Quadratic Placement Revisited M1.3 : Methodology (techniques for Deep-submicron Placement)

The basic \quadratic placement" methodology is rooted in works by Wipper et al. 36], Cheng and Kuh 8], Tsay et al. 32, 33, 34] and many others. It is also, reputedly, an approach that has been used in commercial and in-house tools for placement of standard-cell and gate-array designs. The quadratic placement methodology entails (i) solving recursively generated sparse systems of linear equations, each of which captures one-dimensional placement with a squared-wirelength objective (hence the term \quadratic"), along with (ii) a coupled min-cut partitioning step. In this paper, we dissect the implementation and motivations for quadratic placement: our goal is to provide design and usage intuition for deep-submicron CAD developers and users alike. After providing necessary background discussion, our paper makes the following contributions. First, we discuss performance enhancements. We show that Krylov subspace engines for solving sparse systems of linear equations are signiicantly more eeective than the successive over-relaxation (SOR) engine originally used in 32]. We also show that interaction between the linear system solver and the coupled min-cut partitioning step introduces novel convergence criteria for the solver. Thus, we propose an order convergence approach which can maintain solution quality while using substantially fewer solver iterations. Second, we discuss the motivations and relevance of the quadratic placement approach, in the context of past and future algorithmic technology, performance requirements, and design methodology. Quadratic placement arguably can be boiled down into a variant of classic \min-cut placement". We observe that use of the numerical linear systems solvers with quadratic wirelength objective may historically be due to the pre-1990's weakness of min-cut partitioners: i.e., numerical engines were needed to provide helpful hints to min-cut partitioners. Detailed experiments with modern partitioning technology show that the \quadratic placement approach" might no longer require a \quadratic" systems solver. And while quadratic placement remains the state of the art, there are several methodology drivers in deep-submicron performance-driven design that may soon require more novel approaches to the placement problem.