Minimum r-Star Cover of Class-3 Orthogonal Polygons

We are interested in the problem of covering simple orthogonal polygons with the minimum number of r-stars; an orthogonal polygon is an r-star if it is orthogonally convex and star-shaped. The problem has been considered by Worman and Keil who described an algorithm running in O(n 17 poly-log n) time where n is the size of the input polygon. In this paper, we consider the above problem on simple class-3 orthogonal polygons, i.e., orthogonal polygons that have dents along at most 3 different orientations. By taking advantage of geometric properties of these polygons, we give an output-sensitive O(n + k log k)-time algorithm where k is the size of a minimum r-star cover; this is the first purely geometric algorithm for this problem. Ideas in this algorithm may be generalized to yield faster algorithms for the problem on general simple orthogonal polygons. For such polygons, our algorithm implies a 2-approximation algorithm.