Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Local Spanners Revisited   $\newcommand{\Emph}[1]{{\it{#1}}}
\newcommand{\FF}{\mathcal{F}}\newcommand{\region}{\mathsf{r}}\newcommand{\restrictY}[2]{#1
\cap {#2}}$For a set of points $P \subseteq \mathbb{R}^2$, and a family of
regions $\FF$, a $\Emph{local~t-spanner}$ of $P$, is a sparse graph $G$ over
$P$, such that, for any region $\region \in \FF$, the subgraph restricted to
$\region$, denoted by $\restrictY{G}{\region} = G_{P \cap \region}$, is a
$t$-spanner for all the points of $\region \cap P$.
  We present algorithms for the construction of local spanners with respect to
several families of regions, such as homothets of a convex region.
Unfortunately, the number of edges in the resulting graph depends
logarithmically on the spread of the input point set. We prove that this
dependency can not be removed, thus settling an open problem raised by Abam and
Borouny. We also show improved constructions (with no dependency on the spread)
of local spanners for fat triangles, and regular $k$-gons. In particular, this
improves over the known construction for axis parallel squares.
  We also study a somewhat weaker notion of local spanner where one allows to
shrink the region a "bit". Any spanner is a weak local spanner if the shrinking
is proportional to the diameter. Surprisingly, we show a near linear size
construction of a weak spanner for axis-parallel rectangles, where the
shrinkage is $\Emph{multiplicative}$.

13
The categories of this paper are: quant-ph
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
24
The categories of this paper are: dg-ga
4
The categories of this paper are: cond-mat
9
The categories of this paper are: nlin
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
14
The categories of this paper are: math-ph
29
The categories of this paper are: funct-an
31
The categories of this paper are: ao-sci
0
The categories of this paper are: hep-ph
25
The categories of this paper are: patt-sol
5
The categories of this paper are: gr-qc
18
The categories of this paper are: econ