows these vertices and three lines, each induced by a corner and its corresponding vertex. Consider the gray region bounded by the lines ‘(L, l i ) and ‘(T,t k ). All points from P inside the gray region must be enclosed by the concave chain connecting L and T . Therefore the polygonal chain enclosing the