Before revision:  Our algorithm follows the general high-level approach of Cabello, but differs in the way it is implemented. Our main technical contribution is an explicit and efficient construction of additively weighted Voronoi diagrams on planar graphs (under the assumption that all Voronoi sites lie on a constant number of faces). The idea of using Voronoi diagrams is also borrowed from Cabello, except that he uses abstract Voronoi diagrams (as a black box) , which makes the implementation more involved, more expensive, and randomized.
After revision:  Our use of Voronoi diagrams to compute the diameter follows that of Cabello, but he used abstract Voronoi diagrams (as a black box) , which makes the implementation more involved, more expensive, and randomized.