This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. 1 Diffusion-Based Placement Migration with Application on Legalization Haoxing Ren, Member, IEEE, David Z. Pan, Senior Member, IEEE, Charles J. Alpert, Fellow, IEEE, Paul G. Villarrubia and Gi-Joon Nam, Member, IEEE Abstract— Placement migration is the movement of cells within an existing placement to address a variety of post-placement design closure issues, such as timing, routing congestion, signal integrity, and heat distribution. To fix a design problem, one would like to perturb the design as little as possible while preserving the integrity of the original placement. This work presents a new diffusion-based placement method based on a discrete approximation to the closed-form solution of the continuous diffusion equation. It has the advantage of smooth spreading, which helps preserve neighborhood characteristics of the original placement. Applying this technique to placement legalization demonstrates significant improvements in wire length and timing compared to other commonly used techniques. I. I NTRODUCTION During placement and physical synthesis of VLSI circuits, one is commonly faced with tasks such as cell spreading, legalization of overlapping cells, and manipulation of the placement to address objectives like power and routing con- gestion. These tasks share a common theme of starting with an initial placement that is “good” and perturbing it so that it is improved in some way while still preserving the essential characteristics (cell ordering, wirelength, etc.) of the original placement. We call these sets of tasks “placement migration”. Some specific examples of placement migration include the following: • During physical synthesis, one may insert buffers and repower gates, thereby creating overlapping cells. The new instance needs to be legalized, but one wants to avoid moving any cell too far away from its original location. • After placement, it may be necessary to make Engineer- ing Change Orders (ECO) or insert decoupling capacitors which requires spreading to resolve induced overlaps. • Post routing congestion analysis may identify several hot spots of congestion or crosstalk noise. Placement This work is supported in part by SRC under contract 2005-TJ-1321, IBM Faculty Award and IBM Degree Work Study Program. H. Ren is with IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, email: haoxing@us.ibm.com D. Z. Pan is with ECE Department, University of Texas at Austin, Austin, TX 78712, email: dpan@ece.utexas.edu C. J. Alpert is with IBM Austin Research Laboratory, Austin, TX 78758, email: alpert@us.ibm.com P. Villarrubia is with IBM Corporation, Austin, TX 78759, emails: pgvil- lar@us.ibm.com G. Nam is with IBM Austin Research Laboratory, Austin, TX 78758, email: gnam@us.ibm.com Copyright (c) 2007 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending an email to pubs-permissions@ieee.org. migration can locally spread out cells in these congested or noisy regions [1]. • A global analytic or force-directed placer may use place- ment migration to spread out the cells while attempting to preserve the ordering induced by the overlapping analytic solution. This work presents a new technique for placement migration based on the physical process of diffusion. Diffusion is a well-understood physical process that moves elements (such as dopants) from a state with non-zero potential energy to a state of equilibrium. The process can be modeled by breaking down the movements into several small finite time steps, then moving each element the distance it would be expected to move during that time step. Our approach to placement migration follows this model; it moves each cell a small amount in a given time step according to its local density gradient. The more time steps the process is run, the closer the placement gets towards achieving equilibrium. The primary advantage to this approach is that it spreads the placement smoothly which is more likely to preserve the integrity of the original placement. Among the various placement migration applications, le- galization is perhaps the most straightforward. Therefore, the remainder of the paper will discuss diffusion in this context. The paper is organized as follows. Section II formulates the legalization problem and reviews previous techniques. Section III describes the mathematical formulation for diffusion in the continuous space. Section IV presents the numerical algorithm required to simulate the diffusion process. Section V gives the diffusion-based legalization algorithm. Section VI introduces a robust local diffusion algorithm which utilizes local diffusion and dynamic density update to achieve better quality results than the original diffusion algorithm and runs faster. It only “diffuse” cells as necessary to make a placement legal. Ex- perimental results are shown in section VII, followed by the conclusion in section VIII. A preliminary version of this paper was presented in DAC 2005 [2]. II. LEGALIZATION FORMULATION AND OVERVIEW A placement is illegal if cells overlap or are not aligned with circuit rows. The term “legalization” describes the problem of taking an illegal placement and perturbing the layout so that it is legal. The objective of legalization is for this perturbation to be minimal in order to preserve the desired characteristics of the given illegal placement.