On the crossing distribution problem

VLSI layout design is typically decomposed into four steps: placement, global routing, routing region definition, and detailed routing. The crossing distribution problem occurs prior to detailed routing [Groenveld 1989; Mared-Sadowska and Sarrafzadeh 1995; Wang and Shung 1992]. A crossing is defined as the intersection of two nets. The problem of net crossing distribution is important in layout design, such as design of dense chips, multichip modules (MCM), critical net routing, and analog circuits [Groenveld 1989; Sarrafzadah 1995; Wang and Shung 1992]. It is observed that nets crossing each other are more difficult to route than those that do not cross. The layout of crossing nets has to be realized in more than two layers and requires a larger number of vias In this paper we study the crossing distribution problem of two-terminal nets between two regions. We present an optimal O(n2) time algorithm for two-sided nets, where n is the number of nets. Our results are superior to previous ones [Markek-Sadowska and Sarrafzadeh 1995; Wang and Shung 1992]. We give an optimal O(n2) time algorithm for the crossing distribution problem with one-sided nets. We solve optimally the complete version of the crossing distribution problem for two-terminal nets in two regions that has not been studied before.