关于Theta图的平均边数
On the Average Number of Edges in Theta Graphs
摘要：
Theta图是一种重要的几何图，具有许多应用，包括无线网络、运动规划、实时动画和最小生成树构造。给出了平面上齐次Poisson点过程θ图的平均度的封闭形式表达式。然后，我们表明，基本上相同的界限-与消失的误差项-持有θ图的有限组的点，均匀分布在一个正方形。最后，我们证明了在一个正方形中均匀分布的点的θ图中的边数集中在其期望值周围。
英文摘要：
Theta graphs are important geometric graphs that have many applications, including wireless networking, motion planning, real-time animation, and minimum-spanning tree construction. We give closed form expressions for the average degree of theta graphs of a homogeneous Poisson point process over the plane. We then show that essentially the same bounds---with vanishing error terms---hold for theta graphs of finite sets of points that are uniformly distributed in a square. Finally, we show that the number of edges in a theta graph of points uniformly distributed in a square is concentrated around its expected value.
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