A Randomized Greedy Method for Rectangular-Pattern Fill Problems

In this paper, a class of problems of physical-design optimization with rectangular shapes is presented. Efficient solutions for this class of optimization problems are needed for rectangular metal fill/negative-fill insertion during postrouting optimization. These problems are also shown to be closely related to the problem of floorplanning with rectangular macrocells. The solution must satisfy certain constraints for minimum and maximum pattern densities within a moving rectangular window while optimizing a given objective function such as the area or the aspect ratio of the resultant bounding box. It is shown in this paper that the general class of such problems defined with a gridless formulation is at least NP-hard. This proof, with minor modifications, also extends to the formulation in the gridded domain. A greedy randomized algorithm for one of our problems in the gridded domain is proposed, along with a proof that this algorithm can achieve the given objective with a very high probability while satisfying the constraints. Experimental results based on an implementation of our algorithm are provided and are shown to agree with our theoretical proof.