toward better wireload models in the presence of obstacles* chung-kuan cheng, andrew b. kahng, bao...
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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, [email protected]
*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
6
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
20
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
31
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
32
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
33
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
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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
38
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
39
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
41
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
42
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
43
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
44
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
45
Multiple Obstacle Analysis
Two obstacles with disjoint spans
Two obstacles with identical x- or y-spans
Two obstacles with covering x- or y-spans