Understanding force-directed placement Andrew Kennings Electrical - - PowerPoint PPT Presentation

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Understanding force-directed placement

Andrew Kennings Electrical and Computer Engineering University of Waterloo

Force-directed placement... (UTexas@Austin) 2

Outline of talk

• Introduction.
• Implementation details.
• Force computation.
• Force scaling.
• Overlap assessment.
• Stability issues.
• Improvements.
• Median improvement.
• Multi-level clustering.
• Unification with partitioning.
• Numerical results.
• Current and future work.

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Introduction

• Force-directed placement is an alternative placement method

compared to simulated annealing-based or top-down partitioning-based methods.

• It has two main thrusts:
• Quadratic optimization to pull connected cells together.
• Force computation to push cells apart.
• Force-directed placement is interesting…
• Seems fairly generic (no reason mixed-size blocks can’t

be handled w/ o changing the algorithm).

• Seems amenable to physical re-synthesis and incremental

placement due to “continuous cell trajectories”.

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The mathematics

• Assume that cells are allowed to overlap. Connected cells can

be kept close together (and hence indirectly minimize some measure of wire length) by solving a QP: Simple to solve by differentiating, where we find a positive- definite system of linear equations: Clearly fast to solve (advantage), but result has a lot of cell

• verlap (disadvantage).

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Spreading forces

• We can perturb the optimality conditions to change the

resulting solution (i.e., the cell positions) slowly over many iterations: The perturbations are chosen based on the current (overlapping) cell positions in order to remove the cell overlap. Hence, over many iterations, the cells converge to non-

• verlapping positions.

The above equation resembles a force equation, we have force-directed placement.

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Force-directed placement in action

• This slide is inserted to visually

give an idea of what happens during force-directed placement.

• Cell trajectories (top) and

cumulative forces in cells (bottom) are shown as the force-directed placement progresses.

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Towards implementation

• The best known example of force-directed placement is

probably Kraftwerk (1998).

• It sounds very simple but we found several difficulties during

implementation:

• How to actually compute forces efficiently?
• How to properly weight the forces computed at each

iteration?

• How to track the progression of the algorithm and to

decide when to stop?

• How to identify and handle stability problems?
• Briefly touch on each of these topics.

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Efficient force computation

• We compute forces by building (bottom-up) a (Barnes-Hut)

quad-tree once over the entire placement area.

• Then, prior to each QP:
• Each cell’s area is inserted top-down into bins of the

quad-tree at all levels of the quad-tree.

• Force on quad-tree bins is computed using interaction

lists, near neighbor computations, and so forth.

• Forces on individual cells is then computed by summing

up forces from overlapped bins.

• Hence, we use a particle-mesh-particle methodology for force

computation.

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Force scaling

• The quad-tree provides forces (directions) and proportional

lengths.

• Investigation turned out that forces need to be scaled in two

ways:

• For mixed-size designs, larger cells overlap more quad-

tree bins and therefore get very large forces relative to smallish cells.

• Forces are not computed in any sort of scale compared to

the QP spring forces.

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Scaling for differing cell sizes

• Very large forces on large cells

empirically resulted in large blocks getting pushed quickly to the outside

• f the placement region.
• Empirically, found that scaling the

force on each cell computed from the quad-tree by the square root of number of overlapped quad-tree bins fixed the problem.

Forces on cells of different sizes computed from the quad-tree before scaling and after scaling

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Scaling for optimality conditions

• Recall that forces are combine into optimality conditions but

constant force weighting (advocated by Kraftwerk) was found to hurt quality. Implemented a state machine to dynamically weighting: Start out small (cells adjust themselves into correct order relative to each other). Increase to encourage fairly rapid spreading. Based on a monitoring of cell overlap, adjust weight up or down as appropriate.

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Overlap assessment

• The proper assessment and monitoring of cell overlap is
• important. Provides:
• indication of algorithm progression.
• indication of problems in convergence.
• mechanism to determine when to stop.
• Have developed two metrics:
• Two metrics since we found that different metrics are

useful at different stages of the algorithm.

• The two methods are combine in a weighted amount to

provide a single number measuring overlap.

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Overlap assessment – Metric # 1

• Metric # 1 is based on a typical occupancy verses capacity

measure of available placement area.

• We scan the quad-tree used for force computation top-down,

and compute:

• Metric is normalized into the range [ 0,1] where 0 means no
• verlap, and 1 means (essentially) total overlap.
• Metric found to be very good early in placement.

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Overlap assessment – Metric # 2

• Metric # 2 is based on a O(n log n) plane-sweep algorithm

that directly measures the union of cell area.

• Metric is normalized into the range [ 0,1] (divide union of cell

area by total cell area) where 1 means no overlap, and 0 means (essentially) total overlap.

• Metric found to be very good late in placement.

Area of cells individually is 188 units, and the area of their union is 170 units. 170/188 = 0.90, or ~10%

• verlap

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Stability issues

• Discovered that placements
• ccasionally collapse back onto

themselves.

• Due to the iterative solver and poor

conditioning of the system of equations.

• Problem easily seen visually,

but harder to detect in softw are

• - detected by good overlap

assessm ent!!!

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Our solution to stability issues

• Change in cell positions between two iterations of placement

is given by:

• Modify by making it more diagonally dominant:
• Improves conditioning and makes it easier to solve.
• Diagonal modification is like adding a fixed point for each

cell that tracks the cell’s location.

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How good is the method so far?

• Aforementioned implementation should have paralleled

Kraftwerk so we tested:

• Placed the ISPD-2002 benchmarks using Kraftwerk via

bX at the University of Michigan.

• Measured the overlap in Kraftwerk placements and tuned
• ur tool to stop at roughly the same amount of overlap.
• Compared wire length.
• Not surprisingly results were very comparable to Kraftwerk,

but far from other state-of-the-art academic tools like Capo and Fengshui.

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Problems of cell ordering

• Force-directed placement begins from an initial QP that

provides rough information about the left/ right (top/ bottom)

• rdering of cells.

A C B

Hypothesized that there is a cell ordering problem. Spreading forces used to remove overlap make it difficult for cells to cross paths once ordered.

Cells A and B connected; attractive force pulling A tow ards B. Cell C creates an obstacle causing a force keeping A and B apart.

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Median improvement

• We implemented a heuristic that directly attempts to

minimize wire length and interleaved it within the force- directed iterations.

• Heuristic called as overlap is continually reduced by ~ 3% .
• Essentially considers each cell, and attempts to compute an

HPWL minimizing range of locations into which the cell can be placed.

• The heuristic can re-introduce small amounts of overlap,

so careful attention is paid to overlap.

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Median improvement

BoxPlace- derived range for HPWL minimization Cell BoxPlace- derived range for HPWL minimization Cell Extended range for HPWL minimization and overlap

• For a cell compute median of its connected cells. Move cell

into this box (while trying to avoid re-introduction of overlap).

• If too much overlap re-introduced expand the box so at least

the cell attempts to get re-placed in the “right direction”.

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Force reorientation

• Basically make a hypothetical call to the median improvement

in order to get a direction that each cell wishes to follow.

• Combine direction with direction computed for cell spreading.
• Tends to help with HPWL minimization, but with some

impact on the required iterations for convergence. Use force re-orientation prior to solving each QP during force- directed placement.

Cell Spreading Force Median Improvement Force Resultant Force pointing in favor of spreading

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Multi-level clustering

• We also implemented multi-level clustering in order to have a

multi-level, force-directed placer.

• Several objectives in this flow:
• Help to improve runtime (clustering/ de-clustering takes

time, but smaller netlists at top levels).

• Help to improve quality of results (clustering keeps

connected cells together, thereby helping to circumvent cell ordering problems).

• Makes sense to use clustering; always used in top-down

partitioning and simulated annealing placers.

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Illustrated clustering flow

• Our clustering uses physical info + connectivity.
• It also (occasionally) performs de- and re-clustering.

Solve QP Physical Cluster Force-directed Placement

• riginal

circuit

• riginal

circuit Decluster Improvement Physical Recluster

• riginal

circuit Force-directed Placement Legalization+Detailed Improvement

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How good are things now?

Results on IBM-MS standard cell circuits (macro cells reduced to standard row height).

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How good are things now?

Results on IBM-MS 2002 mixed-size circuits (no pin offsets).

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How good are things now?

Results on IBM-MS 2004 mixed-size circuits (pin offsets).

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Commentary on algorithm progression

• Force-directed placement with median improvement and

clustering provided excellent quality of results compared to

• ther tools (e.g., Capo9.0 and Fengshui2.6).
• CPU somewhat worse…
• Thought about how we could be both better and faster…
• We did this by considering how the force-directed placer

worked at reducing overlap.

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Iteration count versus overlap

• Multi-level force-directed

placement exhibits three ranges

• f placement progression.
• Getting started.
• Spreading cells.
• Incremental overlap removal.

Large initial overlap; potential for more cell ordering problems and wasted runtime. Could potentially “warm-start” the force-directed method using an alternative placement method.

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Unification with partitioning

• Used partitioning at the top to help

generate better cell ordering and initial placements with less overlap.

• Question: how to properly stop

partitioning and switch to force- directed placement?

• Too much partitioning would

result in a partitioning-based placer.

• Too little partitioning wouldn’t

be useful either.

• The proper hand-off to force-

directed placement represents the unification.

Are Cells Adequately Spread? Bi-Partition Is Circuit Adequately Spread? N

• N
• (Exam

ine each block at the current m in-cut layer) Yes (All partitioning blocks at this layer have been exam ined) M in-Cut Partitioning Solve a QP (w ith cutlines) Yes Enter Force- Directed Placem ent

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Proper hand-off to force-directed placement

• Perform top-down bisection. Terminal

propagation done via QP at after each partitioning iteration.

• Assess overlap after each partitioning
• iteration. Terminate partitioning once

we have hit a reasonable overlap target.

• Need to give force-directed

placem ent the change to do som ething, and do not w ant to degenerate into a top-dow n partitioning-based placer.

Initial placement without partitioning… and with partitioning.

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Numerical results

Results on IBM-MS standard cell circuits (macro cells reduced to standard row height).

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How good are things now?

Results on IBM-MS 2002 mixed-size circuits (no pin offsets).

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How good are things now?

Results on IBM-MS 2002 mixed-size circuits (pin offsets).

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Conclusions

• Implementation of a force-directed placer requires attention

to detail:

• Force computation.
• Force scaling.
• Proper overlap assessment (useful for many reasons).
• Stability issues.
• To be competitive with today’s tools, force-directed placers

require enhancements (vs. basic Kraftwerk description):

• Median improvement (various forms).
• Multi-level clustering.
• Unification with partitioning.

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Current thoughts and work - Fixed objects

• Typical that some objects (e.g., IP) are pre-placed.
• Fixed objects become obstacles for force-directed placers.
• What modifications (if any) are required in order for force-

directed placers to properly account for obstacles?

• Other types of placement constraints.

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Future thoughts and work

• Try to significantly reduce CPU w/ o loss in quality.
• Better parameter selection (i.e., weighting of “this vs. that”

at each force-directed iteration) seems beneficial.

• Want to look at other objectives; e.g.,
• Routability, Timing, Physical Synthesis.

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Future thoughts and work – An aside

• Related to work on better parameter selection…
• Much of the analytic and force-directed methods are much

the same… (mFar, Fastplace, Aplace, etc.)

• All methods (more or less) have an explicit or implicit
• bjective function that is a weighted combination of wire

length and overlap.

• Regardless, the resulting optimization generally has two

search directions; one for wire length and one for overlap.

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Future thoughts and work – An aside

• Hence, the tricks in the methods are:
• How to weight wire length direction vs. overlap direction

to get good results fast.

• How far to step once relative weighting is done, and a

search direction for minimization is computed.

• People need to say more about this type of stuff.
• It is also possible that some methods (because of how
• verlap is handled and the resulting objective function) are

“more capable” at certain problems (e.g., moveable pads,

• bstacles, etc.)

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The end – Thanks!

• Happy to answer any questions…