Highly Efficient Gradient Computation for Density-Constrained Analytical Placement

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 of
n
compared with a naive method, where
n
is 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.