Multi-center congestion estimation and minimization during placement

As technology advances, more and more issues need to be considered in the placement stage, e.g., wirelength, congestion, timing, coupling. It is very hard to consider all of them together at the same time. Thus it is good if we can optimize one cost function without affecting others. In this paper, we will study methods to optimize congestion in placement without inflicting degradations/violations in other objectives or constraint. We give a mathematical equation to predict the overflow within a region using a normal distribution approximation. According to experiments, this equation does give a good estimation of overflow. We used this equation to find the smallest regions which have enough routing resource to alleviate the congestion and propose the flexible expansion scheme in our multi-center congestion reduction (MC2R) algorithm. Experimental results show that generally there is a correlation between the amount of reduction in congestion and the amount of change made to the placement: the more we change the placement, the more reduction in congestion we will get. However, the flexible expansion scheme is very effective in helping us reduce congestion while making only little change to the placement. Comparing to the full expansion scheme (49% congestion reduction and 6.5% change in placement), the flexible expansion scheme together with MC2 R algorithm can reduce congestion by almost the same amount (42%) with much less change made to the placement. (1.8%).