Routing vertex disjoint Steiner-trees in a cubic grid and connections to VLSI

Consider a planar grid of sizew×n. The vertices of the grid are called terminals and pairwise disjoint sets of terminals are called nets. We aim at routing all nets in a cubic grid (above the original grid holding the terminals) in a vertex-disjoint way. However, to ensure solvability, it is allowed to extend the length and the width of the original grid to w′ = sw and n′ = sn by introducing s − 1 pieces of empty rows and columns between every two consecutive rows and columns containing the terminals. Hence the routing is to be realized in a cubic grid of size (s · n) × (s · w) × h. The objective is to minimize the height h. It is easy to show that the required height can be as large as h = (max(n,w)) in the worst case. In this paper we show that if s ≥ 2 then a routing with height h = 6 max(n,w) can always be found in polynomial time. Furthermore, the constant factor ‘6’ can be improved either by increasing the value of s or by limiting the number of terminals in a net. Possible trade-offs between s and h are discussed and the various constructions presented are compared by measuring the volumes of the routings obtained. © 2006 Elsevier B.V. All rights reserved.