An efficient algorithm for decomposing a polygon into star-shaped polygons

In this paper we show how a theorem in plane geometry can be converted into an O(n log n) algorithm for decomposing a polygon into star-shaped subsets. The computational efficiency or this new decomposition contrasts with the heavy computational burden of existing methods. 1.0 Introduction The decomposition of a simple planar polygon into simpler components plays an important role in syntactic pattern recognition. Some examples of possible decompositions are decompositions into convex polygons [1], [2], decompositions into spiral polygons [3] and decompositions into monotone polygons [4]. A survey of these methods and many other additional references are contained in Pavlidis [5]. A star-shaped polygon is one in which the entire polygon is visible from at least one fixed point of the polygon. In this note we consider decompositions into star-shaped polygons and give an efficient algorithm for this problem. A similar decomposition has previously been suggested by Maruyama [6] in his thesis. His solution involves the creation of new Steiner points, yields overlapping star-shaped components and requires a complicated and expensive computation. The main part of this paper describes an O(n log n) algorithm for decomposing a polygon with n vertices into disjoint star-shaped polygons. This decomposition does not involve the creation of any new vertices and can always yield a decomposition with at most [n/3] star-shaped polygons ([x] here denotes the greatest integer less than or equal to x). It does not, however, normally give a decomposition into the minimum number of star-shaped polygons. On the other hand, our procedure is extremely flexible and can easily be modified to give a set of radically different decompositions.