Near-Optimal Placement Using a Quadratic Objective Function

Placement algorithms for IC layout which are optimal are known to be NP-complete 5. As a result, heuristics such as pairwise-interchange techniques must be employed to generate satisfactory placements. Unfortunately, with these algorithms, there is generally no way of knowing just how far away the result is from optimum. With the quadratic metric used in this study, however, a useful absolute lower bound can be calculated for the score. Experiments show that with this metric, at least for homogenous interchangeable devices confined to a square grid, pairwise interchange suffices to move the placement very close to the global optimum over a range of 100 to 1600 devices; in particular, asymptotic approach to optimality is observed with increasing size. In addition, a theoretical model is developed which explains the observed deviation from optimality.