用限制定向凸包分离平面上的双色点集
Separating bichromatic point sets in the plane by restricted orientation convex hulls
摘要：
我们探讨了在平面上的点集的可分性的限制方向凸包，这是一个依赖于方向，可能断开，和非凸封闭的形状，概括了凸包。设$R$和$B$是平面上两个不相交的红点和蓝点集，$\mathcal{O}$是通过原点的一组$k \geq 2$线。研究了当$R$的$\mathcal{O}$ -凸包不含$B$点时，计算$\mathcal{O}$的直线的方向集的问题.对于$k=2$正交线，我们有直线凸包。在最佳的$O(n \log n)$时间和$O(n)$空间，$n = \vert R \vert + \vert B \vert$，我们计算的旋转角的集合，使得在同时旋转后的线的$\mathcal{O}$周围的原点在相同的方向，$R$的直线凸包不包含点的$B$。我们把这个结果推广到$\mathcal{O}$是由具有任意取向的$k \geq 2$线构成的情况。在$\mathcal{O}$的线的逆时针循环顺序中，令$α_i$为顺时针旋转$i$第条线所需的角度，使得其与其后继线重合。我们在$O(1/Θ\cdot N \log N)$时间和$O(1/Θ\cdot N)$空间中解决了这种情况下的问题，其中$Θ= \min \{ α_1,\ldots,α_k \}$和$N=\max\{k,\vert R \vert + \vert B \vert \}$。我们最后考虑这样的情况，其中$\mathcal{O}$由$k=2$线形成，其中一条线是固定的，而第二条线旋转了从$0$到$π$的角度。我们表明，这最后一种情况下也可以解决最佳的$O(n\log n)$时间和$O(n)$空间，其中$n = \vert R \vert + \vert B \vert$。
英文摘要：
We explore the separability of point sets in the plane by a restricted-orientation convex hull, which is an orientation-dependent, possibly disconnected, and non-convex enclosing shape that generalizes the convex hull. Let $R$ and $B$ be two disjoint sets of red and blue points in the plane, and $\mathcal{O}$ be a set of $k \geq 2$ lines passing through the origin. We study the problem of computing the set of orientations of the lines of $\mathcal{O}$ for which the $\mathcal{O}$-convex hull of $R$ contains no points of $B$. For $k=2$ orthogonal lines we have the rectilinear convex hull. In optimal $O(n \log n)$ time and $O(n)$ space, $n = \vert R \vert + \vert B \vert$, we compute the set of rotation angles such that, after simultaneously rotating the lines of $\mathcal{O}$ around the origin in the same direction, the rectilinear convex hull of $R$ contains no points of $B$. We generalize this result to the case where $\mathcal{O}$ is formed by $k \geq 2$ lines with arbitrary orientations. In the counter-clockwise circular order of the lines of $\mathcal{O}$, let $α_i$ be the angle required to clockwise rotate the $i$th line so it coincides with its successor. We solve the problem in this case in $O(1/Θ\cdot N \log N)$ time and $O(1/Θ\cdot N)$ space, where $Θ= \min \{ α_1,\ldots,α_k \}$ and $N=\max\{k,\vert R \vert + \vert B \vert \}$. We finally consider the case in which $\mathcal{O}$ is formed by $k=2$ lines, one of the lines is fixed, and the second line rotates by an angle that goes from $0$ to $π$. We show that this last case can also be solved in optimal $O(n\log n)$ time and $O(n)$ space, where $n = \vert R \vert + \vert B \vert$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
