Randomly Generating Triangulations of a Simple Polygon

Abstract

In this paper, we present an $O(n^2 + |E|^{\frac{3}{2}})$ time algorithm for generating triangulations of a simple polygon at random with uniform distribution, where n and |E| are the number of vertices and diagonal edges in the given polygon, respectively. The current best algorithm takes O(n ⁴) time. We also derive algorithms for computing the expected degree of each vertex, the expected number of ears, the expected number of interior triangles, and the expected height of the corresponding tree in such a triangulated polygon. These results are not known for simple polygon. All these algorithms are dominated by the $O(n^2 + |E|^{\frac{3}{2}})$ time triangulation counting algorithm. If the results of the triangulation counting algorithm are given, then the triangulation generating algorithm takes O(n log n) time only. All these algorithms are simple and easy to be implemented.