A comparison of various terminal-gate relationships for interconnect prediction in VLSI circuits

Over the years, different interpretations of Rent's rule and different ways of estimating the Rent parameters have emerged. In general, these parameters are extracted from the average terminal-gate relationship for a set of circuit modules. We show that this relationship (the Rent characteristic) strongly depends on the definition of the circuit modules. These can be generated in many different ways, either from the topology of the circuit graph or, in a geometric way, by cutting regions from a circuit layout. The resulting Rent parameters can be quite far apart. This paper discusses the fundamental differences between the topological and the two geometric interpretations of the Rent characteristic that are expected to be most appropriate for current wirelength estimation techniques. Our discussion is based on experimental data, as well as on a theoretical model that can be used to estimate certain geometric Rent characteristics from the topological Rent parameters. Using this model, we derive a theoretical lower limit to the value of the average geometric Rent exponent. We also study the impact of the placement approach and placement quality on the geometric Rent characteristics.