Minimum r-Star Cover of Class-3 Orthogonal Polygons

Abstract

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 star-shaped. The problem has been considered by Worman and Keil [13] who described an algorithm running in $$O(n^{17} \hbox {poly-log}\, n)$$ 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.