Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Voronoi Diagrams in the Hilbert Metric   The Hilbert metric is a distance function defined for points lying within a
convex body. It generalizes the Cayley-Klein model of hyperbolic geometry to
any convex set, and it has numerous applications in the analysis and processing
of convex bodies. In this paper, we study the geometric and combinatorial
properties of the Voronoi diagram of a set of point sites under the Hilbert
metric. Given any convex polygon $K$ bounded by $m$ sides, we present two
algorithms (one randomized and one deterministic) for computing the Voronoi
diagram of an $n$-element point set in the Hilbert metric induced by $K$. Our
randomized algorithm runs in $O(m n + n (\log n)(\log m n))$ expected time, and
our deterministic algorithm runs in time $O(m n \log n)$. Both algorithms use
$O(m n)$ space. We show that the worst-case combinatorial complexity of the
Voronoi diagram is $\Theta(m n)$.

35
The categories of this paper are: mtrl-th
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
10
The categories of this paper are: q-bio
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
30
The categories of this paper are: q-alg
11
The categories of this paper are: hep-lat
36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
37
The categories of this paper are: supr-con
19
The categories of this paper are: eess
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex