Before revision: We present a deterministic algorithm that computes\tilde \tilde  the diameter of a directed planar graph with real arc lengths in O(n^{5/3}) time.
After revision: We present an explicit and efficient construction of additively weighted Voronoi diagrams on planar graphs. Let G be a planar graph with n vertices and b sites that lie on a constant number of faces. We show how to preprocess G in\tilde \tilde the diameter of a directed planar graph with real arc lengths in O(n^{5/3}) time.