Engineering Details of a Stable ForceDirected Placer

1
Engineering Details of a Stable Force-Directed
Placer

• Kristofer Vorwerk
• Andrew Kennings
• Anthony Vannelli
• Electrical and Computer Engineering
• University of Waterloo

2
Outline

• Introduction
• Implementation Details
• Force Computation
• Assessment of Cell Spreading
• Stability Problems
• Improving Quality
• Median Improvement
• Dynamic Force Weighting
• Numerical Results
• Conclusion

3
Introduction

• Force-directed placement is a placement
  methodology with two main thrusts
• Quadratic optimization to pull connected cells
  together.
• Addition of separating forces (like Coulomb
  forces) to push cells apart.
• Force-directed placement is of interest because
  ...
• It is generic (mixed-size blocks are handled
  seamlessly).
• It can be augmented to handle interesting
  problems, like physical re-synthesis.
• From an academic standpoint, it remains a
  relatively untapped means of placement.

4
Background (1 of 3)

• Assuming cell overlap is allowed, we can keep
  connected cells together via the solution of a
  quadratic program (x-direction only)

• Matrix Q (the Hessian) is a result of connections
  between pairs of movable cells.
• The cost vector, c, is a result of connections
  between movable cells and fixed cells (e.g., I/O
  cells). The constant, d, is simply discarded.

5
Background (2 of 3)

• The benefit of the QP formulation is that its
  solution is simple and fast, obtained by solving
  a single linear system
• Its disadvantage is that the resultant placement
  is highly overlapping.

6
Background (3 of 3)

• Cells must be encouraged to spread throughout the
  placement area.
• We can perturb the optimality conditions by
  adding spreading forces.
• That is, we solve a sequence of optimization
  problems where each has a solution given by
  solving a single system of equations

• Continue until cells are distributed throughout
  the placement area.

7
Force-Directed Placement Caveats

• Some aspects of force-directed placement have not
  been well discussed in the literature
• How to efficiently compute spreading forces?
• How to know when to stop placement?
• How to deal with potential instability issues?
• How to achieve high-quality results on-par with
  other modern approaches (like simulated annealing
  or min-cut partitioning)?

8
Implementation Force Computation (1 of 3)

• How do we compute forces on individual cells to
  achieve a uniform distribution of cells
  throughout the placement area?
• Use ideas from n-body simulation (Barnes-Hut Quad
  Tree).
• We begin by building a Quad Tree over the
  placement area

9
Implementation Force Computation (2 of 3)

• Prior to each placement iteration, we do the
  following
• Given the current placement of cells, insert cell
  area into the Quad Tree.
• Using Barnes-Hut computation (interaction lists,
  near neighbors, etc.) compute the force on each
  bin in the lowest level of the Quad Tree.
• Compute the force on each cell by summing the
  forces on the bins with which the cell overlaps.
• This provides a force on each cell.
• The Quad Tree also tracks the distribution of
  cell area throughout the placement region in
  different granularities. This is useful later

10
ImplementationForce Computation (3 of 3)

• It is necessary to somehow scale the forces on
  cells.
• Large cells overlap with more bins in the lowest
  level of the Quad Tree and therefore tend to
  receive larger forces.
• Large cells can (potentially) get pushed too
  much, too soon.
• We employ the simple scaling scheme of dividing
  the force on each cell by roughly the number of
  bins it overlaps.
• Finally, we normalize so that the largest force
  is equal to unity.

Scaling forces results in more uniform force
magnitudes on individual cells.
11
ImplementationOverlap Assessment

• It is important to assess the distribution of
  cells throughout the placement area as the
  algorithm progresses
• Need to know when to stop.
• Need to know if any problems arise (e.g., a
  re-collapsing of cells onto each other).
• Two metrics developed for overlap assessment
• Capacity vs. occupancy.
• Measurement of overlap.

12
ImplementationCapacity/Occupancy Metric

• First metric measures capacity violations using
  the Quad Tree employed earlier to compute
  spreading forces.
• Scans each bin (i,j) in each level l of the Quad
  Tree and computes

• Found to provide a good global assessment of
  overlap, but poor measure near the end of the
  algorithm.

13
ImplementationKlees Measure

• Second metric uses a O(n log n) plane-sweep
  algorithm to assess overlap.
• Divides area of the union of cells by total cell
  area.
• Found to be poor early in placement (simply too
  much overlap), but very accurate toward the end
  of placement.
• We combine the two metrics to accurately monitor
  cell distribution.

• Area of cells individually is 188 units, and the
  area of their union is 170 units.
• 170/188 0.90, or 10 overlap

14
ImplementationStability (1 of 3)

• Placements would occasionally collapse back onto
  itself indicating some sort of problem.
• The algorithm would recover, but required
  additional iterations.
• The problem, it turns out, has to do with the
  conditioning of the Hessian.
• I.e., placement de-stabilization only occurred on
  some benchmarks that proved to have poorly
  conditioned matrices.

15
ImplementationStability (2 of 3)

• Comparing optimality conditions for two
  consecutive placement iterations implies changes
  in node positions are given by

• Conditioning can be improved by adding positive
  entries along the diagonals of Q i.e.,

• The diagonal entries correspond to zero-area
  fixed cells that move along with their associated
  real cell.
• These zero area cells cause the real cells to
  favor following their spreading force and are
  like friction.

16
ImplementationStability (3 of 3)

• Empirically, it has been found that friction
  need only be applied to 5 of the least
  diagonally dominant rows (columns) of Q.
• Does not appear to inversely impact the quality
  of results, while helping to stabilize the
  placement.
• Side benefit of improving the convergence of the
  iterative solver.
• We were only able to implement this stabilization
  method because of the reliability of assessing
  cell overlap via previously described heuristics.
• It is the assessment of overlap via the metrics
  that allow for the detection of the instability
  problem in the first place.

17
Improvements

• The aforementioned ideas were implemented, and
  placements were obtained reliably.
• Results, however, were not comparable with other
  state-of-the-art placement methods based on
  Simulated Annealing and/or Min-Cut Partitioning
  (off by about 7 to 10).
• Results were, however, comparable to other
  force-directed methods (e.g., Kraftwerk).
• Our force-directed placement method needed some
  updating in order to be considered
  state-of-the-art.

18
ImprovementsBoxPlace (1 of 3)

• Hypothesis was that there was a left/right
  problem with cells.
• Solving the global QP provides only approximate
  information regarding cell positions.
• Furthermore, due to spreading forces, it can be
  hard for cells to cross paths.

B
C
A
Cells A and B connected attractive force pulling
A towards B. Cell C creates an obstacle causing
a force keeping A and B apart.
19
ImprovementsBoxPlace (2 of 3)

• Idea of median improvement is
• Pick a candidate cell and its connected edges.
• Compute the min/max position of each edge,
  ignoring the current cell.
• Find the median location.
• Move the candidate cell into the box.
• Overcomes blocking cells improves final
  results by several percent.
• In doing these movements, need to carefully
  monitor overlap to avoid re-introducing too much
  overlap.

20
ImprovementsBoxPlace (3 of 3)

• Another method to improve wire length is to
  modify the spreading forces directly using the
  target location from a hypothetical application
  of median improvement.
• Re-orient the forces (somewhat) so that they
  point not only in the direction of overlap
  removal, but also in the direction of wire length
  minimization.
• We combine the forces in a 60/40 mix (in favor
  of spreading).
• More iterations are required, however, for the
  placement to converge.

21
ImprovementsDynamic Force Weighting

• Recall the perturbed optimality conditionsforces
  are added in each iteration with a weight a

• Other force-directed placers appear to use a
  constant weighting we have found that this does
  not work too well.
• We have found that dynamically adapting weights
  is important.
• Weighting starts off very low, increases as cells
  spread, and decreases if instability is detected.
• In effect, we implement a state-machine (FSM)
  that adapts weights according to the assessment
  of cell spreading.

22
Numerical Results (1 of 4)
Mixed-size IBM benchmark statistics (from Adya
Markov)
23
Numerical Results (2 of 4)

• We compare to Kraftwerk, another force-directed
  placer, using the ISPD2002 mixed-size benchmarks.
• HPWL results show a fairly uniform improvement
  across designs.
• Prior to enhancements using median improvement,
  we were about 1.0 with Kraftwerk.
• Runtimes are comparable (but are on machines of
  slightly different speeds).

Note FDP results presented here are legalized
(i.e., no overlap). Kraftwerk results are not
legal.
24
Numerical Results (3 of 4)

• We also compare to Capo 8.0 flows and mPG on
  mixed-size circuits.
• Note that placements from Capo Flow 2-A contain
  some overlap.
• Our quality is on-par with Capo 8.0 Dragon for
  standard cells.

25
Numerical Results (4 of 4)

• We have since improved our tools handling of
  mixed-size designs.
• These results are obtained using recent
  modifications.
• Quality of results comparable to Capo 9.0 (though
  runtime is higher).

26
Conclusions

• Presented numerous details regarding the
  implementation of force-directed placement.
• Numerical results of the first generation tool
  were comparable to state-of-the-art.
• Some additional work has since taken place which
  now yields 13 better HPWL on mixed-size
  designs. This is comparable to the latest Capo
  9.0 flow.
• Work is continually being done to improve quality
  and performance.
• Source code is freely available for academic use,
  and can be downloaded at http//gibbon.uwaterloo.
  ca