An O(n log n) algorithm for finding a shortest central link segment

A central link segment of a simple n-vertex polygon P is a segment s inside P that minimizes the quantity max(x epsilon P) min(y epsilon s) d(L)(x, y), where d(L)(x, y) is the link distance between points a: and y of P. In this paper we present an O(n log n) algorithm for finding a central link segment of P. This generalizes previous results for finding an edge or a segment of P from which P is visible. Moreover, in the same time bound, our algorithm finds a central link segment of minimum length. Constructing a central link segment has applications to the problems of finding an optimal robot placement in a simply connected polygonal region and determining the minimum value k for which a given polygon is k-visible from some segment