关于直线排列中圆的带
On the zone of a circle in an arrangement of lines
摘要：
令$\mathcal L$是平面中的一组$n$线，令$C$是平面中的凸曲线，如圆或抛物线。$\mathcal L$中的$C$的“区域”（表示为$\mathcal Z(C,\mathcal L)$）被定义为排列$\mathcal A(\mathcal L)$中由$C$包围的所有小区的集合。Edelsbrunner等人（1992）证明了$\mathcal Z(C,\mathcal L)$的复杂度（边或顶点的总数）至多为$O(nα(n))$，其中$α$是逆阿克曼函数。他们通过将$\mathcal Z(C,\mathcal L)$的边序列翻译成序列$S$来避免子序列$ababa$。$\mathcal Z(C,\mathcal L)$的最坏情况复杂度是否仅为线性是一个长期存在的问题。由于将问题松弛为伪线确实有一个$Θ(nα(n))$界，所以对于直线的情况，$O(n)$的任何证明都必须使用几何参数。在本文中，我们提出了一些这样的几何参数。我们表明，如果$C$是一个圆，那么某些配置的直线段上的端点$C$是不可能的。特别是，我们表明，存在一个Hart-Sharir序列，不能出现作为一个子序列的$S$。Hart-Sharir序列基本上是构造无$ababa$的超线性长度序列的唯一已知方法。因此，如果可以证明超线性长度的无$ababa$序列的每个族最终包含所有Hart-Sharir序列，则可以得出只要$C$是圆，则$\mathcal Z(C,\mathcal L)$的复杂度为$O(n)$。
英文摘要：
Let $\mathcal L$ be a set of $n$ lines in the plane, and let $C$ be a convex curve in the plane, like a circle or a parabola. The "zone" of $C$ in $\mathcal L$, denoted $\mathcal Z(C,\mathcal L)$, is defined as the set of all cells in the arrangement $\mathcal A(\mathcal L)$ that are intersected by $C$. Edelsbrunner et al. (1992) showed that the complexity (total number of edges or vertices) of $\mathcal Z(C,\mathcal L)$ is at most $O(nα(n))$, where $α$ is the inverse Ackermann function. They did this by translating the sequence of edges of $\mathcal Z(C,\mathcal L)$ into a sequence $S$ that avoids the subsequence $ababa$. Whether the worst-case complexity of $\mathcal Z(C,\mathcal L)$ is only linear is a longstanding open problem. Since the relaxation of the problem to pseudolines does have a $Θ(nα(n))$ bound, any proof of $O(n)$ for the case of straight lines must necessarily use geometric arguments. In this paper we present some such geometric arguments. We show that, if $C$ is a circle, then certain configurations of straight-line segments with endpoints on $C$ are impossible. In particular, we show that there exists a Hart-Sharir sequence that cannot appear as a subsequence of $S$. The Hart-Sharir sequences are essentially the only known way to construct $ababa$-free sequences of superlinear length. Hence, if it could be shown that every family of $ababa$-free sequences of superlinear-length eventually contains all Hart-Sharir sequences, it would follow that the complexity of $\mathcal Z(C,\mathcal L)$ is $O(n)$ whenever $C$ is a circle.
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