Generating Empty Convex Polygon Randomly from a Subset of Given Point Set

Abstract

In computational geometry, problem of generating random geometric objects are very interesting and extensively studied problems. In this paper we propose a new algorithm to generate an empty convex polygon from a subset of given point set. Let $S = \{ p_{1} ,p_{2} , \ldots ,p_{n} \}$ be the given point set lying in ${\mathbb{R}}^{2}$. The proposed algorithm generates a random empty convex polygon consisting of k vertices in S. In preprocessing phase we compute Convex Hull Layers CL(S) in $O(n \log n)$ time. By using the visibility relationship in Convex Layers, the proposed algorithm generates a random empty convex polygon in $O( \log n + k)$ time which is improved over the existing solution for this problem.