Voronoi diagrams on planar graphs, and computing the diameter in deterministic Õ (n^5/3) time

We present an 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 Õ (nb2) time1 so that one can compute any additively weighted Voronoi diagram for these sites in Õ(b) time. We use this construction to compute the diameter of a directed planar graph with real arc lengths in O(n5/3) time. This improves the recent breakthrough result of Cabello (SODA'17), both by improving the running time (from Õ (n11/6)), and by providing a deterministic algorithm. It is in fact the first truly subquadratic deterministic algorithm for this problem. Our use of Voronoi diagrams to compute the diameter follows that of Cabello, but he used abstract Voronoi diagrams, which makes his diameter algorithm more involved, more expensive, and randomized. As in Cabello's work, our algorithm can also compute the Wiener index of a planar graph (i.e., the sum of all pairwise distances) within the same bound. Our construction of Voronoi diagrams for planar graphs is of ndependent interest. It has already been used to obtain fast exact distance oracles for planar graphs [Cohen-Addad et al., FOCS'17].