Placement legalization for heterogeneous cells of non-integer multiple-heights

It is intuitively clear that a circuit to be implemented by selectively utilizing standard cells of various non-integer multiple-heights (NIMH) (e.g., mixed use of 6-track, 7.5-track, and 9-track cells) is able to provide a better opportunity in optimizing power, performance, and area over that by using cells of single-height only or of integer multiple-heights only. However, from the cell placement legalization point of view, the issues to be addressed for placement legalization on NIMH designs are very complex. And this paper primarily focuses on introducing novel ideas for row placement when utilizing NIMH cells, which involves determining the row pattern. The most inter-dependent and critical tasks, which are rather unique to the NIMH cell placement legalization problem, are (task 1) for the cells of the same height, distributing and assigning them to a set of distinct rows on a die and (task 2) determining the location of the rows containing cells of the same height. We solve the legalization problem by, starting from an initial row placement, iteratively solving the two tasks by formulating task 1 into an instance of row-capacity constrained network flow problem, followed by solving task 2 which leads to an optimal vertical displacement of the cells in the rows. Meanwhile, through experiments, it is shown that our network flow driven global cell assignment to rows for NIMH cell placement problem tightly linking the optimal determination of row location is able to reduce the total amount of cell displacements from the global (initial) placement by 9.5% and 51.2% in comparison with that produced by a greedy approach and the conventional state-of-the-art NIMH cell placement legalization method, respectively.