We propose an electrostatics-based placement algorithm for large-scale mixed-size circuits (ePlace-MS). ePlace-MS is generalized, flat, analytic and nonlinear. The density modeling method eDensity is extended to handle the mixed-size placement. We conduct detailed analysis on the correctness of the gradient formulation and the numerical solution, as well as the rationale of dc removal and the advantages over prior density functions. Nesterov's method is used as the nonlinear solver, which shows high yet stable performance over mixed-size circuits. The steplength is set as the inverse of Lipschitz constant of the gradient function, while we develop a backtracking method to prevent overestimation. An approximated nonlinear preconditioner is developed to minimize the topological and physical differences between large macros and standard cells. Besides, we devise a simulated annealer to legalize the layout of macros and use a second-phase global placement to reoptimize the standard cell layout. All the above innovations are integrated into our mixed-size placement prototype ePlace-MS, which outperforms all the related works in literature with better quality and efficiency. Compared to the leading-edge mixed-size placer NTUplace3, ePlace-MS produces up to 22.98% and on average 8.22% shorter wirelength over all the 16 modern mixed-size benchmark circuits with the same runtime.

We develop a flat, analytic, and nonlinear placement algorithm, ePlace , which is more effective, generalized, simpler, and faster than previous works. Based on the analogy between placement instance and electrostatic system, we develop a novel placement density function eDensity , which models every object as positive charge and the density cost as the potential energy of the electrostatic system. The electric potential and field distribution are coupled with density using a well-defined Poisson's equation, which is numerically solved by spectral methods based on fast Fourier transform (FFT). Instead of using the conjugate gradient (CG) nonlinear solver in previous placers, we propose to use Nesterov's method which achieves faster convergence. The efficiency bottleneck on line search is resolved by predicting the steplength using a closed-form equation of Lipschitz constant. The placement performance is validated through experiments on the ISPD 2005 and ISPD 2006 benchmark suites, where ePlace outperforms all state-of-the-art placers (Capo10.5, FastPlace3.0, RQL, MAPLE, ComPLx, BonnPlace, POLAR, APlace3, NTUPlace3, mPL6) with much shorter wirelength and shorter or comparable runtime. On average, of all the ISPD 2005 benchmarks, ePlace outperforms the leading placer BonnPlace with 2.83% shorter wirelength and runs 3.05× faster; and on average, of all the ISPD 2006 benchmarks, ePlace outperforms the leading placer MAPLE with 4.59% shorter wirelength and runs 2.84× faster.

ePlace is a generalized analytic algorithm to handle large-scale standard-cell and mixed-size placement. We use a novel density function based on electrostatics to remove overlap and Nesterov's method to minimize the nonlinear cost. Steplength is estimated as the inverse of Lipschitz constant, which is determined by our dynamic prediction and backtracking method. An approximated preconditioner is proposed to resolve the difference between large macros and standard cells, while an annealing engine is devised to handle macro legalization followed by placement of standard cells. The above innovations are integrated into our placement prototype ePlace, which outperforms the leading-edge placers on respective standard-cell and mixed-size benchmark suites. Specifically, ePlace produces 2.83%, 4.59% and 7.13% shorter wirelength while runs 3.05×, 2.84× and 1.05× faster than BonnPlace, MAPLE and NTUplace3-unified in average of ISPD 2005, ISPD 2006 and MMS circuits, respectively.

This paper presents a novel mathematical formulation to the placement of arbitrarily sized rectangles that simultaneously account for the topological and geometrical characteristics of the placement problem. Special attention is paid to maintain the continuity of the formulation. Two examples are given to demonstrate the performance of the algorithm. Identical placement is obtained from two different initial placements of an example which has 8 movable components and 29 signal nets. For a practical example with 15 movable blocks and 142 signal nets, a final placement which is manual solution comparable is obtained using random initial positions.