A communication network is a graph in which each node has only local information about the graph and nodes communicate by passing messages along its edges. Here, we consider the {\it geometric communication network} where the nodes also occupy points in space and the distance between points is the Euclidean distance. Our goal is to understand the communication cost needed to solve several fundamental geometry problems, including Convex Hull, Diameter, Closest Pair, and approximations of these problems, in the asynchronous CONGEST KT1 model. This extends the 2011 result of Rajsbaum and Urrutia for finding a convex hull of a planar geometric communication network to networks of arbitrary topology.

中文翻译：

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几何通信网络中的通信成本

通信网络是一个图，其中每个节点仅具有关于图的本地信息，并且节点通过沿其边缘传递消息进行通信。在这里，我们考虑{\ it几何通信网络}，其中节点也占据空间中的点，并且点之间的距离是欧几里得距离。我们的目标是了解在异步CONGEST KT1模型中解决几个基本几何问题所需的通信成本，包括凸包，直径，最近对以及这些问题的近似值。这将Rajsbaum和Urrutia的2011年研究结果扩展为寻找平面几何通信网络的凸包到任意拓扑的网络。