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A Morphing Approach To Address Placement Stability

Philip Chong

Christian Szegedy

March 20, 2007 2

Overview

• Motivation/Background• Our Algorithm• Results/Conclusions• Future Work

March 20, 2007 3

Motivation

• Design process has become tremendously complex– Timing, power, noise/crosstalk, manufacturability, etc. play

increasingly important roles

• Placement must account for all these metrics• Pure “top-down” design flow impossible

– Impossible to predict physical effects prior to placement

• Iteration and incremental design are critical for closure• Need to have stability in placement

– Also need stability throughout entire design flow

March 20, 2007 4

Stability

• An algorithm is stable if, when given two similar inputs, it produces two similar outputs– A small change in the input only causes a small change in the

output

• Stability is necessary to be able to obtain closure in an incremental design flow– E.g. a single buffer insertion should not cause a significant

change in the layout

March 20, 2007 5

Our Work

• We have developed a global placement approach called grid morphing which focuses on stability and is targeted towards an incremental design flow

March 20, 2007 6

Existing Work

• Brenner, Vygen, “Legalizing A Placement With Minimum Total Movement”, IEEE Trans. CAD, Dec. 2004– Optimizes total absolute perturbation; we optimize relative

perturbation

• Ren et al, “Diffusion-based Placement Migration”, DAC 2005– Focus on final placement stage and less efficient; we look at

global placement and are more efficient

• Xiu et al, “Large-scale Placement By Grid-Warping”, DAC 2004– Inefficient non-linear formulation and restricted to

quadrisectioning; we have a more efficient approach which optimizes over a global grid

March 20, 2007 7

Overall Placement Flow

Start

Initial QP Solution

Grid Morphing

Local Optimization

Evenly Spread?

Final Placement

Stop

N

March 20, 2007 8

Grid Morphing Overview

• All cell movements are computed using an ordinary bilinear transform

March 20, 2007 9

Grid Morphing Example

InitialPlacement

After FirstMorphing

After SecondMorphing

After FirstIteration

March 20, 2007 10

Morphing Formulation

• Find new control points for the grid which minimizes distortion relative to the original grid

• Tiles in new grid must be large enough to fit their contained gates

• Difficulty: Straightforward Lagrangian relaxation using subgradient method yields poor convergence

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March 20, 2007 11

Our Approach

• Based on an intuitive physical model• Imagine each tile is a flexible container for some gas

under pressure proportional to its placement utilization• Walls between containers move to equalize the pressure

between them• Containers under high pressure expand to occupy a

larger volume, containers under low pressure are squished to occupy a smaller volume

March 20, 2007 12

Control Point Movement

• First move border points according to pressures• Then move control points according to border point

movements

March 20, 2007 13

Algorithmic Improvements

• Multilevel Morphing– Improve runtime: Start with a coarse grid, solve the morphing

problem, then use the solution as an initial solution for a finer grid

• Maximum Blowup Control– Improve stability: Limit the Aj values so that no one grid tile

expands too greatly in one iteration

March 20, 2007 14

Algorithmic Improvements (cont.)

• Netlength Guided Morphing– Improve netlength: First estimate sensitivity of wirelength to grid

control point positions (i.e. how much wirelength degrades when a control point is moved), then add penalty terms to the optimization objective function weighted by the sensitivities

• Local Optimization– Improve netlength: After each iteration perform greedy

optimization of netlength

March 20, 2007 15

Comparison With MetaPlacer

MetaPlacer

Our Placer

Original 10% Resized 20% Resized

March 20, 2007 16

Placement Results

• Net perturbation (Alpert et al, ASPDAC 2005):

• Run times: MetaPlacer 2-3 hours, Our Placer <1 hour

Placement Net Len Avg Pert RMS Pert Max PertMeta Orig 1.10E+11Meta 10% 1.16E+11 1.19E+06 9.87E+06 3.49E+09Meta 20% 1.20E+11 1.20E+06 9.08E+06 2.88E+09

Our Orig 1.02E+11Our 10% 1.06E+11 6.34E+05 3.59E+06 1.01E+09Our 20% 1.09E+11 6.45E+05 3.74E+06 1.18E+09

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March 20, 2007 17

Perturbation Distributions

MetaPlacer Our Placer

5218 1790

• Often makes sense to reduce largest perturbations at the expense of introducing many tiny movements

March 20, 2007 18

Perturbation Maps

MetaPlacer

Our Placer

10% Resized 20% Resized

March 20, 2007 19

Future Work

• More testcases– Wider variety of designs– Examine effect of different kinds of incremental changes– Comparisons with other placers

• Incorporate different metrics– Timing, routability, crosstalk, manufacturability, etc.