Timing-Based Placement Considering Uncertainty Due to Process Variations

In the nanometer regime, the effects of variations are having an increasing impact on the delay, power, and yield characteristics of devices. In this paper, we propose the use of fuzzy and stochastic mathematical programming techniques for variation aware timing-based incremental placement. The uncertainty due to process variations in these techniques, are modeled using fuzzy numbers and probabilistic constraints, respectively. The objective is to minimize the critical path delay of the circuit in the presence of variations considering gate and interconnect delays. In the fuzzy approach, the average and worst case deterministic optimizations are performed to identify the bounds and convert the uncertain fuzzy problem into a crisp nonlinear problem. The stochastic optimization framework, on the other hand, transforms the probabilistic constraints into a second-order conic program (SOCP) with explicit mean and variance values. The fuzzy and stochastic approaches tested on ITC´99 benchmark circuits yielded around 12.60% and 10.53% improvements in timing, when compared to optimization with the worst case process variations setting.