Voronoi Diagrams on Planar Graphs, and Computing the Diameter in Deterministic Õ(n^5/3) Time

Abstract

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 inÕ(nb 2 ) time 1 so that one can compute any additively weighted Voronoi diagram for these sites inÕ(b) time.We use this construction to compute the diameter of a directed planar graph with real arc lengths inÕ(n 5/3 ) time. This improves the recent breakthrough result of Cabello (SODA'17), both by improving the running time (from O(n 11/6 )), 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 independent interest. It has already been used to obtain fast exact distance oracles for planar graphs FOCS'17].

Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension

“…Since planar graphs have distance VC-dimension at most four [10] then, it follows from the planar separator theorem of Lipton and Tarjan [44] that it is the case for planar graphs. Therefore, Theorem 5 gives us a new subquadratic-time algorithm for diameter computation on unweighted planar graphs, but with a slower running-time than for the algorithms presented in [14,35]. More generally, the following separator theorem is from Alon et al:…”

“…Planar graphs. Finally, in a recent breakthrough paper [14], Cabello presented the first truly subquadratic algorithm for diameter computation on planar graphs (see also [35] for improvements on his work). For that he combined r-divisions: a recursive decomposition technique for planar graphs and other hereditary graph classes with sublinear balanced separators, with a clever use of additively weighted Voronoi diagrams.…”

“…We stress that for graphs with millions of nodes and edges, quadratic time is already too prohibitive.The conditional lower-bound of [52] also holds for sparse graphs i.e., with only m = O(n) edges [1]. However it does not hold for many well-structured graph classes [1,11,13,20,14,21,23,25,31,33,35,49]. Our work proposes some new advances on the characterization of graph families for which we can compute the diameter in truly subquadratic time.…”

Better Tradeoffs for Exact Distance Oracles in Planar Graphs

“…The dual representation VD * (S, ω) can then be obtained in additional O(r) time by following the constructive description above. There are more efficient algorithms [4,12] when one wants to construct many different additively weighted Voronoi diagrams for the same set of sites S. The basic approach is to invest superlinear time in preprocessing P , but then construct VD(S, ω) for multiple choices of ω inÕ(|S|) time each (instead of O(r)). Since the focus of this paper is on the tradeoff between space and query-time, and not on the preprocessing time, the particular algorithm used for constructing the Voronoi diagrams is less important.…”

“…Replacing S with S and h with h enforces the assumption. Note that since this transformation changes P , it is not suitable when working with Voronoi diagrams constructed by algorithms that preprocess P , such as the ones in [4,12] mentioned in Section 2. The transformation, however, is suitable, when computing Voronoi diagrams naively, as we do in this work.…”