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# Local Spanners Revisited

$\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\rad}{r} \newcommand{\restrictY}[2]{#1 \cap {#2}} \newcommand{\G}{\Mh{G}}% \newcommand{\region}{\Mh{\mathsf{r}}}% \newcommand{\Mh}[1]{#1} \newcommand{\PS}{P} \newcommand{\FF}{\Mh{\mathcal{F}}}% \newcommand{\query}{q} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\IntRange}[1]{[ #1 ]} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)} \newcommand{\polylog}{\mathrm{polylog}} \newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\pt}{p} \newcommand{\distY}[2]{\left\| {#1} - {#2} \right\|} \newcommand{\ptq}{q} \newcommand{\pts}{s}$
Stav Ashur and Sariel Har-Peled.
For a set of points $\PS \subseteq \Re^2$ and a family of regions $\FF$, a \emph{local $t$-spanner} of $\PS$ is a sparse graph $\G$ over $\PS$, such that for any region $\region \in \FF$ the subgraph restricted to $\region$, denoted by $\restrictY{\G}{\region}$, is a $t$-spanner for all the points of $\region \cap \PS$. 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 notions of weaker local spanners where one is allowed to shrink the region a bit''. Surprisingly, we show a near linear size construction of a weak spanner for axis-parallel rectangles, where the shrinkage is \emph{multiplicative}. Any spanner is a weak local spanner if the shrinking is proportional to the diameter of the region.
PDF. : arXiv