Toward Better Wireload Models in the Presence of Obstacles*

Chung-Kuan Cheng, Andrew B. Kahng, Bao Liu and Dirk Stroobandt†

UC San Diego CSE Dept.

†Ghent University ELIS Dept.

e-mail: {kuan,abk,bliu}@cs.ucsd.edu, dstr@elis.rug.ac.be

*This work was supported in part by the MARCO Gigascale Silicon Research Center and a grant from Cadence Design Systems, Inc.

2

Presentation Outline

Motivation and Background Wirelengths and Obstacles Two-terminal Nets with a Single Obstacle Two-terminal Nets with Multiple Obstacles Model Applications Conclusion

3

Motivation and Background

Impasse of interconnect delay and placement To break impasse: use wireload models Wireload models benefit from wirelength estimation

techniques IP blocks in SOC design form routing obstacles Wirelength estimation cannot be blind to routing

obstacles

4

A Priori or Online Wirelength Estimation

A Priori WLE

Placement

Global Routing

Detailed Routing

OK?

done

Online WLE

Synthesis

5

Presentation Outline

Motivation and Background Wire Lengths and Obstacles Two-terminal Nets with a Single Obstacle Two-terminal Nets with Multiple Obstacles Model Applications Conclusion

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Problem Formulation

Given obstacles and n terminals uniformly distributed in a rectangular routing region that lie outside the obstacles

Find the expected rectilinear Steiner minimal length of the n-terminal net

M

N

7

Effects of Routing Obstacles on Expected Wirelength

Detours that have to be made around the obstacles

Changes due to redistribution of interconnect terminals

M

N

8

Definitions of Wirelength Components

Intrinsic wirelength Li is expected wirelength without any obstacle

Point redistribution wirelength Lp is expected wirelength with transparent obstacles

Resultant wirelength Lr is expected wirelength with opaque obstacles

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Wirelength Components

P1

Intrinsic Li

P2

M

N

P2’

P1’

Point Redistribution Lp

W

H

Resultant Lr

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Summary of Wirelength Components

Redistribution effect equals Lp-Li (in the presence of transparent obstacles)

Blockage effect equals Lr-Lp (in the presence of opaque obstacles)

Resultant Lr

Redistribution Lp

Intrinsic Li

Blockage effect

Redistribution effect

Blockage effect is ALWAYS POSITIVERedistribution effect CAN BE NEGATIVE

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Presentation Outline

Motivation and Background Wire Lengths and Obstacles Two-terminal Nets with a Single Obstacle Two-terminal Nets with Multiple Obstacles Model Applications Conclusion

12

Intrinsic Wirelength of Two-terminal Nets

3

|)||(|

)2(

0 0 0 0

1212

0 0 0 0

12122121

NM

dxdxdydy

dxdxdydyyyxx

L

N N M M

N N M M

i

M

NAverage expected wirelength between two terminals is one

third of the half perimeter of the layout region without obstacles

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Point Redistribution WL of Two-terminal Nets

HW )2()2( ip LL

Observation 1: The redistribution effect Lp-Li (the difference of average expected wirelength with and without transparent obstacles) mainly increases with the obstacle area

M

N

W

H

where =f(M,N,a,b)

(a,b)

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Detour Wirelengthof Two-terminal Nets

N

M

W

H

p1 p2

p2

p2

Detour WL dependence on position of, e.g., P2

Linear for P2 with y coordinate b-H/2 < yP2 < yP1 and 2b-yP1 < yP2 < b+H/2

Constant for all P2 with y coordinate yP1 < yP2 < 2b-yP1

Detour WL

yP2

(a,b)

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Resultant Wirelength of Two-terminal Nets

33

2

33

)(3

))2/)(2/()2/)(2/((2

)(

WH

WH

WHMN

HbHbMWaWaN

LELL dpr

Observation 2: The blockage effect Lr-Lp (the difference of average expected wirelength with transparent and opaque obstacles) mainly increases with the largest obstacle dimension

where =f(M,N,W,H,a) and =g(M,N,W,H,b)

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Experimental Setup

Random point generator Visibility graph

each terminal and obstacle corner is a vertex

each “visible” pair of vertices is connected by an edge

Graph Steiner minimal tree heuristic*

1

1

p1

p2

p3

* L.Kou, G.Markowsky and L.Berman,“A Fast Algorithm for Steiner

Trees”, Acta Informatica, 15(2), 1981, pp.141-145

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Redistribution Effect vs. Obstacle Dimension

Observation 1: The redistribution effect Lp-Li mainly increases with the obstacle area

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Blockage Effect vs. Obstacle Dimension

Observation 2: The blockage effect Lr-Lp mainly

increases with the largest obstacle dimension

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Redistribution Effect of Ten-terminal Nets

Observation 3: For multi-terminal nets the redistribution effect increases with the number of terminals and with the obstacle area

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Blockage Effect of Ten-terminal Nets

Observation 3: For multi-terminal nets the blockage effect increases with the number of terminals and with the difference between obstacle dimensions

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Experiment Setting for Obstacle Displacement

1

1

0.5

0.2

0.5

0.2

0.5

0.2

0.5

0.2

0.5

0.2

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Redistribution Wirelength vs. Obstacle Displacement

Observation 4: The closer the obstacle is to the routing region boundary the smaller is the redistribution effect

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Blockage Effect vs. Obstacle Displacement

The closer the obstacle is to the routing region boundary the smaller is the blockage effect

Observation 5: Displacement along the longest obstacle side has little effect on blockage effect

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Effect of Layout Region Aspect Ratio

Observation 6: The redistribution effect does not depend on the aspect ratios of the region and the obstacle: it dominates when the aspect ratios are similar

The blockage effect is very dependent on the aspect ratios: it dominates when the aspect ratios are different

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Observation 7: L-shaped region has negative redistribution effect (Lp < 0.67) and no blockage effect (Lr = Lp)

Experimental Setting forL-shaped Routing Region

W

H

1

1

P1

P2

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Effect of L-shaped Routing Region

Observation 8: The more the L-shaped region deviates from a rectangle the less its total wirelength

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Presentation Outline

Motivation and Background Wire Lengths and Obstacles Two-terminal Nets with a Single Obstacle Two-terminal Nets with Multiple Obstacles Model Applications Conclusion

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Additive Property for Multiple Obstacles

Redistribution effect can be obtained by “polynomial expansion”

=x -x =x

-x -x +x =x

- 2 (x +x ) +x

+ 2x

x

+x x

22121

2221 )()(2)( OOOOIIOOI

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Additive Property for Multiple Obstacles

Blockage effect for m WixHi obstacles with non-

overlapping x- and y-spans:

m

i

nii

ip

m

iir

nii

pr

HWNM

nLnLHWNMnLnL

1

,1

,

)(

))()(()()()(

x + x

37

Effect of Obstacle Numberm \ A 0.1 0.3 0.5 0.7 0.91 1.0183 1.0178 1.0427 1.0427 1.0427

2 1.0184 1.0432 1.0378 1.0342 1.0333

4 1.0280 1.0793 1.0784 1.0739 1.0756

6 1.0459 1.1210 1.1204 1.1119 1.1140

8 1.0609 1.1521 1.1446 1.1452 1.1524

10 1.0709 1.1776 1.1716 1.1696 1.1663

Randomly generating a given number m of obstacles with a prescribed total obstacle area A

Observation 11: The total wirelength increases as the number of obstacles increases while the total obstacle area remains the same

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Presentation Outline

Motivation and Background Wire Lengths and Obstacles Two-terminal Nets with a Single Obstacle Two-terminal Nets with Multiple Obstacles Model Applications Conclusion

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Analyze Individual Wires

Redistribution effect is an average effect over all wires

Blockage effect is different for each wire with different length

Which wire of what length suffers blockage effect the most?

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Blockage Distribution

Blockage effect makes a lot of differences for medium-sized wires (~30% wires make detour, up to a 60% increase in wirelength)

Can be combined with different wirelength distribution models

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Presentation Outline

Motivation and Background Wire Lengths and Obstacles Two-terminal Nets with a Single Obstacle Two-terminal Nets with Multiple Obstacles Model Applications Conclusion

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Conclusion

The first work to consider routing obstacle effect in wirelength estimation

Distinguish two routing obstacle effects Theoretical expressions for 2-terminal nets and a

single obstacle Lookup table for multi-terminal nets and additive

property for multiple obstacles Help to guide SOC design and improve wireload

models

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Future Directions

Continuous study on multi-obstacle cases for finding equivalent obstacle relationships

Combination with different wirelength distributions which count placement optimization effect

Effects of channel capacity and routing sequence

Wirelength estimation for skew-balanced clock trees

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Discrete Analysis Approach

Site density function f(l) is the number of wires with length l

generating polynomial V(x)= f(l)xl

Complete expression for intrinsic, redistribution and resultant wirelenghts

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Multiple Obstacle Analysis

Two obstacles with disjoint spans

Two obstacles with identical x- or y-spans

Two obstacles with covering x- or y-spans