有界角生成树：带角约束的网络建模
Bounded-Angle Spanning Tree: Modeling Networks with Angular Constraints
摘要：
我们引入了一个新的结构，一组点在平面上和一个角度$α$，这是类似的味道有界度MST。我们将这种结构命名为$α$-MST.令$P$是平面上的一组点，令$0 < α\le 2π$是一个角度。$P$的$α$-ST是由$P$导出的完全欧几里德图的生成树，具有附加性质，即对于每个点$p \in P$，包含与$p$相邻的所有边的围绕$p$的最小角度至多是$α$。$P$的$α$-MST则是最小权重的$P$的$α$-ST。对于$α< π/3$，$α$ -ST并不总是存在，而对于$α\ge π/3$，它总是存在。在本文中，我们研究的问题，计算一个$α$-MST的$α$的几个共同的价值观。受无线网络的启发，我们制定了定向天线的问题。对于每个点$p \in P$，我们将角度$α$和顶点$p$的楔形$W_p$相关联。目标是为每个楔形$W_p$分配方向和半径$r_p$，使得所得到的图是连通的并且其MST是$α$-MST。(We如果是$p \in W_q$、$q \in W_p$和$|pq| \le r_p, r_q$，则在$p$和$q$之间绘制边。）毫不奇怪，计算$α$-MST的问题是NP难的，至少对于$α=π$和$α=2π/3$是这样。我们提出了$α= π/2, 2π/3, π$的常数因子近似算法。我们的主要结果之一是一个令人惊讶的定理，$α= 2π/3$，其中，除了有趣的从几何的角度来看，有重要的应用。例如，该定理保证给定平面中$3n$点的任何集合$P$以及将点划分为$n$三元组的任何划分，可以{ \em独立}定向每个三元组的楔形，使得由$P$诱导的图是连通的。我们将定理应用于{ \em天线转换}问题。
英文摘要：
We introduce a new structure for a set of points in the plane and an angle $α$, which is similar in flavor to a bounded-degree MST. We name this structure $α$-MST. Let $P$ be a set of points in the plane and let $0 < α\le 2π$ be an angle. An $α$-ST of $P$ is a spanning tree of the complete Euclidean graph induced by $P$, with the additional property that for each point $p \in P$, the smallest angle around $p$ containing all the edges adjacent to $p$ is at most $α$. An $α$-MST of $P$ is then an $α$-ST of $P$ of minimum weight. For $α< π/3$, an $α$-ST does not always exist, and, for $α\ge π/3$, it always exists. In this paper, we study the problem of computing an $α$-MST for several common values of $α$. Motivated by wireless networks, we formulate the problem in terms of directional antennas. With each point $p \in P$, we associate a wedge $W_p$ of angle $α$ and apex $p$. The goal is to assign an orientation and a radius $r_p$ to each wedge $W_p$, such that the resulting graph is connected and its MST is an $α$-MST. (We draw an edge between $p$ and $q$ if $p \in W_q$, $q \in W_p$, and $|pq| \le r_p, r_q$.) Unsurprisingly, the problem of computing an $α$-MST is NP-hard, at least for $α=π$ and $α=2π/3$. We present constant-factor approximation algorithms for $α= π/2, 2π/3, π$. One of our major results is a surprising theorem for $α= 2π/3$, which, besides being interesting from a geometric point of view, has important applications. For example, the theorem guarantees that given any set $P$ of $3n$ points in the plane and any partitioning of the points into $n$ triplets, one can orient the wedges of each triplet {\em independently}, such that the graph induced by $P$ is connected. We apply the theorem to the {\em antenna conversion} problem.
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