# The Geodesic Diameter of Polygonal Domains

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## Abstract

This paper studies the geodesic diameter of polygonal domains having \(h\) holes and \(n\) corners. For simple polygons (i.e., \(h=0\)), the geodesic diameter is determined by a pair of corners of a given polygon and can be computed in linear time, as shown by Hershberger and Suri. For general polygonal domains with \(h \ge 1\), however, no algorithm for computing the geodesic diameter was known prior to this paper. In this paper, we present the first algorithms that compute the geodesic diameter of a given polygonal domain in worst-case time \(O(n^{7.73})\) or \(O(n^7 (\log n + h))\). The main difficulty unlike the simple polygon case relies on the following observation revealed in this paper: two interior points can determine the geodesic diameter and in that case there exist at least five distinct shortest paths between the two.

### Keywords

Polygonal domain Shortest path Geodesic diameter Exact algorithm Convex function Lower envelope## Notes

### Acknowledgments

We thank Hee-Kap Ahn, Jiongxin Jin, Christian Knauer, and Joseph Mitchell for fruitful discussion. We also thank Joseph O’Rourke for pointing out the reference [21]. Work by S.W. Bae was supported by National Research Foundation of Korea (NRF) Grant funded by the Korea government (MEST) (No. 2010-0005974 and NRF-2013R1A1A1A05006927). M. Korman received the support of the Secretary for Universities and Research of the Ministry of Economy and Knowledge of the Government of Catalonia, the European Union, and ESF EUROCORES Programme EuroGIGA-ComPoSe IP04-MICINN Project EUI-EURC-2011-4306. Work by Y. Okamoto was supported by Global COE Program “Computationism as a Foundation for the Sciences” and Grant-in-Aid for Scientific Research from Ministry of Education, Science and Culture, Japan, and Japan Society for the Promotion of Science.

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