Overloaded regions in the routing space of a physical network are resolved via a routing procedure composed of a topological routing phase and a geometric routing phase. The overloads are resolved in the topological routing phase where the constraints of routing are less prohibitive. Multiple topological transformations directed toward resolving the overloads are executed in the topological routing phase prior to a geometric arrangement being realized. The topological transformations may be applied concurrently by way of a multi-threaded embodiment of the invention.