Optimal Empty Pseudo-Triangles in a Point Set

Given n points in the plane, we study three optimization 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 vertices. Then, we can compute the minimum perimeter or maximum area pseudo-triangle in O(n3 ) time, and the pseudo-triangle with minimum longest concave chain in O(n2 log n) time. If the convex vertices are not given, we achieve running times of O(n log n) for minimum perimeter, O(n6 ) for maximum area, and O(n2 log n) for minimum longest concave chain. In any case, we use only linear space.

• Type: Conference
• Year: 2009
• Where: CCCG
• Authors: Hee-Kap Ahn, Sang Won Bae, Iris Reinbacher
• Keywords: CCCG 2007, CCCG 2009, Concave Chain, Longest Concave Chain, Minimum Longest Concave