Optimal Phase Con ict Removal for Layout of Dark Field Alternating Phase Shifting Masks

We describe new, e cient algorithms for layout modi cation and phase assignment for dark eld alternating-type phase-shifting masks in the single-exposure regime. We make the following contributions. First, we suggest new 2-coloring and compaction approach that simultaneously optimizes layout and phase assignment which is based on planar embedding of an associated con ict graph. We also describe additional approaches to co-optimization of layout and phase assignment for alternating PSM. Second, we give optimal and fast algorithms to minimize the number of phase con icts that must be removed to ensure 2-colorability of the con ict graph. We reduce this problem to the T -join problem which asks for a minimum weight edge set A such that a node u is incident to an odd number of edges of A i u belongs to a given node subset T of a weighted graph. Third, we suggest several practical algorithms for the T -join problem. In sparse graphs, our algorithms are faster than previously known methods. Computational experience with industrial VLSI layout benchmarks shows the advantages of the new algorithms.