An exact algorithm for the bottleneck 2-connected k-Steiner network problem in Lp planes

We present the first exact algorithm for constructing minimum bottleneck 2-connected Steiner networks containing at most k Steiner points, where k > 2 is a constant integer. The objective of the problem is – given a set of n terminals embedded in the Euclidean plane – to find the locations of the Steiner points, and the topology of a 2-connected graph Nk spanning the Steiner points and the terminals, such that the length of the bottleneck (the longest edge ofNk) is minimised. The problem is motivated by the modelling of relay-augmentation for optimisation of energy consumption in wireless transmission networks. Our algorithm employs Voronoi diagrams and properties of block cut-vertex decompositions of graphs to find an optimal solution in O(h(k)n log n) steps, where h(k) is a function of k only.