具有少量交叉的几何(1+ε)-生成树
Geometric $(1+\varepsilon)$-Spanners with Few Crossings
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中文总结 AI 辅助
针对平面上n个点和ε>0，构造具有O(n/ε)条边且交叉数有改进的(1+ε)-生成树，每条边交叉数为Õ(1/ε³)，总数为Õ(n/ε⁴)，并给出一些特定点集下(1+ε)-生成树交叉数的下界情况。
中文摘要 AI 辅助
对于平面上的n个点和ε>0，我们构造了一个具有O(n/ε)条边的(1+ε)-生成树，其中每条边有Õ(1/ε³)个交叉，因此交叉总数为Õ(n/ε⁴)，此外任意两条交叉边的长度比为O(1/ε²)。我们的生成树构造显著改进了(1+ε)-生成树交叉数的先前上限，且是首个对任意常数ε>0每条边确保O(1)交叉的生成树构造。相比之下，我们还构造了特定的点集情况。
英文摘要
For $n$ points in the plane and an $\varepsilon>0$, we construct a $(1+\varepsilon)$-spanner with $O(n/\varepsilon)$ edges in which every edge has $\tilde{O}(1/\varepsilon^3)$ crossings, hence the total number of crossings is $\tilde{O}(n/\varepsilon^4)$, furthermore the ratio between the lengths of any two crossing edges is $O(1/\varepsilon^2)$. Our spanner construction substantially improves on the previous upper bound for the number of crossings in a $(1+\varepsilon)$-spanner, and it is the first spanner construction that ensures $O(1)$ crossings per edge for any constant $\varepsilon>0$. In contrast, we construct: $n$ points in the plane for which every $(1+\varepsilon)$-spanner has $Ω(n/\varepsilon^3)$ crossings, $n$ points for which every $(1+\varepsilon)$-spanner has an edge with $Ω(1/\varepsilon^{5/2})$ crossings, and 4 points for which every $(1+\varepsilon)$-spanner contains two crossing edges where one is $Ω(1/\varepsilon)$ times longer than the other.
