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, where alpha(.) 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^{12+epsilon}) time. Therefore, our algorithms are much faster than the algorithms for the Euclidean problems. Our algorithms are based on several interesting observations on L_1 shortest paths in polygonal domains.
@InProceedings{wonbae_et_al:LIPIcs.STACS.2016.14, author = {Won Bae, Sang and Korman, Matias and Mitchell, Joseph S. B. and Okamoto, Yoshio and Polishchuk, Valentin and Wang, Haitao}, title = {{Computing the L1 Geodesic Diameter and Center of a Polygonal Domain}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {14:1--14:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.14}, URN = {urn:nbn:de:0030-drops-57151}, doi = {10.4230/LIPIcs.STACS.2016.14}, annote = {Keywords: geodesic diameter, geodesic center, shortest paths, polygonal domains, L1 metric} }
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