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, Sharir and Rote [Disc. & Comput. Geom. 89] 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 (explicitly posed by Mitchell [Handbook of Computational Geometry, 2000]). In this paper we affirmatively answer this question and present a linear time algorithm to solve this problem.

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Keywords 1-center problem, Facility location, Geodesic distance, Simple polygons
Persistent URL dx.doi.org/10.4230/LIPIcs.SOCG.2015.209
Conference 31st International Symposium on Computational Geometry, SoCG 2015
Ahn, H.K. (Hee Kap), Barba, L. (Luis), Bose, P, De Carufel, J.-L. (Jean-Lou), Korman, M. (Matias), & Oh, E. (Eunjin). (2015). A Linear-Time Algorithm for the Geodesic Center of a Simple Polygon. Presented at the 31st International Symposium on Computational Geometry, SoCG 2015. doi:10.4230/LIPIcs.SOCG.2015.209