角单调图的构造与局部路由
Angle-Monotone Graphs: Construction and Local Routing
摘要：
一个平面上的几何图是宽度为$γ$的角单调图，如果每对顶点都由宽度为$γ$的角单调路连接，该路使得路中任意两条边的角相差至多$γ$。角单调图具有良好的生成性质。我们证明了平面上的每个点集都有一个宽度为$90^\circ$的角单调图，因此有一个生成比$\sqrt 2$和一个次二次边数。这回答了Dehkordi、Frati和Gudmundsson提出的一个开放性问题。我们展示了如何构造，对于任何点集的大小$n$和任何角度$α$，$0 < α< 45^\circ$，一个角度单调图的宽度$(90^\circ+α)$与$O(\frac{n}α)$边。在此基础上，给出了一个局部路由算法，在这些图中找到宽度为$(90^\circ+α)$的角单调路径。作为路径长度与欧几里德距离之比的路由比至多为$1/\cos(45^\circ + \fracα{2})$，即，范围为$\sqrt 2 \approx 1.414$至$2.613$。对于特殊情况$α= 30^\circ$，我们得到的$Θ_6$ -图和我们的路由算法实现了已知的路由比2，同时找到宽度为$120^\circ$的角度单调路径。
英文摘要：
A geometric graph in the plane is angle-monotone of width $γ$ if every pair of vertices is connected by an angle-monotone path of width $γ$, a path such that the angles of any two edges in the path differ by at most $γ$. Angle-monotone graphs have good spanning properties. We prove that every point set in the plane admits an angle-monotone graph of width $90^\circ$, hence with spanning ratio $\sqrt 2$, and a subquadratic number of edges. This answers an open question posed by Dehkordi, Frati and Gudmundsson. We show how to construct, for any point set of size $n$ and any angle $α$, $0 < α< 45^\circ$, an angle-monotone graph of width $(90^\circ+α)$ with $O(\frac{n}α)$ edges. Furthermore, we give a local routing algorithm to find angle-monotone paths of width $(90^\circ+α)$ in these graphs. The routing ratio, which is the ratio of path length to Euclidean distance, is at most $1/\cos(45^\circ + \fracα{2})$, i.e., ranging from $\sqrt 2 \approx 1.414$ to $2.613$. For the special case $α= 30^\circ$, we obtain the $Θ_6$-graph and our routing algorithm achieves the known routing ratio 2 while finding angle-monotone paths of width $120^\circ$.
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