Finding a Minimum-Sum Dipolar Spanning Tree in R3

In this paper we consider finding a minimum-sum dipolar spanning tree in Ropf³, and present an algorithm that takes O(n² log² n) time using O(n²) space, thus almost matching the best known results for the planar case. To achieve this, we prove an interesting result related to the complexity of the common intersection of n balls in Ropf³, of possible different radii, that are all tangent to a given point p. The problem has applications in communication networks, when the goal is to minimize the distance between two hubs or servers as well as the distance from any node in the network to the closer of the two hubs, and could lead to reduction in power consumption for devices like PDAs, sensors, cell phones and laptops.