$\frac{2π}{3}$ - MST的4-逼近
A 4-Approximation of the $\frac{2π}{3}$-MST
摘要：
有界角（最小）生成树首先在具有定向天线的无线网络的上下文中引入。它们让人联想到有界度生成树，后者受到了极大的关注。令$P = \{p_1,\ldots,p_n\}$为平面中的一组$n$点，令$Π$为多边形路径$(p_1,\ldots,p_n)$，令$0 < α< 2π$为角度。$P$的$α$ -生成树（$α$-ST）是$P$上的完全欧几里德图的生成树，具有以下性质：对于每个顶点$p_i \in P$，所有与$p_i$关联的边所跨越的（最小）角度至多为$α$。$α$ -最小生成树（$α$-MST）是具有最小权重的$P$的$α$-ST，其中$α$-ST的权重是其边的长度之和。在本文中，我们考虑的问题，计算的$α$-MST，为重要的情况下，$α= \frac{2π}{3}$。我们提出了一个简单的4-近似算法，从而改进了Aschner和Katz和Biniaz等人的结果，他们分别提出了近似比为6和$\frac{16}{3}$的算法。为了得到这个结果，我们设计了一个简单的$O(n)$ -时间算法来构造$P$的一个$\frac{2π}{3}$ -ST\，${\cal T}$，使得${\cal T}$的权值至多是$Π$的两倍，而且${\cal T}$是$Π$的一个3跳的移位.后一个结果是最优的，因为对于任何$\varepsilon > 0$，存在多边形路径，对于该多边形路径，每个$\frac{2π}{3}$的权重大于路径权重的$2-\varepsilon$倍。
英文摘要：
Bounded-angle (minimum) spanning trees were first introduced in the context of wireless networks with directional antennas. They are reminiscent of bounded-degree spanning trees, which have received significant attention. Let $P = \{p_1,\ldots,p_n\}$ be a set of $n$ points in the plane, let $Π$ be the polygonal path $(p_1,\ldots,p_n)$, and let $0 < α< 2π$ be an angle. An $α$-spanning tree ($α$-ST) of $P$ is a spanning tree of the complete Euclidean graph over $P$, with the following property: For each vertex $p_i \in P$, the (smallest) angle that is spanned by all the edges incident to $p_i$ is at most $α$. An $α$-minimum spanning tree ($α$-MST) is an $α$-ST of $P$ of minimum weight, where the weight of an $α$-ST is the sum of the lengths of its edges. In this paper, we consider the problem of computing an $α$-MST, for the important case where $α= \frac{2π}{3}$. We present a simple 4-approximation algorithm, thus improving upon the previous results of Aschner and Katz and Biniaz et al., who presented algorithms with approximation ratios 6 and $\frac{16}{3}$, respectively. In order to obtain this result, we devise a simple $O(n)$-time algorithm for constructing a $\frac{2π}{3}$-ST\, ${\cal T}$ of $P$, such that ${\cal T}$'s weight is at most twice that of $Π$ and, moreover, ${\cal T}$ is a 3-hop spanner of $Π$. This latter result is optimal in the sense that for any $\varepsilon > 0$ there exists a polygonal path for which every $\frac{2π}{3}$-ST has weight greater than $2-\varepsilon$ times the weight of the path.
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