We present a rigorous framework that defines a class of net weighting schemes in which unconstrained minimization is successively performed on a weighted objective. We show that, provided certain goals are met in the unconstrained minimization, these net weighting schemes are guaranteed to converge to the optimal solution of the original timing-constrained placement problem. These are the first results that provide conditions under which a net weighting scheme will converge to a timing optimal placement. We then identify several weighting schemes that satisfy the given convergence properties and implement them, with promising results: a modification of the weighting scheme given in [11]results in consistently improved delay over the original, 4% on average, without increase in computation time.

Recent analytical global placers use density constraints to approximate nonoverlap constraints, and these show very successful results. This paper unifies a wide range of density smoothing techniques called global smoothing and presents a highly efficient method for computing the gradient of such smoothed densities used in several well-known analytical placers. This method reduces the complexity of the gradient computation by a factor ofncompared with a naive method, wherenis the number of modules. Furthermore, with this efficient gradient computation, it is able to support an efficient nonlinear programming-based placement framework, which supersedes the existing force-directed placement methods. Experiments show that replacing the approximated gradient computation in mPL6 with the exact gradient computation improves wire length by 15% on the IBM-HB+ benchmark and by 3% on average on the modified International Symposium on Physical Design 2005 (ISPD'05) and ISPD'06 placement contest benchmarks with movable macros. The results also show that the augmented Lagrangian method outperforms the quadratic penalty method with the exact gradient computation.