Quadratic Placement Revisited

Abstract

The "quadratic placement" methodology is rooted in [6] [14] [16] and is reputedly used in many commercial and in-house tools for placement of standard-cell and gate-array designs. The methodology iterates between two basic steps: solving sparse systems of linear equations, and repartitioning. This work dissects the implementation and motivations for quadratic placement. We first show that (i) Krylov subspace engines for solving sparse systems of linear equations are more effective than the traditional successive over-relaxation (SOR) engine [15] and (ii) order convergence criteria can maintain solution quality while using substantially fewer solver iterations. We then discuss the motivations and relevance of the quadratic placement approach, in the context of past and future algorithmic technology, performance requirements, and design methodology. We provide evidence that the use of numerical linear systems solvers with quadratic wirelength objective may 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. Finally, we note emerging methodology drivers in deep-submicron design that may require new placement approaches to the placement problem.

Quadratic placement revisited

“…The behavior observed with golem3 is similar to that observed with case3 and avq large (see[3] for the complete set of experiments). Both avq large and golem3 are available from http://www.cbl.ncsu.edu (the CAD Benchmarking Laboratory).…”

“…Note that the benefit of using a later iterate as opposed to a relatively early iterate is marginal; there is indeed a potential gain if we can apply order convergence criteria. We have also observed that not only do the cutsizes remain similar from iterate to iterate, but the solutions themselves do not vary significantly (see [3] for more details, including data showing Hamming distances for various pairs of partitioning solutions). Figure 4: Iteration number versus minimum cut obtained for 5 runs of ML-FM using the iterate as a pre-seed.…”

New theoretical results on quadratic placement

“…The latter ones are used frequently for the layout of many of the most complex ASICs. Previously, quadratic placement was mainly used as an initial solution for a subsequent min-cut strategy; see [27,28].…”

Quadratic placement with single-iteration linear system solver

“…Amongst several placement techniques, quadratic placement [1] has taken most of academic efforts lately since it is in general faster and achieves comparable results than other state-of-the-art strategies. In this paper, we present a new standard cell placement tool based on quadratic placement technique.…”