欧氏平面上与流相关的二次Steiner树问题
A Flow-dependent Quadratic Steiner Tree Problem in the Euclidean Plane
摘要：
我们介绍了一个流依赖的版本的二次斯坦纳树问题的平面。一组嵌入式源和汇的问题的一个实例要求有向树$T$跨越这些节点和有限数量的斯坦纳点，使$\displaystyle\sum_{e \in E(T)}f(e)|e|^2$是一个最小值，其中$f(e)$是边$e$上的流。边缘是无电容的，并且流量是相加确定的，即，离开节点$u$的边缘上的流将是进入$u$的所有边缘上的流的总和。我们研究这个问题的动机是它的效用作为一个模型的无线传感器网络的中继增强。在这些情况下，人们试图通过引入额外的中继来优化网络中的功耗-这主要是由于通信，并且在自由空间中，与传输距离的平方成比例。我们证明了几个几何和组合的结构上的最佳和局部最优解-树（根据各种策略，用于界定Steiner点的数量），并描述了一个几何线性时间算法，用于构建这样的树与已知的拓扑结构。
英文摘要：
We introduce a flow-dependent version of the quadratic Steiner tree problem in the plane. An instance of the problem on a set of embedded sources and a sink asks for a directed tree $T$ spanning these nodes and a bounded number of Steiner points, such that $\displaystyle\sum_{e \in E(T)}f(e)|e|^2$ is a minimum, where $f(e)$ is the flow on edge $e$. The edges are uncapacitated and the flows are determined additively, i.e., the flow on an edge leaving a node $u$ will be the sum of the flows on all edges entering $u$. Our motivation for studying this problem is its utility as a model for relay augmentation of wireless sensor networks. In these scenarios one seeks to optimise power consumption -- which is predominantly due to communication and, in free space, is proportional to the square of transmission distance -- in the network by introducing additional relays. We prove several geometric and combinatorial results on the structure of optimal and locally optimal solution-trees (under various strategies for bounding the number of Steiner points) and describe a geometric linear-time algorithm for constructing such trees with known topologies.
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