Computing the k -Visibility Region of a Point in a Polygon

Two points p and q in a simple polygon P are k -visible when the line segment pq crosses the boundary of P at most k times. Given a query point q , a positive integer k , and a polygon P , we design an algorithm that computes the region of P that is k -visible from q in O ( n k ) time, where n denotes the number of vertices of P . This region is called the k -visibility region of q . This is the first algorithm parameterized in terms of k , resulting in an asymptotically faster worst-case running time compared to previous algorithms when k is o(logn) , and bridging the gap between the O ( n )-time algorithm for computing the 0-visibility region of q in P and the O(nlog n) -time algorithm for computing the k -visibility region of q in P for any k . We also design a data structure of size O ( n 5 ) that supports visibility queries, returning the k -visible region of P for any arbitrary query point q in O(logn+m) time, where m denotes the number of vertices on the boundary of the output visibility region.