欧几里得非交叉Steiner近似最优稀疏性-spanner
Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity
AI总结：
本文提出了一种欧几里得非交叉Steiner(1+ε)-spanner，具有近最优稀疏性，改进了之前的结果并建立了下界。
摘要：
对于点集P⊆R²，欧几里得非交叉Steiner(1+ε)-spanner是一种平面直线图，对于任何两点a,b∈P，包含一条路径，其长度不超过1+ε倍的欧几里得距离。我们为任何平面中的n个点集构造了一个欧几里得非交叉Steiner(1+ε)-spanner，具有O(n/ε^{3/2})条边。这一结果改进了近三十年前获得的O(n/ε⁴)上界。我们还建立了几乎匹配的下界：存在平面中的n个点，使得任何欧几里得非交叉Steiner(1+ε)-spanner都有Ω_μ(n/ε^{3/2-μ})条边，对于任何μ>0。我们的下界使用了最近的Szemerédi-Trotter定理的泛化到几何测度理论中的盘-管相交。
英文摘要：
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.
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