摘要：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.
