Faster Approximate Diameter and Distance

8 We present an algorithm that computes a (1 + ε)-approximation of the diameter of a weighted, 9 undirected planar graph of n vertices with non-negative edge lengths inO ( n logn ( logn+ (1/ε)5 )) 10 expected time, improving upon the O ( n ( (1/ε)4 log4 n+ 2O(1/ε) )) -time algorithm of Weimann 11 and Yuster [ICALP 2013]. Our algorithm makes two improvements over that result: first and 12 foremost, it replaces the exponential dependency on 1/ε with a polynomial one, by adapting and 13 specializing Cabello’s recent abstract-Voronoi-diagram-based technique [SODA 2017] for approx14 imation purposes; second, it shaves off two logarithmic factors by choosing a better sequence of 15 error parameters during recursion. 16 Moreover, using similar techniques, we improve the (1 + ε)-approximate distance oracle of 17 Gu and Xu [ISAAC 2015] by first replacing the exponential dependency on 1/ε on the prepro18 cessing time and space with a polynomial one and second removing a logarithmic factor from the 19 preprocessing time. 2