On the three-dimensional channel routing

3D channel routing is a fundamental problem in the physical design of 3D integrated circuits. The 3D channel is a 3D grid G and the terminals are vertices of G located in the top and bottom layers. A net is a set of terminals to be connected. The object of the 3D channel routing problem is to connect the terminals in each net with a tree (wire) in G using as few layers as possible and as short wires as possible in such a way that wires for distinct nets are disjoint. This paper shows that any set of n 2-terminal nets can be routed in a 3D channel with O(√n) layers using wires of length O(√n). We also show that there exists a set of n 2-terminal nets that requires a 3D channel with Ω(√n) layers to be routed.