A differential equation for placement analysis

Abstract
A first-order differential equation for placement analysis is derived by considering the competing processes that generate and terminate wires crossing a circuit partition. The solution of this equation provides an estimate for the number of wires needed by a circuit partition for external communication and corresponds to the information normally associated with Rent's rule. The rate model is shown to account not only for the simple power-law form of Rent's rule for small partition sizes but also for deviations from power-law behavior observed for larger partition sizes. The accuracy of the model is validated by comparing solutions of the differential equation with experimental data extracted from a variety of netlists. The netlists, ranging from 10000 to 68000 cells, were optimized using a commercial placement tool. The accurate modeling of terminal-cell data results in a more robust predictive model for the distribution of wire lengths. This improved model accurately captures the change in the distribution of wires as the level of circuit placement optimization ranges from random to highly optimized placement. In addition, the new model provides an explanation for the experimentally observed inflection point and local maximum in the wire length distribution of some netlists.