A Robust Two-Step Modulus-Based Matrix Splitting Iteration Method for Mixed-Size Cell Circuit Legalization Problem

Abstract

As semiconductor manufacturing moves into the nanotechnology era, mixed-size standard cell circuit designs have become mainstream, but they also pose significant challenges to placement. To address the mixed-size standard cell circuit legalization problem, in this paper, we first formulate it as a quadratic programming problem with nonnegative constraints, resulting in an equivalent linear complementarity problem (LCP). In particular, the system matrix involved in the LCP is positive semi-definite, so we perturb it to a positive-definite matrix. Then, we cast the linear complementarity problem as an equivalent implicit fixed-point equation. To improve the convergence rate of the class of modulus-based matrix splitting iteration method (MMS), we develop a robust two-step MMS to solve the fixed-point equation. Finally, we analyze the convergence property and illustrate the approximation of the solution between the perturbed LCP and the original LCP. Numerical experiments for the legalization problem are provided, and the results reveal that our approach is competitive with the existing state-of-the-art methods for solving mixed-size cell circuit legalization problems.