关于双色点集粗糙度的新结果
New results on the coarseness of bicolored point sets
摘要：
设$S$为平面中$n$点的双色（红色和蓝色）集合。如果存在凸集$C$使得$I=C\cap S$，则$S$的子集$I$是岛.一个岛的差异是它所包含的红色点的数量减去蓝色点的数量的绝对值。$S$的凸划分是将$S$划分为具有成对不相交凸包的岛。凸划分的差异是其最小差异岛的差异。$S$的粗糙度是具有最大偏差的$S$的凸划分的偏差。Bereg等人最近定义了这一概念[CGTA 2013]。在本文中，我们研究以下问题：给定一个集合$S$的$n$点在平面上的一般位置，如何将他们中的每一个（红色或蓝色），使所产生的2色点集具有小粗糙度？证明了每个$n$ -点集$S$都可以着色，使得其粗度为$O(n^{1/4}\sqrt{\log n})$ .这个界几乎是紧的，因为存在$n$ -点集，使得每一个2-着色至少给出$Ω(n^{1/4})$粗糙度.此外，我们证明了存在一种近似算法用于计算2色点集的粗糙度，其比率介于$1/128$和$1/64$之间，解决了Bereg等人[CGTA 2013]提出的一个开放问题。我们所有的结果考虑$k$ -可分离的岛屿的$S$，对于一些$k$，这是由相交的$S$与最多$k$半平面。
英文摘要：
Let $S$ be a 2-colored (red and blue) set of $n$ points in the plane. A subset $I$ of $S$ is an island if there exits a convex set $C$ such that $I=C\cap S$. The discrepancy of an island is the absolute value of the number of red minus the number of blue points it contains. A convex partition of $S$ is a partition of $S$ into islands with pairwise disjoint convex hulls. The discrepancy of a convex partition is the discrepancy of its island of minimum discrepancy. The coarseness of $S$ is the discrepancy of the convex partition of $S$ with maximum discrepancy. This concept was recently defined by Bereg et al. [CGTA 2013]. In this paper we study the following problem: Given a set $S$ of $n$ points in general position in the plane, how to color each of them (red or blue) such that the resulting 2-colored point set has small coarseness? We prove that every $n$-point set $S$ can be colored such that its coarseness is $O(n^{1/4}\sqrt{\log n})$. This bound is almost tight since there exist $n$-point sets such that every 2-coloring gives coarseness at least $Ω(n^{1/4})$. Additionally, we show that there exists an approximation algorithm for computing the coarseness of a 2-colored point set, whose ratio is between $1/128$ and $1/64$, solving an open problem posted by Bereg et al. [CGTA 2013]. All our results consider $k$-separable islands of $S$, for some $k$, which are those resulting from intersecting $S$ with at most $k$ halfplanes.
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