Polygonizations of point sets in the plane

We examine the different ways a set ofn points in the plane can be connected to form a simple polygon. Such a connection is called apolygonization of the points. For some point sets the number of polygonizations is exponential in the number of points. For this reason we restrict our attention to star-shaped polygons whose kernels have nonempty interiors; these are callednondegenerate star-shaped polygons.We develop an algorithm and data structure for determining the nondegenerate star-shaped polygonizations of a set ofn points in the plane. We do this by first constructing an arrangement of line segments from the point set. The regions in the arrangement correspond to the kernels of the nondegenerate star-shaped polygons whose vertices are the originaln points. To obtain the data structure representing this arrangement, we show how to modify data structures for arrangements of lines in the plane. This data structure can be computed inO(n ⁴) time and space. By visiting the regions in this data structure in a carefully chosen order, we can compute the polygon associated with each region inO(n) time, yielding a total computation time ofO(n ⁵) to compute a complete list ofO(n ⁴) nondegenerate star-shaped polygonizations of the set ofn points.