Approximation Algorithms for Finding the Optimal Bridge Connecting Two Simple Polygons

Given two simple polygons P and Q we de ne the weight of a bridge p q with p P and q Q where de nes the boundary of the polygon between the two polygons as gd p P d p q gd q Q where d p q is the Euclidean distance between the points p and q and gd a A is the geodesic distance between a and its geodesic farthest neighbor on A An optimal bridge of minimum weight can be found in O n log n time as described in We present an approximation scheme that given any positive integer k constructs a bridge with objective function value within k of optimal in O kn log kn time thus solving an open problem stated in We also present a fully polynomial time approximation scheme that for any generates a bridge with objective function within of optimal in O kn log kn time where k d log e In contrast to the exact algorithm given in our algorithm does not use complicated data structures and is amenable to e cient implementations We also show that the claim of that if p q is the optimal bridge then gd p q d p q is incorrect