集合可视化中的着色生成图
Colored Spanning Graphs for Set Visualization
摘要：
我们研究了一个算法问题，这是出于墨水最小化稀疏集可视化。我们的输入是平面上的一组点，它们可以是蓝色、红色或紫色。蓝色点只属于蓝色集合，红色点只属于红色集合，紫色点属于两个集合。\emph{red-blue-purple spanning graph}（RBP生成图）是连接点的边的集合，使得由红色和紫色点诱导的子图是连通的，并且由蓝色和紫色点诱导的子图是连通的。我们研究了最小RBP生成图的几何性质和与计算它们相关的算法问题。具体来说，我们表明，一般的问题可以解决在多项式时间使用拟阵技术。此外，我们讨论了更有效的算法的情况下，点位于一条线或一个圆，也描述了一个快速的$(\frac 12ρ+1)$近似算法，其中$ρ$是施泰纳比。
英文摘要：
We study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the blue set, red points belong exclusively to the red set, and purple points belong to both sets. A \emph{red-blue-purple spanning graph} (RBP spanning graph) is a set of edges connecting the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected. We study the geometric properties of minimum RBP spanning graphs and the algorithmic problems associated with computing them. Specifically, we show that the general problem can be solved in polynomial time using matroid techniques. In addition, we discuss more efficient algorithms for the case in which points are located on a line or a circle, and also describe a fast $(\frac 12ρ+1)$-approximation algorithm, where $ρ$ is the Steiner ratio.
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