Geometric Path Problems with Violations

In this paper, we study variants of the classical geometric shortest path problem inside a simple polygon, where we allow a part of the path to go outside the polygon. Let P be a simple polygon consisting of n vertices and let s, t be a pair of points in P. Let (Formula presented.) represent the interior of P and let (Formula presented.) represent the exterior of P, i.e. (Formula presented.) and (Formula presented.). For an integer (Formula presented.), we define a k-violation path from s to t to be a path connecting s and t such that its intersection with (Formula presented.) consists of at most k segments. There is no restriction in terms of the number of segments of the path within P. The objective is to compute a path of minimum Euclidean length among all possible (Formula presented.)-violation paths from s to t. In this paper, we study this problem for (Formula presented.) and propose an algorithm that computes the shortest one-violation path in (Formula presented.) time. We show that for rectilinear polygons, the minimum length rectilinear one-violation path can be computed in (Formula presented.) time. We extend the concept of one-violation path to a one-stretch violation path. In this case, the path between s and t is composed of (a) a path in P from s to a vertex u of P, (b) a path in (Formula pres