Local Spanners Revisited

TL;DR

This paper investigates local spanners for point sets with respect to various regions, establishing bounds on their size and dependencies, and introduces improved constructions for specific shapes, addressing open problems in the field.

Contribution

It provides algorithms for constructing local spanners for multiple region families, proves the necessity of logarithmic dependency on spread, and offers improved constructions for fat triangles and regular polygons.

Findings

Edge count depends logarithmically on point set spread, and this cannot be avoided.

Improved local spanner constructions for fat triangles and regular k-gons.

Near-linear size weak spanner for axis-parallel rectangles with multiplicative shrinkage.

Abstract

$\newcommand{\Emph}[1]{{\it{#1}}} \newcommand{\FF}{\mathcal{F}}\newcommand{\region}{\mathsf{r}}\newcommand{\restrictY}[2]{#1 \cap {#2}}$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 $\Emph{multiplicative}$.