Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Voronoi diagrams on planar graphs, and computing the diameter in
  deterministic $\tilde{O}(n^{5/3})$ time   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 $\tilde O(nb^2)$ time (footnote: The $\tilde O$ notation hides
polylogarithmic factors.) so that one can compute any additively weighted
Voronoi diagram for these sites in $\tilde O(b)$ time.
  We use this construction to compute the diameter of a directed planar graph
with real arc lengths in $\tilde{O}(n^{5/3})$ time. This improves the recent
breakthrough result of Cabello (SODA'17), both by improving the running time
(from $\tilde{O}(n^{11/6})$), and by providing a deterministic algorithm. It is
in fact the first truly subquadratic {\em 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 compute, for every vertex $v$, both
the farthest vertex from $v$ (i.e., the eccentricity of $v$), and the sum of
distances from $v$ to all other vertices. Hence, our algorithm can also compute
the radius, median, and Wiener index (sum of all pairwise distances) of a
planar graph within the same time bounds. Our construction of Voronoi diagrams
for planar graphs is of independent interest.

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The categories of this paper are: plasm-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: physics
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The categories of this paper are: alg-geom
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The categories of this paper are: q-alg
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The categories of this paper are: q-bio
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The categories of this paper are: math-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: nlin
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The categories of this paper are: hep-lat
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The categories of this paper are: mtrl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: funct-an
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The categories of this paper are: bayes-an
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The categories of this paper are: gr-qc
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The categories of this paper are: cond-mat