A Euclidean noncrossing Steiner $(1+ε)$-spanner for a point set $P\subset\mathbb{R}^2$ is a planar straight-line graph that, for any two points $a, b \in P$, contains a path whose length is at most $1+ε$ times the Euclidean distance between $a$ and $b$. We construct a Euclidean noncrossing Steiner $(1+ε)$-spanner with $O(n/ε^{3/2})$ edges for any set of $n$ points in the plane. This result improves upon the previous best upper bound of $O(n/ε^{4})$ obtained nearly three decades ago. We also establish an almost matching lower bound: There exist $n$ points in the plane for which any Euclidean noncrossing Steiner $(1+ε)$-spanner has $Ω_μ(n/ε^{3/2-μ})$ edges for any $μ>0$. Our lower bound uses recent generalizations of the Szemerédi-Trotter theorem to disk-tube incidences in geometric measure theory.

翻译：对于点集 $P\subset\mathbb{R}^2$，欧几里得非交叉斯坦纳 $(1+ε)$-生成树是一种平面直线图，对于任意两点 $a, b \in P$，该图包含一条路径，其长度至多为 $a$ 与 $b$ 间欧几里得距离的 $1+ε$ 倍。我们为平面上任意 $n$ 个点的集合构造了一个具有 $O(n/ε^{3/2})$ 条边的欧几里得非交叉斯坦纳 $(1+ε)$-生成树。该结果改进了近三十年前获得的最佳上界 $O(n/ε^{4})$。我们还建立了一个几乎匹配的下界：存在平面上的 $n$ 个点，使得对于任意 $μ>0$，任何欧几里得非交叉斯坦纳 $(1+ε)$-生成树都至少有 $Ω_μ(n/ε^{3/2-μ})$ 条边。我们的下界利用了测度几何中关于圆盘-管关联的塞迈雷迪-特罗特定理的最新推广形式。