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As minimum feature size and pitch spacing further scale down, triple patterning lithography is a likely 193 nm extension along the paradigm of double patterning lithography for 14-nm technology node. Layout decomposition, which divides input layout into several masks to minimize the conflict and stitch numbers, is a crucial design step for double/triple patterning lithography. In this paper, we present a systematic study on triple patterning layout decomposition problem, which is shown to be NP-hard. Because of the NP-hardness, the runtime required to exactly solve it increases dramatically with the problem size. We first propose a set of graph division techniques to reduce the problem size. Then, we develop integer linear programming (ILP) to solve it. For large layouts, even with the graph-division techniques, ILP may still suffer from serious runtime overhead. To achieve better trade-off between runtime and performance, we present a novel semidefinite programming (SDP)-based algorithm. Followed by a mapping process, we can translate the SDP solutions into the final decomposition solutions. Experimental results show that the graph division can reduce runtime dramatically. In addition, SDP-based algorithm can achieve great speedup even compared with accelerated ILP, with very comparable results in terms of the stitch number and the conflict number.

Layout Decomposition for Triple Patterning Lithography