Survivable minimum bottleneck networks

We show that the survivable bottleneck Steiner tree problem in normed planes can be solved in polynomial time when the number of Steiner points is constant. This is a fundamental problem in wireless ad-hoc network design where the objective is to design networks with efficient routing topologies. Our result holds for a general definition of survivability and for any norm whose ball is specified by a piecewise algebraic curve of bounded degree and with a bounded number of pieces. In particular, under the Euclidean and rectilinear norms our algorithm constructs an optimal solution in O(n log n) steps, where n is the number of terminals and k is the number of Steiner points. Our algorithm is based on the construction of generalised Voronoi diagrams and relative neighbourhood graphs.