An O(nlog log n)-Time Algorithm for Triangulating a Simple Polygon

Given a simple n-vertex polygon, the triangulation problem is to partition the interior of the polygon into $n - 2$ triangles by adding $n - 3$ nonintersecting diagonals. We propose an O(nlog log n)-time algorithm for this problem, improving on the previously best bound of O(nlog n) and showing that triangulation is not as hard as sorting. Improved algorithms for several other computational geometry problems, including testing whether a polygon is simple, follow from our result.