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A Flow-dependent Quadratic Steiner Tree Problem in the Euclidean Plane

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.