A new Incremental Placement Algorithm and its application to CongestionAware Divisor Extraction

1
A new Incremental Placement Algorithm and its
application toCongestion-Aware Divisor Extraction

• Satrajit Chatterjee (satrajit_at_cs.berkeley.edu)
• Robert Brayton (brayton_at_eecs.berkeley.edu)

ICCAD / Nov 9th 2004
2
Outline

• Motivation
• General Approaches
• Layout-Aware Decomposition
• Example
• Algorithm
• Experimental Results
• Conclusions

3
The Problem Wires

• Timing Closure
• Wire delays are a significant fraction of the
  total circuit delay
• Routing Difficulty
• Designs are increasing in wiring complexity
• Wire not scaling as much as logic
• Tighter timing constraints on routes
• Because delays from wires are significant

4
The Problem Flow
5
Approaches to Placement-Aware Synthesis
2
1

• Maintaining point placement through-out the flow
• Pedram and Bhat 91
• Gosti et al. 01
• Kutzschebauch
• and Stok 02

Structural (graph) properties of netlists Kudva
et al. 02
Empirical study
Timing Closure
Congestion
6
Approaches to Placement-Aware Synthesis
2
1

• Maintaining point placement through-out the flow
• Pedram and Bhat 91
• Gosti et al. 01
• Kutzschebauch
• and Stok 02

Structural (graph) properties of netlists Kudva
et al. 02
Empirical study
This Work
Timing Closure
Focus Improving routability Approach Maintain
point-placement of netlist during decomposition
(All other logic optimization and mapping is not
layout aware)
7
Approaches to Placement-Aware Synthesis
2
1

• Maintaining point placement through-out the flow
• Pedram and Bhat 91
• Gosti et al. 01
• Kutzschebauch
• and Stok 02

Structural (graph) properties of netlists Kudva
et al. 02
Empirical study
We look at common divisor extraction since that
leads to large changes in the netlist
This Work
Timing Closure
Focus Improving routability Approach Maintain
point-placement of netlist during decomposition
(All other logic optimization and mapping is not
layout aware)
8
Conventional Extraction A Review
The Fast-extract algorithm

• Make a list of candidate divisors
• While good divisors exist
• Pick the divisor with highest literal-gain
• Perform division
• Update gains of all affected divisors

No looking at wires
literal-gain reduction in num of literals if
divisor is used
Example



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Layout-Aware Extraction

• Associate a point placement (x, y) with each node
  of the netlist

Use this to estimate wire-lengths using HPWL model
Use wire-length estimates to drive extraction
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An Example

• Same literal savings, but for wire-length A is
  better than B

A
B
Some divisors are better for wire-length than
others i.e. have better wire-gain than
others So in this example, wire-gain(t) gt
wire-gain(u)
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Cost Function

• gain(d) kwire-gain(d) (1 k)literal-gain(d)

conventional literal savings
new wire-length savings

• When k 0 it is the same as conventional
  extraction
• When k 1 it is completely wire-length driven

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Computing the wire-gain

• Wire-gain of a divisor depends on where it is
  placed

a
a
tx
tx
tab
c
bcy
c
bcy
tab
tc
tc
b
b
We place the new divisor optimally where the
increase in the total HPWL is minimum
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Division Illustrated
Expand
divisor d
Shrink
wire-gain(d) total wirelength before shrink
total wirelength after expand
Reduction in wirelength if d is used
To maximize wire-gain, we want to minimize
expansion of bounding boxes
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Problem Abstraction
Where to place P so that expansion is minimum?
P(x,y)
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Problem is Separable

• The semi-perimeter of each rectangle is the sum
  of the vertical component and the horizontal
  component.
• If the y-coordinate of P is changed, the
  horizontal components of cost doesnt change.

cost (x, y) costx(x) costy(y)
Can minimize costx(x) and costy(y) independently
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Some analysis ..

• .. that we shall skip and explain the algorithm
  directly with an example

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An example
y-coordinate
What is the y-coordinate that minimizes the
vertical expansion of the rectangles?
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The Algorithm
y-coordinate
1
2
3
4
5
(12, 12) is optimal region
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7
8
9
10
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Step 1 Sort edges in order of increasing
y-coordinate Step 2 Compute the median points
that is region of optimality
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The Algorithm
y-coordinate
1
2
3
4
5
Subtract bottom edges above median
-4
-4
12
6
7
8
9
Add top edges below median
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10
22
11
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Optimal Cost
Step 3 Total cost of expansion is just sum of
y-coordinates with suitable signs!
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Layout-aware Extraction Algorithm

• Place given netlist
• Make a list of candidate divisors
• For each candidate divisor d compute optimum
  location and
• gain(d) kwire-gain(d) (1
  k)literal-gain(d)
• While good divisors exist
• Pick the divisor with highest gain
• Perform division
• Update gains of all affected divisors
• Every 500 iterations re-place the current
  netlist and update gains

21
Experimental Flow
Technology Mapping for Area in SIS
Technology Independent
sweep eliminate simplify eliminate sweep eliminate
5 simplify fx / lx-0.5 / lx-1.0 sweep eliminate s
weep mfs w 22
Two different back-ends
Congestion Driven Placement
Wire-length Driven Placement
Global Routing
Measure Congestion
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Experimental Flow

Technology Mapping for Area in SIS

• Only layout-aware step i.e. layout-aware only
  locally
• Both earlier and later steps do not use any
  layout information
• So the final placement step regenerates the
  placement information from scratch

Technology Independent
sweep eliminate simplify eliminate sweep eliminate
5 simplify sweep eliminate sweep mfs w 22
Two different back-ends
Congestion Driven Placement
Wire-length Driven Placement
lx-0.5 means k 0.5
fx / lx-0.5 / lx-1.0
gain(d) kwire-gain(d) (1
k)literal-gain(d)
Global Routing
Measure Congestion
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Experimental Flow
Technology Mapping for Area in SIS
Technology Independent
sweep eliminate simplify eliminate sweep eliminate
5 simplify fx / lx-0.5 / lx-1.0 sweep eliminate s
weep mfs w 22
Two different back-ends
Congestion Driven Placement
Wire-length Driven Placement
Global Routing
Measure Congestion
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Results Congestion Driven Placement

• Measure of congestion is the percentage of
  over-capacity cells during routing
• Wirelength doesnt go down much since placement
  is congestion-driven
• Area goes down slightly (even assuming constant
  whitespace)

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Results Wire-length Driven Placement

• Measure of congestion is the maximum number of
  nets crossing a cell edge
• Wirelength doesnt go down much in lx-1.0 because
  of placer instability
• Area goes down slightly (even assuming constant
  whitespace)

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Variation in k
Vary k to get different netlists
gain(d) kwire-gain(d) (1
k)literal-gain(d)
Measure number of literals
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Variation in k

• Even in purely wire-length driven mode, literals
  does not increase much
• Usually literals decreases (compared to purely
  literal-driven approach) but not always
• Doesnt look like it is due to scheduling
  randomization

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Variation in k
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Congestion
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Number of literals
Two netlists with approximately same size, but
one has about 40 less congestion than the other
Connection with structural metrics like adhesion?
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Conclusions

• Layout-aware decomposition at the technology
  independent level can significantly reduce
  congestion
• About 20 reduction on average is seen for
  percentage of over capacity global router cells
• No area penalty (usually even fewer literals)
• Only one step in the technology independent flow
  was layout-aware yet it worked
• Need to explore connection with structural metrics

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The End
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Additional Slides
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COG and optimal location can be very different
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Development of the Algorithm
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Some Results

• Separability
• The semi-perimeter of each rectangle is the sum
  of the vertical component and the horizontal
  component.
• If the y-coordinate of P is changed, the
  horizontal components of cost doesnt change.

cost (x, y) costx(x) costy(y)
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Some Results

• Convexity of costx(x)
• Sum of piece-wise linear convex functions
• Contiguous optimum region
• Optima occur on projections of the edges

Costx

x
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Some Results

• Claim Number of rectangles entirely on one side
  of the optimum region equals number of rectangles
  entirely on the other side.
• Proof Suppose not. Then by moving from the
  optimum region towards the side with more
  rectangles we can reduce the cost. Contradiction.

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Some Results

• Claim If Yy1, .. y2m is the ordered list of
  projections then (ym, ym1) are the y-coordinates
  of the endpoints of the optimum region.
• Proof
• Every rectangle lies either entirely above,
  entirely below or straddles the optimum region.
• Number above Number below.
• Every straddler contributes one coordinate to
  above and below

Looking at costy(y)
y1
y1
..
y6
(This proof assumes that all the projected points
are distinct, but can be easily extended to the
general case.)
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Algorithm

• Determining optimum region easy
• Compute the median of the projections of the
  edges
• How to find cost of optimum region?

Inside the optimum region Nabove Nbelow, so
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Algorithm Y-Optimum Region
Input List L of tuples (y, t) Projection of
horizontal edges indicating begin whether
top or bottom edge i 1, cost 0 m
length(L) / 2 for each (y, t) in L in order
do if (i m and t bottom) then cost
cost y else if (i m 1 and t top)
then cost cost y end if if (i m)
then yopt y else if (i m 1) then
yopt y end if i i 1 end for return
((yopt, yopt), cost) end