On crossing minimization problem

In this paper, we consider a problem related to global routing postoptimization: the crossing minimization problem (CMP). Given a global routing representation, the CMP is to minimize redundant crossings between every pair of nets. In particular, there are two kinds of CMP: constrained CMP (CCMP) and unconstrained CMP (UCMP). These problems have been studied previously where an O(m²n) algorithm was proposed for CCMP, and where an (mn²+ξ² ) algorithm was proposed for UCMP where m is the total number of modules, n is the number of nets, and ξ is the number of crossings defined by an initial global routing topology. We present a simpler and faster O(mn) algorithm for CCMP and an O[n(m+ξ)] time algorithm for UCMP. Both algorithms improve over the time bounds of the previously proposed algorithms. The novel part of our algorithm is that it uses the plane embedding information of globally routed nets in the routing area to construct a graph-based framework and obtain a good junction terminal assignment that minimizes the number of crossings