Cut Redistribution and Insertion for Advanced 1-D Layout Design via Network Flow Optimization

Abstract

Given a 1-D layout with horizontal wires, cut redistribution technique is used for sliding the line-end cuts in order to merge them vertically or resolve spacing conflicts. A pair of movable cuts could have three possible relations, left-of, right-of, and merge-into. We observe that if the left-right-merge orderings of cuts are fixed, the cut redistribution can be formulated as a network flow problem, which can be solved efficiently. We also find that inserting cuts can resolve the spacing conflicts in some circumstances. According to the observations, we proposed a network-flow-based cut redistribution and insertion method for 1-D layout design. First, the orderings of the cuts are generated according to a predefined distribution. The hyperparameters of the distribution are tuned with Bayesian optimization. In order to resolve the conflicts, the cuts are redistributed and extra cuts are also inserted with network-flow-based method. Afterward, we adopt the complementary e-beam lithography to eliminate the unresolved conflicts. Finally, two wire length reduction techniques have been proposed for two different processes of e-beam, respectively. Experimental results show that compared with the existing integer linear programming-based method that does not consider the cut insertion, our method can achieve $132\times $ speedup and higher quality.