On approximation behavior of the greedy triangulation for convex polygons

Abstract

We prove that the greedy triangulation heuristic for minimum weight triangulation of convex polygons yields solutions within a constant factor from the optimum. For interesting classes of convex polygons, we derive small upper bounds on the constant approximation factor. Our results contrast with Kirkpatrick's Ω(n) bound on the approximation factor of the Delaunay triangulation heuristic for minimum weight triangulation of convexn-vertex polygons. On the other hand, we present a straightforward implementation of the greedy triangulation heuristic for ann-vertex convex point set or a convex polygon takingO(n ²) time andO(n) space. To derive the latter result, we show that given a convex polygonP, one can find for all verticesv ofP a shortest diagonal ofP incident tov in linear time. Finally, we observe that the greedy triangulation for convex polygons having so-called semicircular property can be constructed in timeO(n logn).