  In Placement Legalization, it is often assumed that (almost) all standard
cells possess the same height and can therefore be aligned in cell rows, which
can then be treated independently. However, this is no longer true for recent
technologies, where a substantial number of cells of double- or even arbitrary
multiple-row height is to be expected. Due to interdependencies between the
cell placements within several rows, the legalization task becomes considerably
harder. In this paper, we show how to optimize quadratic cell movement for
pairs of adjacent rows comprising cells of single- as well as double-row height
with a fixed left-to-right ordering in time $\mathcal{O}(n\cdot\log(n))$,
whereby $n$ denotes the number of cells involved. Opposed to prior works, we
thereby do not artificially bound the maximum cell movement and can guarantee
to find an optimum solution. Experimental results show an average percental
decrease of over $26\%$ in the total quadratic movement when compared to a
legalization approach that fixes cells of more than single-row height after
Global Placement.
