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

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## Abstract

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.

## Keywords

Geodesic distance Facility location Simple polygons## Mathematics Subject Classification

65D18## Notes

### Acknowledgments

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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