On Triangulating Palm Polygons in Linear Time

No one has yet been able to triangulate a simple polygon of n vertices in O (n) time. The fastest algorithm to date, due to Tarjan and van Wyk, runs in 0 (n loglogn) time. On the other hand several classes of simple polygons do admit linear-time triangulation. Some examples of such famous classes are: star-shaped, monotone, spiral, edge visible, and weakly externally visible polygons. In this paper the notion of geodesic paths is used to characterize all the classes of polygons for which linear time triangulation algorithms are known. First we introduce a new class of polygons, termed palm polygons, which subsumes many known classes of polygons for which linear time triangulation algorithms are known, and present an algorithm for triangulating palm polygons in 0(n) time. Then a class of polygons termed crab polygons is defined and shown to contain all classes of existing polygons for which linear time triangulation algorithms are known. As a by product of this characterization we obtain a new very simple linear time algorithm for triangulating star-shaped polygons.