The top-down “quadratic placement” methodology is rooted in such works as [36, 9, 32]
and is reputedly the basis of commercial and in-house VLSI placement tools. This
methodology iterates between two basic steps: solving sparse systems of linear equations
to achieve a continuous placement solution, and “legalization” of the placement by
transportation or partitioning. Our work, which extends [5], studies implementation
choices and underlying motivations for the quadratic placement methodology. We first
recall some observations from [5], e.g., that (i) Krylov subspace engines for solving
sparse linear systems are more effective than traditional successive over-relaxation
(SOR) engines [33] and (ii) that correlation convergence criteria can maintain solution
quality while using substantially fewer solver iterations. The focus of our investigation is
the coupling of numerical solvers to iterative partitioners that is a hallmark of the
quadratic placement methodology. We provide evidence that this coupling may have
historically been motivated by the pre-1990’s weakness of min-cut partitioners, i.e.,
numerical engines were needed to provide helpful hints to weak min-cut partitioners. In
particular, we show that a modern multilevel FM implementation [2] derives no benefit
from such coupling. We also show that most insights obtained from study of individual
min-cut partitioning instances (within the top-down placement) also hold within the
overall context of a complete top-down placer implementation.
Methods of orthogonal design for solving systems of linear equations for two-dimensional Krylov subspace
