AI 中文总结
研究平面上双色点完全二分图的生成子图，针对边数与拉伸问题，给出边数为\(O(\sqrt{1/\varepsilon}\cdot n)\)的\((3 + \varepsilon)-\)生成子图新构造，解决多年开放问题并提出新问题，还研究实线上双色点情况，给出相关生成子图结论。
AI 中文摘要
对于边加权图\(G=(V,E)\)和拉伸参数\(t\geq1\)，\(t -\)生成子图\(H\subseteq G\)需满足对于所有\(u,v\in V\)，\(G\)和\(H\)中的最短路径距离满足\(\delta_H(u,v)\leq t\,\delta_G(u,v)\)。在度量生成子图中，\(V\)是有限度量空间且\(G\)是边权重对应端点间距离的完全图。当\(G\)是平面上\(n\)个点的完全图时，对于任意\(t>1\)可构造\(O(n)\)大小的\(t -\)生成子图。当\(G = K(R,B)\)是平面上\(n\)个双色点的完全二分图时，一般不存在边数为\(o(n^2)\)且拉伸\(t<3\)的生成子图构造。Bose等人构造了边数为\(O(n\log n)\)的\((3 + \varepsilon)-\)生成子图。本文主要结果是构造了边数为\(O(\sqrt{1/\varepsilon}\cdot n)\)的\((3 + \varepsilon)-\)生成子图，解决了一个17年多的开放问题并提出新研究问题。还研究了实线上\(n\)个双色点的\(G = K(R,B)\)的生成子图，证明其最小生成树是7 - 生成子图，并构造了边数至多为\(2n - 3\)的3 - 生成子图。
英文摘要
For an edge-weighted graph $G=(V,E)$ and a stretch parameter $t\geq 1$, a $t$-spanner is a subgraph $H\subseteq G$ such that the shortest path distances in $G$ and $H$ satisfy $δ_H(u,v)\leq t\, δ_G(u,v)$ for all $u,v\in V$. In metric spanners, $V$ is a finite metric space, and $G$ is the complete graph with edge weights corresponding to the distances between the endpoints. When $G$ is the complete graph on $n$ points in the plane, $O(n)$-size $t$-spanners are possible for any $t>1$: For every $\varepsilon>0$, there is an $(1+\varepsilon)$-spanner with $O(n/\varepsilon)$ edges (i.e., the stretch can be arbitrarily close to 1). When $G=K(R,B)$ is the complete bipartite graph on $n$ bichromatic points in the plane, in general, no spanner construction can guarantee stretch $t<3$ with $o(n^2)$ edges. Bose et al.~(SICOMP 2009) constructed a $(3+\varepsilon)$-spanner with $O(n\log n)$ edges for any constant $\varepsilon>0$. Our main result is a new construction for a $(3+\varepsilon)$-spanner with $O(\sqrt{1/\varepsilon}\cdot n)$ edges. Eliminating the $O(\log n)$ factor resolves a problem left open for more than 17 years, and raises a new research problem about optimizing the dependence on $\varepsilon$. We also study spanners for $G=K(R,B)$ on $n$ bichromatic points on the real line: In this case, we show that the MST of $K(R,B)$ is a 7-spanner, and we construct a 3-spanner with at most $2n-3$ edges.
Comments19 pages, 7 figures
