加性加权点集的生成子
Spanners of Additively Weighted Point Sets
摘要：
研究了加权点集的几何空间的计算问题。加权点集是一组对$(p,r)$，其中$p$是平面中的点，而$r$是实数。两点$(p_i,r_i)$和$(p_j,r_j)$之间的距离定义为$|p_ip_j|-r_i-r_j$。我们表明，在所有的$r_i$是正数和$|p_ip_j|\geq r_i+r_j$的情况下，所有的$i,j$（在这种情况下，点可以被看作是不相交的磁盘在平面上），姚图的一个变种是一个$(1+ε)$ -图，具有线性数量的边缘。我们还证明了附加加权Delaunay图（附加加权Voronoi图的面对偶）具有常数的生成比。加法加权Delaunay图的直线嵌入可能不是平面图。我们展示了如何计算一个平面嵌入，也有一个恒定的跨度比。
英文摘要：
We study the problem of computing geometric spanners for (additively) weighted point sets. A weighted point set is a set of pairs $(p,r)$ where $p$ is a point in the plane and $r$ is a real number. The distance between two points $(p_i,r_i)$ and $(p_j,r_j)$ is defined as $|p_ip_j|-r_i-r_j$. We show that in the case where all $r_i$ are positive numbers and $|p_ip_j|\geq r_i+r_j$ for all $i,j$ (in which case the points can be seen as non-intersecting disks in the plane), a variant of the Yao graph is a $(1+ε)$-spanner that has a linear number of edges. We also show that the Additively Weighted Delaunay graph (the face-dual of the Additively Weighted Voronoi diagram) has constant spanning ratio. The straight line embedding of the Additively Weighted Delaunay graph may not be a plane graph. We show how to compute a plane embedding that also has a constant spanning ratio.
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