Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Higher-Order Voronoi Diagram of Line Segments   Surprisingly, the order-$k$ Voronoi diagram of line segments had received no
attention in the computational-geometry literature. It illustrates properties
surprisingly different from its counterpart for points; for example, a single
order-$k$ Voronoi region may consist of $\Omega(n)$ disjoint faces. We analyze
the structural properties of this diagram and show that its combinatorial
complexity for $n$ non-crossing line segments is $O(k(n-k))$, despite the
disconnected regions. The same bound holds for $n$ intersecting line segments,
when $k\geq n/2$. We also consider the order-$k$ Voronoi diagram of line
segments that form a planar straight-line graph, and augment the definition of
an order-$k$ Voronoi diagram to cover non-disjoint sites, addressing the issue
of non-uniqueness for $k$-nearest sites. Furthermore, we enhance the iterative
approach to construct this diagram. All bounds are valid in the general $L_p$
metric, $1\leq p\leq \infty$. For non-crossing segments in the $L_\infty$ and
$L_1$ metrics, we show a tighter $O((n-k)^2)$ bound for $k>n/2$.

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The categories of this paper are: nucl-th
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-th
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The categories of this paper are: astro-ph
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The categories of this paper are: eess
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The categories of this paper are: funct-an
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The categories of this paper are: hep-lat
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: bayes-an
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The categories of this paper are: econ
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The categories of this paper are: q-fin
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The categories of this paper are: hep-ex
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The categories of this paper are: chem-ph