Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Truly Optimal Euclidean Spanners   Euclidean spanners are important geometric structures, having found numerous
applications over the years. Cornerstone results in this area from the late 80s
and early 90s state that for any $d$-dimensional $n$-point Euclidean space,
there exists a $(1+\epsilon)$-spanner with $nO(\epsilon^{-d+1})$ edges and
lightness $O(\epsilon^{-2d})$. Surprisingly, the fundamental question of
whether or not these dependencies on $\epsilon$ and $d$ for small $d$ can be
improved has remained elusive, even for $d = 2$. This question naturally arises
in any application of Euclidean spanners where precision is a necessity.
  The state-of-the-art bounds $nO(\epsilon^{-d+1})$ and $O(\epsilon^{-2d})$ on
the size and lightness of spanners are realized by the {\em greedy} spanner. In
2016, Filtser and Solomon proved that, in low dimensional spaces, the greedy
spanner is near-optimal. The question of whether the greedy spanner is truly
optimal remained open to date.
  The contribution of this paper is two-fold. We resolve these longstanding
questions by nailing down the exact dependencies on $\epsilon$ and $d$ and
showing that the greedy spanner is truly optimal. Specifically, for any $d=
O(1), \epsilon = \Omega({n}^{-\frac{1}{d-1}})$:
  - We show that any $(1+\epsilon)$-spanner must have $n
\Omega(\epsilon^{-d+1})$ edges, implying that the greedy (and other) spanners
achieve the optimal size.
  - We show that any $(1+\epsilon)$-spanner must have lightness
$\Omega(\epsilon^{-d})$, and then improve the upper bound on the lightness of
the greedy spanner from $O(\epsilon^{-2d})$ to $O(\epsilon^{-d})$.
  We then complement our negative result for the size of spanners with a rather
counterintuitive positive result: Steiner points lead to a quadratic
improvement in the size of spanners! Our bound for the size of Steiner spanners
is tight as well (up to lower-order terms).

36
The categories of this paper are: cmp-lg
34
The categories of this paper are: chem-ph
13
The categories of this paper are: quant-ph
22
The categories of this paper are: chao-dyn
28
The categories of this paper are: alg-geom
32
The categories of this paper are: atom-ph
20
The categories of this paper are: acc-phys
2
The categories of this paper are: math
6
The categories of this paper are: astro-ph
11
The categories of this paper are: hep-lat
19
The categories of this paper are: eess
30
The categories of this paper are: q-alg
7
The categories of this paper are: hep-th
24
The categories of this paper are: dg-ga
18
The categories of this paper are: econ
26
The categories of this paper are: bayes-an