论文标题
飞机上有色点套装的彩虹多边形
Rainbow polygons for colored point sets in the plane
论文作者
论文摘要
给定平面中的彩色点设置，完美的彩虹多边形是一个简单的多边形，在内部或边界上完全包含每种颜色的一个点。 Let $\operatorname{rb-index}(S)$ denote the smallest size of a perfect rainbow polygon for a colored point set $S$, and let $\operatorname{rb-index}(k)$ be the maximum of $\operatorname{rb-index}(S)$ over all $k$-colored point sets in general position;也就是说，每个$ k $颜色的点seet $ s $都有一个完美的彩虹多边形，最多只能使用$ \ operatatorName {rb-index}（k）$ vertices。在本文中，我们确定$ \ operatoTorname {rb-index}（k）$最高$ k = 7 $的值，这是$ \ operatotorname {rb-index}（k）\ neq k $的第一种情况，我们证明了$ k \ ge 5 $，\ ge [\ freac [\ freac \ freac \ frac {40 \ lfloor（rfloor）/\ rfloor（k） -8} {19}％birgit：\ leq \ operatorName {rb-index}（k）\ leq 10 \ bigg \ bigg \ lfloor \ frac {k} {7} \ bigg \ big \ big \ big \ rfloor + 11。 $ 10 \ lfloor \ frac {k} {7} \ rfloor + 11 $顶点可以在$ o（n \ log n）$时间中计算。
Given a colored point set in the plane, a perfect rainbow polygon is a simple polygon that contains exactly one point of each color, either in its interior or on its boundary. Let $\operatorname{rb-index}(S)$ denote the smallest size of a perfect rainbow polygon for a colored point set $S$, and let $\operatorname{rb-index}(k)$ be the maximum of $\operatorname{rb-index}(S)$ over all $k$-colored point sets in general position; that is, every $k$-colored point set $S$ has a perfect rainbow polygon with at most $\operatorname{rb-index}(k)$ vertices. In this paper, we determine the values of $\operatorname{rb-index}(k)$ up to $k=7$, which is the first case where $\operatorname{rb-index}(k)\neq k$, and we prove that for $k\ge 5$, \[ \frac{40\lfloor (k-1)/2 \rfloor -8}{19} %Birgit: \leq\operatorname{rb-index}(k)\leq 10 \bigg\lfloor\frac{k}{7}\bigg\rfloor + 11. \] Furthermore, for a $k$-colored set of $n$ points in the plane in general position, a perfect rainbow polygon with at most $10 \lfloor\frac{k}{7}\rfloor + 11$ vertices can be computed in $O(n\log n)$ time.
