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 2 k + 3 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. 
Survivable minimum bottleneck networks