Discrete & Computational Geometry

, Volume 56, Issue 4, pp 836–859 | Cite as

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

  • Hee-Kap Ahn
  • Luis Barba
  • Prosenjit Bose
  • Jean-Lou De Carufel
  • Matias Korman
  • Eunjin Oh


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.


Geodesic distance Facility location Simple polygons 

Mathematics Subject Classification




H.-K. Ahn and E. Oh: Supported by the NRF Grant 2011-0030044 (SRC-GAIA) funded by the Korea government (MSIP). M. K. was supported in part by the ELC Project (MEXT KAKENHI Nos. 12H00855 and 15H02665).


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Hee-Kap Ahn
    • 1
  • Luis Barba
    • 2
    • 3
  • Prosenjit Bose
    • 2
  • Jean-Lou De Carufel
    • 2
  • Matias Korman
    • 4
  • Eunjin Oh
    • 1
  1. 1.Department of Computer Science and EngineeringPOSTECHPohangKorea
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada
  3. 3.Département d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium
  4. 4.Tohoku UniversitySendaiJapan

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