  Lightness and sparsity are two natural parameters for Euclidean
$(1+\varepsilon)$-spanners. Classical results show that, when the dimension
$d\in \mathbb{N}$ and $\varepsilon>0$ are constant, every set $S$ of $n$ points
in $d$-space admits an $(1+\varepsilon)$-spanners with $O(n)$ edges and weight
proportional to that of the Euclidean MST of $S$. In a recent breakthrough, Le
and Solomon (2019) established the precise dependencies on $\varepsilon>0$, for
constant $d\in \mathbb{N}$, of the minimum lightness and sparsity of
$(1+\varepsilon)$-spanners, and observed that Steiner points can substantially
improve the lightness and sparsity of a $(1+\varepsilon)$-spanner. They gave
upper bounds of $\tilde{O}(\varepsilon^{-(d+1)/2})$ for the minimum lightness
in dimensions $d\geq 3$, and $\tilde{O}(\varepsilon^{-(d-1)/2})$ for the
minimum sparsity in $d$-space for all $d\geq 1$.
  In this work, we improve several bounds on the lightness and sparsity of
Euclidean Steiner $(1+\varepsilon)$-spanners. We establish lower bounds of
$\Omega(\varepsilon^{-d/2})$ for the lightness and
$\Omega(\varepsilon^{-(d-1)/2})$ for the sparsity of such spanners in Euclidean
$d$-space for all constant $d\geq 2$. Our lower bound constructions generalize
previous constructions by Le and Solomon, but the analysis substantially
simplifies previous work, using new geometric insight, focusing on the
directions of edges. Next, we show that for every finite set of points in the
plane and every $\varepsilon\in (0,1]$, there exists a Euclidean Steiner
$(1+\varepsilon)$-spanner of lightness $O(\varepsilon^{-1})$; this matches the
lower bound for $d=2$. We generalize the notion of shallow light trees, which
may be of independent interest, and use directional spanners and a modified
window partitioning scheme to achieve a tight weight analysis.

Computational Geometry;Discrete Mathematics