Three layers are enough

In this paper we show that any channel routing problem of density d involving two-terminal nets can always be solved in the knock-knee mode in a channel of width equal the density d with three conducting layers. An algorithm is described which produces a layout of n nets with the following properties: (i) it has minimal width d; (ii) it can be realized with three layers; (iii) it has at most 3n vias; (iv) any two wires share at most four grid points.