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Computing the \(L_1\) Geodesic Diameter and Center of a Polygonal Domain

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Abstract

For a polygonal domain with h holes and a total of n vertices, we present algorithms that compute the \(L_1\) geodesic diameter in \(O(n^2+h^4)\) time and the \(L_1\) geodesic center in \(O((n^4+n^2 h^4)\alpha (n))\) time, respectively, where \(\alpha (\cdot )\) denotes the inverse Ackermann function. No algorithms were known for these problems before. For the Euclidean counterpart, the best algorithms compute the geodesic diameter in \(O(n^{7.73})\) or \(O(n^7(h+\log n))\) time, and compute the geodesic center in \(O(n^{11}\log n)\) time. Therefore, our algorithms are significantly faster than the algorithms for the Euclidean problems. Our algorithms are based on several interesting observations on \(L_1\) shortest paths in polygonal domains.

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Acknowledgements

Work by S.W. Bae was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2013R1A1A1A05006927), and by the Ministry of Education (2015R1D1A1A01057220). M. Korman is partially supported by JSPS/MEXT Grant-in-Aid for Scientific Research Grant Numbers 12H00855 and 15H02665 and also by The ELC project (MEXT KAKENHI) Grant Number 24106008. J. Mitchell acknowledges support from the US-Israel Binational Science Foundation (Grant 2010074) and the National Science Foundation (CCF-1018388, CCF-1526406). Y. Okamoto is partially supported by JST, CREST, Foundation of Innovative Algorithms for Big Data and JSPS/MEXT Grant-in-Aid for Scientific Research Grant Numbers JP24106005, JP24700008, JP24220003, and JP15K00009. V. Polishchuk is supported in part by Grant 2014-03476 from the Sweden’s innovation agency VINNOVA and the project UTM-OK from the Swedish Transport Administration Trafikverket. H. Wang was supported in part by the National Science Foundation (CCF-1317143).

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Authors and Affiliations

  1. Kyonggi University, Suwon, South Korea

    Sang Won Bae

  2. Tohoku University, Sendai, Japan

    Matias Korman

  3. Stony Brook University, New York, USA

    Joseph S. B. Mitchell

  4. The University of Electro-Communications, Tokyo, Japan

    Yoshio Okamoto

  5. Linköping University, Linköping, Sweden

    Valentin Polishchuk

  6. Utah State University, Logan, UT, USA

    Haitao Wang

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  1. Sang Won Bae
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  2. Matias Korman
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  3. Joseph S. B. Mitchell
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  4. Yoshio Okamoto
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  5. Valentin Polishchuk
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  6. Haitao Wang
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Correspondence to Haitao Wang.

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Editor in Charge: Kenneth Clarkson

A preliminary version of this paper appeared in the Proceedings of the 33rd International Symposium on Theoretical Aspects of Computer Science (STACS 2016).

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Bae, S.W., Korman, M., Mitchell, J.S.B. et al. Computing the \(L_1\) Geodesic Diameter and Center of a Polygonal Domain. Discrete Comput Geom 57, 674–701 (2017). https://doi.org/10.1007/s00454-016-9841-z

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  • DOI: https://doi.org/10.1007/s00454-016-9841-z

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