3-dimensional Channel Routing

Consider two parallel planar grids of size w×n. The vertices of these grids are called terminals and pairwise disjoint subsets of terminals are called nets. We aim at routing all nets in a cubic grid between the two layers holding the terminals. However, to ensure solvability, it is allowed to introduce an empty row/column between every two consecutive rows/columns containing the terminals (in both grids). Hence the routing is to be realized in a cubic grid of size 2n× 2w×h. The objective is to minimize the height h. In this paper we generalize previous results of Recski and Szeszlér [10] and show that every problem instance is solvable in polynomial time with height h = O(max(n,w)). This linear bound is best possible (apart from a constant factor).