Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms 2018
DOI: 10.1137/1.9781611975031.33
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Voronoi Diagrams on Planar Graphs, and Computing the Diameter in Deterministic Õ(n5/3) 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… Show more

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Cited by 23 publications
(27 citation statements)
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References 27 publications
(42 reference statements)
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“…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:…”
Section: Application To H-minor Free Graphsmentioning
confidence: 99%
See 2 more Smart Citations
“…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:…”
Section: Application To H-minor Free Graphsmentioning
confidence: 99%
“…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.…”
Section: Introductionmentioning
confidence: 99%
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“…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.…”
Section: Preliminariesmentioning
confidence: 99%
“…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.…”
Section: The Oraclementioning
confidence: 99%