SLIDE 1
Effective Linear Programming-Based Placement Techniques
Sherief Reda Sherief Reda
UC San Diego UC San Diego
Amit Chowdhary Amit Chowdhary
Intel Corporation Intel Corporation
SLIDE 2 Outline Outline
Motivation
- Modeling of cell spreading
Modeling of cell spreading
Linear programming (LP)-
based placer
Applications of LP-
based placer
Experimental results
Conclusions
SLIDE 3 Motivation Motivation
- Linear programming has been shown to be effective
Linear programming has been shown to be effective in modeling timing objective during placement in modeling timing objective during placement
– – Static timing can be modeled as linear constraints Static timing can be modeled as linear constraints using the notion of differential timing [DAC 2005] using the notion of differential timing [DAC 2005]
- Linear programming can effectively model wirelength
Linear programming can effectively model wirelength as half as half-
- perimeter wirelength (HPWL)
perimeter wirelength (HPWL)
- Cell spreading not modeled in linear programming
Cell spreading not modeled in linear programming yet yet
– – LP has been restricted to incremental placement LP has been restricted to incremental placement
SLIDE 4 Main Contribution Main Contribution
- In this paper, we model cell spreading using linear
In this paper, we model cell spreading using linear constraints constraints
- Designed a global placer based on linear
Designed a global placer based on linear programming programming
– – Efficient modeling of timing and wirelength Efficient modeling of timing and wirelength – – Uses relative placement constraints to spread cells Uses relative placement constraints to spread cells gradually gradually
Our LP-
- based placement approach can be used as
based placement approach can be used as
– – Global placer Global placer – – Whitespace allocator (WSA) Whitespace allocator (WSA)
SLIDE 5 LP-based Placement Approach LP-based Placement Approach
- Place cells using an ideal or exact HPWL model of
Place cells using an ideal or exact HPWL model of wirelength wirelength
- Use this initial placement to establish a
Use this initial placement to establish a relative relative
- rdering
- rdering of cells
- f cells
- Transform
Transform relative order relative order of cells into
linear constraints constraints
– – Solve the corresponding LP problem to spread cells Solve the corresponding LP problem to spread cells while maintaining relative order while maintaining relative order
- Iterate till cells are spread out
Iterate till cells are spread out
SLIDE 6
LP-based Placement Approach LP-based Placement Approach
Placement after iteration 1
Placement after iteration 3 Legalized placement Placement after iteration 2 Placement after iteration 4
Initial WL-optimal placement
SLIDE 7 Modeling of wirelength Modeling of wirelength
leftx, , rightx rightx, , lowery lowery and and uppery uppery variables defined for every net variables defined for every net
- HPWL model of wirelength used
HPWL model of wirelength used
- For every cell at location
For every cell at location ( (x x, ,y y) ) connected to net connected to net
Length of this net is
lowery uppery Net leftx rightx
y uppery y lowery x rightx x leftx ≥ ≤ ≥ ≤
) ( ) ( lowery uppery leftx rightx l − + − =
SLIDE 8 Lower bound on wirelength Lower bound on wirelength
- Length of a net has a lower bound based on the area of
Length of a net has a lower bound based on the area of cells connected to it cells connected to it
- Lower bound on each net spreads out cells
Lower bound on each net spreads out cells – – Helps in defining relative order of cells Helps in defining relative order of cells
- Overall wirelength objective is
Overall wirelength objective is
∑ ≥
u u
Area l
∑ =
netsl
HPWL
SLIDE 9 Modeling of timing Modeling of timing
- Various aspects of static timing analysis can be
Various aspects of static timing analysis can be formulated as linear constraints of cell placements formulated as linear constraints of cell placements [DAC2005] [DAC2005]
– – Delay and transition time for cells Delay and transition time for cells – – Delay and transition time for nets Delay and transition time for nets – – Propagated arrival times Propagated arrival times – – Slack at cell pins Slack at cell pins – – Timing metrics Timing metrics
worst negative slack WNS
total negative slack TNS
SLIDE 10 Defining relative order Defining relative order
For each cell v, define four sets corresponding to the cells
to the left, right, upper, and below v – Quadratic time and space complexity
Reduce space complexity to linear using transitivity
– Still quadratic time
We use fast heuristic methods that capture a good amount
- f the relative order relationships
– Q-adjacency graph – Delaunay triangulation
SLIDE 11 Q-adjacency Graph Q-adjacency Graph
- Simple Idea: Establish an adjacency between each cell and
its closest cell in each of the four quadrant
Complexity is O(M.logM + k.M), where k is a constant that
depends on the input placement
SLIDE 12 Delaunay Triangulation Delaunay Triangulation
- Capture adjacency using Delaunay triangulation
Voronoi diagram Delaunay triangulation (dual of the Voronoi diagram)
- Voronoi diagram
- Partitioning of a plane with n
points into convex polygons such that each polygon contains exactly one generating point and every point in a given polygon is closer to its generating point than to any other point
SLIDE 13 Example of Delaunay and Q-Adjacency Example of Delaunay and Q-Adjacency
- We use relative order from Q-Adjacency as well as
Delaunay triangulation
DEL Q-ADJ + DEL Q-ADJ
SLIDE 14 LP Modeling of Relative Order LP Modeling of Relative Order
u v u v u v
- 2. if u and v do not overlap
make sure a non-overlap does not turn into an overlap
- 1. if u and v overlap in the current placement
then next separation = current separation + minimum additional separation to remove the overlap and make sure u and v relative positions stay the same make sure the amount of overlap reduces
For each adjacency {u, v}:
SLIDE 15
LP-Based Placement Techniques LP-Based Placement Techniques
Our LP-based placement approach can be used in several ways 1. Wirelength-driven global placement 2. Whitespace allocation 3. Timing- and wirelength-driven global placement
SLIDE 16
- 1. Wirelength-driven Global Placement
- 1. Wirelength-driven Global Placement
Relative Placement MidX - Spreading Legalization Detailed Placement Final Placement Unplaced circuit
SLIDE 17 Benchmark statistics Benchmark statistics
16491 16491 18217 18217 14780 14780 18592 18592 16056 16056 1308 1308 7673 7673 11819 11819 20935 20935
# cells # cells
16473 16473 9219 9219 9271 9271 15073 15073 8404 8404 2214 2214 7522 7522 7851 7851 9708 9708
# I O Pads # I O Pads
54% 54% 17817 17817 292 292
WFUB09 WFUB09
52% 52% 1935 1935 146 146
WFUB04 WFUB04
49% 49% 20002 20002 363 363
WFUB05 WFUB05
27% 27% 15960 15960 554 554
WFUB07 WFUB07
55% 55% 12780 12780 356 356
WFUB02 WFUB02
36% 36% 8542 8542 587 587
WFUB03 WFUB03
43% 43% 19525 19525 507 507
WFUB08 WFUB08
41% 41% 53% 53%
WS% WS%
22737 22737 539 539
WFUB06 WFUB06
21901 21901 270 270
WFUB01 WFUB01 # nets # nets # DH cells # DH cells Circuit Circuit
- Circuits chosen from a recent microprocessor
SLIDE 18 Comparison with Other Placers Comparison with Other Placers
Compared results with other placers
- Capo 9.3 (UMich)
- Better of FengShui 2.6 and 5.1 (SUNY)
- APlace 2.0 (UCSD)
- Better of mPL 4.1 and 5.0 (UCLA)
All placements were measured using the same HPWL calculator
SLIDE 19 Results of comparison Results of comparison
Average% Average% WFUB09 WFUB09 WFUB08 WFUB08 WFUB07 WFUB07 WFUB06 WFUB06 WFUB05 WFUB05 WFUB04 WFUB04 WFUB03 WFUB03 WFUB02 WFUB02 WFUB01 WFUB01 Circuit Circuit
1.34%
110 110 802 802 652 652 118 118 107 107 fail fail fail fail 539 539 882 882
APlace APlace 2.0 2.0 0.00% 0.00%
111 111 804 804 693 693 114 114 106 106 94 94 326 326 574 574 888 888
Our Our Placer Placer
117 117 190 190 106 106 128 128 131 131 93 93 110 110 136 136 109 109 699 699 765 765 673 673 651 651 883 883 562 562 349 349 456 456 325 325 872 872 891 891 821 821
0.71% 0.71%
123 123 927 927
Capo Capo 9.3 9.3 10.90% 10.90% 32.05% 32.05%
119 119 117 117 1053 1053 fail fail
mPL mPL 5.0 5.0 FengShui FengShui 5.1 5.1
SLIDE 20 Comparison of Placers Comparison of Placers
5 10 15 20 25 30 35 Average HPWL % Diff MidX Capo9.3 FengShui5.1 APlace2.0 mPL5.0
SLIDE 21
Placements of different placers Placements of different placers
FengShui5.1 HPWL = 891 HPWL = 881 mPL5.0 Capo9.3 HPWL = 823 HPWL = 804 Our Placement APlace2.0 HPWL = 802
SLIDE 22
- 2. Whitespace Allocation
- 2. Whitespace Allocation
In an existing legal placement, we redistribute white space to optimize wirelength
Global Placement + Legalization MidX – Whitespace Allocation Legalization Detailed Placement Final Placement Unplaced circuit
SLIDE 23
Whitespace Allocation: Example Whitespace Allocation: Example
Input global placement: HPWL = 9.43 After MidX and legalization: HPWL = 8.81 After MidX: HPWL = 8.76
SLIDE 24
- 3. Timing-driven global placement
- 3. Timing-driven global placement
Timing-optimal Relaxed Placement MidXT - Spreading Legalization Detailed Placement Final Placement Unplaced circuit
SLIDE 25 Timing-driven global placement Timing-driven global placement
- Start with a timing-optimal relaxed placement obtained
using our differential timing-based placer [DAC 2005]
- Identify timing critical cells from a static timer
- Spread cells using MidXT with a combined objective of
- Minimize total wirelength of all nets
- Minimize total displacement of all timing-critical cells
- We are working on integrating the static timing
constraints in MidXT placer
SLIDE 26
Timing-driven placement: Example Timing-driven placement: Example
Input Placement: TNS = -13.99 Relaxed placement: TNS = -8.82 After MidXT: TNS = -9.40 After legalization: TNS = -10.34
SLIDE 27 Conclusions Conclusions
Extended LP-
- based placement approach to
based placement approach to model cell spreading model cell spreading
– – Relative order amongst adjacent cells are Relative order amongst adjacent cells are transformed into linear constraints transformed into linear constraints
Presented a powerful LP-
based global placer
– – Gradually spreads cells while maintaining Gradually spreads cells while maintaining relative order relative order – – Benchmarked against academic placers Benchmarked against academic placers – – Models timing and wirelength very accurately Models timing and wirelength very accurately
