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Faster Approximate Diameter and Distance Oracles in Planar Graphs

Authors Timothy M. Chan, Dimitrios Skrepetos



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Timothy M. Chan
Dimitrios Skrepetos

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Timothy M. Chan and Dimitrios Skrepetos. Faster Approximate Diameter and Distance Oracles in Planar Graphs. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 25:1-25:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ESA.2017.25

Abstract

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.
Keywords
  • planar graphs
  • diameter
  • abstract Voronoi diagrams

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