Efficient generation of diagonal constraints for 2-D mask compaction

We propose a new efficient constraint generation technique for 2-D mask compaction based on Branch-and-Bound Optimization (BBO). To assure separation in X and Y between rectangles in a layout, we generate the so called "diagonal" constraints in addition to the usual simple constraints in the X and Y directions. A diagonal constraint consist of two constraints, one in each dimension. Our method determines areas around the rectangles beyond which the diagonal (and the simple) constraints need not be generated, producing thus a nearly irredundant system of constraints. During BBO, the diagonal constraints are treated as dual constraints, in which case, one of the two component constraints of the dual can be deactivated. When a such constraint is deactivated, few additional constraints must be be added to assure a legal layout. The overall performance is still considerably improved, because there are fewer dual constraints. Proofs of sufficiency are based on a novel characterization of the 2-D constraint structure.