Exact channel routing using symbolic representation

The layout problem in VLSI-design can be broken up into the subtasks partitioning, placement and routing. As well as the others of these tasks, the routing problem is NP-complete and so in general it is impossible to find an optimal solution to it in an efficient way. However for problem instances of small size, it may be possible not only to generate the best routing with respect to a certain cost function, but even to determine all legal solutions to the problem. We consider the channel routing problem and show how to solve it exactly using symbolic methods i.e. MDDs (Multi-valued Decision Diagrams). The basic operations on MDDs can be carried out very efficiently and thus the necessary time-consuming computations can be speeded up considerably. In our approach, wires in the channel are described as functions of multi-valued variables, and so the grid points and also the set of legal solutions can be represented by MDD-nodes. Furthermore, routability is transformed into satisfiability of these functions. By this we can also decide if a channel is routable. Because satisfiability is trivial for MDDs. It will be shown that the best routing can be extracted from the set of all possible solutions very easily. Finally, experimental results are given to demonstrate the feasibility of our approach