Toward Better Wireload Models in the Presence of Obstacles*

1
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_at_cs.ucsd.edu,
  dstr_at_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
Synthesis
A Priori WLE
Placement
Online WLE
Global Routing
Detailed Routing
OK?
done
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 average expected
  wirelength without any obstacle
• Point redistribution wirelength Lp is average
  expected wirelength with transparent obstacles
• Resultant wirelength Lr is average expected
  wirelength with opaque obstacles

9
Wirelength Components
M
N
10
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
Blockage effect
Redistribution Lp
Redistribution effect
Intrinsic Li
11
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

• Average expected wirelength between two terminals
  is one third of the half perimeter of the layout
  region without obstacles

13
Point Redistribution WL of Two-terminal Nets
W
where af(M,N,a,b)

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

(a,b)
H
M
N
14
Detour Wirelengthof Two-terminal Nets

• Detour WL dependence on position of, e.g., P2
• Linear for P2 with y coordinate b-H/2 lt yP2 lt yP1
  and 2b-yP1 lt yP2 lt bH/2
• Constant for all P2 with y coordinate yP1 lt yP2 lt
  2b-yP1

W
(a,b)
H
M
p1
N
15
Resultant Wirelength of Two-terminal Nets
where af(M,N,W,H,a) and bg(M,N,W,H,b)

• 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

16
Experimental Setup

• Random point generator
• Visibility graph
• each terminal and obstacle corner as a vertex
• each visible pair of vertices is connected by
  an edge
• Graph Steiner minimal tree heuristic

p1
1
p2
p3
1
L.Kou, G.Markowsky and L.Berman,A Fast
Algorithm for Steiner Trees, Acta Informatica,
15(2), 1981, pp.141-145
17
Redistribution Effect vs. Obstacle Dimension

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

18
Blockage Effect vs. Obstacle Dimension

• Observation 2 The blockage effect Lr-Lp mainly
  increases with the largest obstacle dimension

19
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

21
Experiment Setting for Obstacle Displacement
1
1
22
Redistribution Wirelength vs. Obstacle
Displacement

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

23
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

24
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

25
Experimental Setting forL-shaped Routing Region
P1
1
W
H
P2
1

• Observation 7 L-shaped region has negative
  redistribution effect (Lp lt 0.67) and no blockage
  effect (Lr Lp)

26
Effect of L-shaped Routing Region

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

27
Experimental Setting forC-shaped Routing Region
W
H
1
1
28
Blockage Effect in a C-shaped Region
Comparing with

• The blockage effect doubles when an obstacle with
  a high aspect ratio touches the routing region
  boundary (compared with before it touches the
  boundary)

29
Blockage Effect in a C-shaped Region

• In a C-shaped region the blockage effect mainly
  increases with the obstacle dimension that is not
  on the routing region boundary

30
Redistribution Effect in a C-shaped Region

• The redistribution effect does not generally
  increase with the obstacle area when the obstacle
  is on the routing region boundary

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
x
2

x
x
x
33
Additive Property for Multiple Obstacles

• Blockage effect for m WixHi obstacles with
  non-overlapping x- and y-spans

x

x
34
Experiment Setting for Additive Property
Region 1
Region 2'
1
Region 2
1
35
Additive Property in Region 1

• Observation 9 The redistribution effect is
  additive for obstacles with small areas
• Observation 10 The blockage effect is additive
  if there is no x- or y-span overlap between any
  obstacle pair

36
Non-Additive Property in Region 2

• Observation 9 The redistribution effect is
  additive for obstacles with small areas
• Observation 10 The blockage effect is not
  additive for obstacles with overlapping x- or
  y-spans

37
Effect of Obstacle Number

• 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?

40
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)S 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

46
Two obstacles with covering x- or y-spans

• Number of medium-length wires decreases as any of
  the obstacle widths increases.