Faster counting empty convex polygons in a planar point set

Abstract

In this paper, we study the problem of counting the number of convex empty polygons in a given set of n points in general position in the plane. We present two algorithms: one counts the number of all convex empty polygons in O(T) time and the other counts the number of convex empty m-gons for all 3≤m≤k in O(kT) time, where T denotes the number of empty triangles in S. Note that T varies from Ω(n2) to O(n3), while its expected value is known to be Θ(n2) when n input points are chosen uniformly and independently at random from a convex and bounded body.