Counting Non-Convex 5-Holes in a Planar Point Set

Let S be a set of n points in the general position, that is, no three points in S are collinear. A simple k-gon with all corners in S such that its interior avoids any point of S is called a k-hole. In this paper, we present the first algorithm that counts the number of non-convex 5-holes in S. To our best knowledge, prior to this work there was no known algorithm in the literature except a trivial brute force algorithm. Our algorithm runs in time O(T+Q), where T denotes the number of 3-holes, or empty triangles, in S and Q that denotes the number of non-convex 4-holes in S. Note that T+Q ranges from Ω(n2) to O(n3), while its expected number is Θ(n2logn) when the points in S are chosen uniformly and independently at random from a convex and bounded body in the plane