On routability for channel routing problem

TLDR: This paper modify a routing requirement with cyclic conflicts into one without them by dividing trunks into two parts by defining a method of dividing a net n contained in the crowded set into two subnets n' and n'' at some terminal position.

Abstract: The routing requirement in the channel routing problem for automatic wire routing in the interior of LSI is realizable if the constraint graph contains no cycle. Otherwise, the trunks for several appropriate nets must be divided into pieces. In this paper we modify a routing requirement with cyclic conflicts into one without them by dividing trunks into two parts. First we define a directed bipartite graph G to represent the routing requirement. A set of vertices in a strongly connected component in G is called a crowded set. It is shown that a routing requirement is realizable if there exists no crowded set in G. On the other hand, in the presence of a crowded set we define a method of dividing a net n contained in the crowded set into two subnets n' and n'' at some terminal position. If n' and n'' are not contained in any crowded set in the graph modified by division, we say that the division is effective. We then present a necessary and sufficient condition for the existence of effective division. Further, a semi-effective division is defined to deal with the case where there is no effective division and then the routability for the channel routing problem is discussed.