XArchitecture Placement Based on Effective Wire Models

1
X-Architecture Placement Based on Effective Wire
Models

• Tung-Chieh Chen, Yi-Lin Chuang, and Yao-Wen Chang
• Graduate Institute of Electronics Engineering
• Department of Electrical Engineering
• National Taiwan University
• Taipei, Taiwan
• March 20, 2007

2
Outline

• Introduction
• Previous works
• New wire model XHPWL
• Applications
• Min-cut partitioning placement
• Analytical placement
• Conclusion

3
Wiring Dominates Nanometer Design

• As integrated circuit geometries keep shrinking,
  interconnect delay has become the dominant factor
  in determining circuit performance.

For 90 nm technology, interconnect delay will
account for 75 of the overall delay.
Source Cadence Design System
4
Solutions

• Timing optimization techniques
• Wire sizing
• Buffer insertion
• Gate sizing

• New IC technologies
• Copper and low-k dielectrics
• X-architecture

5
Manufacturing the X Architecture

• X-initiative
• was created to advance the usage of the X
  Architecture by ensuring support for the X
  Architecture throughout the design and
  manufacturing cycle.

• Impacts on EDA tools
• Placement and Routing
• Extraction

6
Placement and Routing for X Architecture

• Placement
• Simulated annealing
• Chen et al., Estimation of wirelength reduction
  for?-geometry vs. Manhattan placement and
  routing (SLIP-2003)
• Over-simplified all cells are of unit size
• Partitioning placement
• Ono, Tilak, and Madden, Bisection based
  placement for the X architecture (ASP-DAC-2007)
• X-cutlines does not lead to shorter wirelength
• Routing
• Multilevel routing system
• Ho et al., Multilevel full-chip routing for the
  X-based architecture (DAC-2005)
• Global routing
• Cao et al., DraXRouter global routing in
  X-architecture with dynamic resource assignment
  (ASP-DAC-2006)

7
Partitioning Placement

• Teig and Ganley, US Patent 6,848,091
• Ono, Talik, and Madden, ASP-DAC-2007
• Study showed X-cutlines cannot reduce the X
  wirelength

(a) Manhattan cutlines
(b) X cutlines
Shortest X-wirelength
8
Our Contributions

• Propose a new X-half-perimeter wirelength (XHPWL)
  model
• Develop effective x-architecture placers
• Min-cut partitioning placement
• Using generalized net-weighting method
• Analytical placement
• Smoothing XHPWL by log-sum-exp functions
• Achieve shorter X-routing wirelength than the
  Manhattan HPWL model for both min-cut
  partitioning placement and analytical placement.

9
Outline

• Introduction
• Previous works
• New wire model XHPWL
• Applications
• Min-cut partitioning placement
• Analytical placement
• Conclusion

10
Half-Perimeter Wirelength (HPWL)

• Half of the bounding box perimeter length
• X bounding box (XBB)
• The minimum region enclosing all net terminals
  bounded by 0, 45, 90, 135 degree lines

Manhattan Bounding Box
C
C
B
B
D
D
A
A
XHPWL ½ XBB perimeter length
11
Computing X-Half-Perimeter Wirelength (XHPWL)
12
The XHPWL Function
Obtain the Resulting X Bounding Box
XHPWL(e)
We can apply this new model to both min-cut
partitioning and analytical placement algorithms.
13
Outline

• Introduction
• Previous works
• New wire model XHPWL
• Applications
• Min-cut partitioning placement
• Analytical placement
• Conclusion

14
Partitioning Placement Problem

• Consider a region to be divided into two
  subregions.
• Find the partitioned results with the minimum
  wirelength
• Cells are put at the center of the subregion
• Partition recursively to obtain positions for all
  cells

c2
c1
Minimize wirelength (Minimize interconnect Between
subregions)
15
Min-Cut Partitioning

• Do not change cutlines
• Use net-weighting during min-cut to map
  partitioning objective to the desired wirelength
  objective
• Selvakkumaran and Karypis proposed to use
  net-weighting
• Technical Report, Dept CSE, UMinnesota, 2004
• Chen and Chang proposed a compact form to
  minimize MHPWL
• ICCAD-2005
• Roy and Markov minimizes Manhattan Steiner
  wirelength
• ISPD-2006

16
Generalized Net-Weighting

• Consider a net v1, v2, , vm, t1, t2, , tn
• vi pin in a movable cell
• ti fixed pin
• c1 (c2) is the center of the subregion 1 (2)
• Find the following three wirelength values
• w1 wirelength( c1, t1, t2, , tn )
• w2 wirelength( c2, t1, t2, , tn )
• w12 wirelength( c1, c2, t1, t2, , tn )

wirelength( )
The desired wire function
17
Partitioning Graph and Edge Weights

• Create hypergraph G
• Two fixed pseudo nodes to present the two
  subregions
• Movable nodes to present movable cells
• Introduce 1 or 2 hyperedges for a net
• e1 connecting all movable nodes and the fixed
  pseudo node corresponding to the subregion that
  results in a smaller wirelength
• e2 connecting all movable nodes

18
Relation between Cut-Size and Wirelength

• Theorem wirelength min( w1, w2 ) ncutsize

w2 w1 (w2 w1)
w12 w1 (w12 w1)
w1 w1 0
19
Min-Cut Placement Flow
Select a bin to be partitioned
Create the partitioning graph
Assign net-weights using generalized net-weighting
Find a min-cut bisection result
Add large sub-partitions into the bin list
Non-empty bin list
20
Experiments on Min-Cut Partitioning

• Platform AMD Opteron 2.6GHz
• Min-cut partitioning placer NTUplace1
  (ISPD-2005)
• Benchmarks IBM version 2.0 (8 circuits)
• Three different models (for calculating w1, w2,
  w12)
• MHPWL (Manhattan-half-perimeter wirelength)
• XHPWL (X-half-perimeter wirelength)
• XStWL (X Steiner wirelength)
• Use total X Steiner wirelength to evaluate the
  resulting placement

21
Resulting Wirelengths and CPU times

• XHPWL 1 shorter wirelength, 8 CPU penalty
• XStWL 5 shorter wirelength, 22 CPU penalty

22
X Steiner Wirelength Reductions

• XHPWL reduces up to about 2 wirelength
• XStWL reduces up to about 6 wirelength

0.00
23
Outline

• Introduction
• Previous works
• New wire model XHPWL
• Applications
• Min-cut partitioning placement
• Analytical placement
• Conclusion

24
Analytical Placement

• Minimize W(x) O(x)
• Wire forces dW(x) / dx
• Spreading forces dO(x) / dx

W(x) wirelength function O(x) overlap
function
Wire forcesMinimize wirelengths
Spreading forcesMinimize overlaps
25
Wire Forces in Analytical Placement

• Pins on the boundary receive forces to reduce the
  bounding box size.

C
B
D
A
Wirelength Forces and the Manhattan
Bounding Box
B has a wire force. C and D change their force
directions.
26
Smoothing XHPWL

• The wire function needs to be smooth enough for
  analytical placement to facilitate the minimizing
  process
• XHPWL is not smooth

27
Log-Sum-Exp Function

• Use the log-sum-exp function to smooth the
  max-abs function

28
XHPWL-LSE Function

• The smoothed version of the XHPWL function

29
Wire Forces

• Forces are given by the gradient of the wire
  function

Vertical
Horizontal
30
Analytical Placement Flow
Find an initial placement
Minimize a W ß O
Find wire forces (dW/dx) and spreading forces
(dO/dx)
Move cells
Cannot further minimizing
Update a and ß
Spreading enough
31
Experiments on Analytical Placement

• Platform AMD Opteron 2.6GHz
• Analytical placer NTUplace3 (ICCAD-2006)
• Benchmarks IBM version 2.0 (8 circuits)
• Three different models
• MHPWL (Manhattan-half-perimeter wirelength)
• XHPWL (X-half-perimeter wirelength)
• Use total X Steiner wirelength to evaluate the
  resulting placement

32
Resulting Wirelengths and CPU times

• 3 less X-Steiner wirelength on average
• 15 more CPU time on average

33
X Steiner Wirelength Reductions

• XHPWL can consistently reduce X-Steiner
  wirelengths.
• Up to about 5 reduction

0.00
34
Outline

• Introduction
• Previous works
• New wire model XHPWL
• Applications
• Min-cut partitioning placement
• Analytical placement
• Conclusion

35
Summary of Wirelength Reductions

• Using both X placement and X routing can reduce
  11 to 12 wirelength on average

36
Conclusions

• The XHPWL model is effective to minimize the
  X-architecture wirelength
• The generalized net-weighting method for min-cut
  partitioning placement can incorporate different
  wire models.
• The smoothing XHPWL, XHPWL-LSE, is proposed for
  analytical placement
• Using both X placement and X routing can reduce
  11 to 12 wirelength on average
• With only 8 to 22 CPU time penalty

37
Thank You!
Resulting Placement IBM01