A Linear-Time Algorithm for the Geodesic Center of a Simple Polygon

Let P be a closed simple polygon with n vertices. For any two points in P , the geodesic distance between them is the length of the shortest path that connects them among all paths contained in P . The geodesic center of P is the unique point in P that minimizes the largest geodesic distance to all other points of P . In 1989, Pollack et al. (Discrete Comput Geom 4(1): 611–626, 1989 ) showed an O ( n log n ) -time algorithm that computes the geodesic center of P . Since then, a longstanding question has been whether this running time can be improved. In this paper we affirmatively answer this question and present a deterministic linear-time algorithm to solve this problem.