ε-网大小的紧下界
Tight lower bounds for the size of epsilon-nets
摘要：
根据Haussler和Welzl（1987）的一个著名定理，任何有界VC-维的值域空间都存在一个大小为$O\left(\frac{1}{\eps}\log\frac1{\eps}\right)$的$\eps$ -网。Pach和Woeginger（1990）利用概率技巧证明了存在VC维为2的值域空间，可以达到上面的界。唯一已知的小VC维的值域空间，其中值域是某些欧氏空间中的几何对象，并且最小$\eps$ -网的大小在$\frac1{\eps}$中是超线性的，由Alon（2010）发现。在他的例子中，最小的$\eps$ -网的大小是$Ω\left(\frac{1}{\eps}g(\frac{1}{\eps})\right)$，其中$g$是一个增长极其缓慢的函数，与逆阿克曼函数密切相关。\smallskip我们表明，存在几何定义的范围空间，已经VC-维$2$，其中最小的$\eps$ -网的大小是$Ω\left(\frac{1}{\eps}\log\frac{1}{\eps}\right)$。我们还构造了平面上平行轴矩形诱导的值域空间，其中最小$\eps$ -网的大小为$Ω\left(\frac{1}{\eps}\log\log\frac{1}{\eps}\right)$。根据Aronov，Ezra，and Sharir（2010）的定理，这个界限是紧的。
英文摘要：
According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VC-dimension admits an $\eps$-net of size $O\left(\frac{1}{\eps}\log\frac1{\eps}\right)$. Using probabilistic techniques, Pach and Woeginger (1990) showed that there exist range spaces of VC-dimension 2, for which the above bound can be attained. The only known range spaces of small VC-dimension, in which the ranges are geometric objects in some Euclidean space and the size of the smallest $\eps$-nets is superlinear in $\frac1{\eps}$, were found by Alon (2010). In his examples, the size of the smallest $\eps$-nets is $Ω\left(\frac{1}{\eps}g(\frac{1}{\eps})\right)$, where $g$ is an extremely slowly growing function, closely related to the inverse Ackermann function. \smallskip We show that there exist geometrically defined range spaces, already of VC-dimension $2$, in which the size of the smallest $\eps$-nets is $Ω\left(\frac{1}{\eps}\log\frac{1}{\eps}\right)$. We also construct range spaces induced by axis-parallel rectangles in the plane, in which the size of the smallest $\eps$-nets is $Ω\left(\frac{1}{\eps}\log\log\frac{1}{\eps}\right)$. By a theorem of Aronov, Ezra, and Sharir (2010), this bound is tight.
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