We present an algorithm that computes a (1+varepsilon)-approximation of the diameter of a weighted, undirected planar graph of n vertices with non-negative edge lengths in O(nlog n(log n + (1/varepsilon)^5)) expected time, improving upon the O(n((1/varepsilon)^4 log^4(n) + 2^{O(1/varepsilon)}))-time algorithm of Weimann and Yuster [ICALP 2013]. Our algorithm makes two improvements over that result: first and foremost, it replaces the exponential dependency on 1/varepsilon with a polynomial one, by adapting and specializing Cabello's recent abstract-Voronoi-diagram-based technique [SODA 2017] for approximation purposes; second, it shaves off two logarithmic factors by choosing a better sequence of error parameters during recursion. Moreover, using similar techniques, we improve the (1+varepsilon)-approximate distance oracle of Gu and Xu [ISAAC 2015] by first replacing the exponential dependency on 1/varepsilon on the preprocessing time and space with a polynomial one and second removing a logarithmic factor from the preprocessing time.
@InProceedings{chan_et_al:LIPIcs.ESA.2017.25, author = {Chan, Timothy M. and Skrepetos, Dimitrios}, title = {{Faster Approximate Diameter and Distance Oracles in Planar Graphs}}, booktitle = {25th Annual European Symposium on Algorithms (ESA 2017)}, pages = {25:1--25:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-049-1}, ISSN = {1868-8969}, year = {2017}, volume = {87}, editor = {Pruhs, Kirk and Sohler, Christian}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2017.25}, URN = {urn:nbn:de:0030-drops-78382}, doi = {10.4230/LIPIcs.ESA.2017.25}, annote = {Keywords: planar graphs, diameter, abstract Voronoi diagrams} }
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