Intrinsic Shortest Path Length: A New, Accurate A Priori Wirelength Estimator Andrew B. KahngSherief...

Intrinsic Shortest Path Length: A New, Accurate A Priori Wirelength Estimator

Andrew B. Kahng Sherief Reda [email protected] [email protected]

VLSI CAD LaboratoryUniversity of CA, San Diego

http://vlsicad.ucsd.edu/~sreda

• Previous work and motivation

• Intrinsic Shortest Path Length (ISPL) definition

• Validation of ISPL as wirelength estimator

• Practical Applications:

• A Priori Total wirelength estimation

• A Priori Global interconnect prediction

• Relationship to Rent parameter

Outline

Definition and Applications

• A priori wirelength estimation is the process of estimating and predicting the wirelength characteristics of VLSI netlists without knowledge of the netlist placement or floorplanning.

Applications that benefit from a priori wirelength estimation:

• Physical driven synthesis Faster timing convergence

• Early system planning

• Determining amount of necessary whitespace

• VLSI netlist characterization/reverse engineering/creation

Previous Work

Previous approaches:

• Correlators: • If some measure e correlates with net length l then e can be used

in relevant applications, e.g., clustering. Typically no analytical modeling between l and e.

• Examples: mutual contraction and edge separability.

• Average wirelength estimators: • Rent parameter-based.• Predict aggregate wirelength characteristics, e.g., wirelength

distribution and total wirelength.

Motivation

Wanted:

• Estimator has intuitive physical meaning

• Handles hypergraphs transparently

• Individual net length estimator: If l1 > l2 then e1 > e2

• Analytical modeling between li and ei, e.g., li = f(ei)

• Estimator and wirelength have similar distributions

• Total wirelength estimation

• Practical runtime for calculation

A Motivating Observation

a

b

• Nodes a and b are directly connected by an edge.

• Does this mean a and b will be placed spatially close in a good placement?

Input Netlist

Observation:

Unlikely. Despite edge {a, b}, a and b are “structurally” far from each other

Intrinsic Shortest Path Length (ISPL)

a

bInput Netlist

• “structural proximity” shortest path

• shortest path between nodes a and b that does not include {a, b}.

Will edge {a, b} be short?

• analyze the “structural proximity” of a and b

• Example: ISPL of {a, b} = 8. {a, b} and its ISP form a cycle

• BUT: Netlists are hypergraphs a transparent mechanism is needed

To estimate the Intrinsic Shortest Path Length ISPL of edge {a, b} :

delete {a, b} and calculate the shortest path length (number of edges) between a and b

ISPL in Hypergraphs• Set the “distance” or weight of a k-pin hyperedge by k/2

u

v

ISPL of {u, v} = 1+1.5+1 = 3.5

a b

c

h

• The ISPL of a k-pin hyperedge, h = {a, b, c}, is calculated as follows:

1. Delete h

2. Calculate the ISPL for every pair of nodes that belong to h: {a, b}, {b, c}, and {a, c}

3. The ISPL of h is maximum among all values calculated in Step 2

Runtime requirement: )log( 2 nmO n: number of nodes, m number of edges








Outline

Validation of ISPL as Wirelength Estimator

To validate our observation:

1. Correlation between the placed net length and net ISPL?

2. Correlation between the effect of net pin count on average net length and average net ISPL?

3. Correlation between the average/total netlist wirelength and the average/total ISPL over a range of benchmarks?

4. Given two individual nets of some netlist, can we predict which individual net will be placed with greater wirelength?

5. Relationship between the distribution, or profile, of ISPL and the wirelength distribution?

Validation: 1. ISPL and Net Length

Objective: validation of the relationship between ISPL and net length:

Given a netlist (ibm01):

1. Calculate the ISPL of every hyperedge

2. Place the netlist using some placer (Dragon)

3. Plot ISPL versus Half-Perimeter Wirelength (HPWL) of every net

0200400600800

10001200140016001800

0 5 10 15 20 25

ISPL

HP

WL

100 buckets

0

200

400

600

800

1000

1200

1400

0 5 10 15 20

ISPL

HP

WL

30 buckets

As ISPL increases, HPWL increases Correlation coefficient 0.91

0

2000

4000

6000

8000

10000

12000

0 5 10 15 20 25 30 35

ISPL

HP

WL

• Reduce data clutter by dividing data into buckets and averaging the results within each bucket

Validation: 1. ISPL and Net Length

Circuit ISPL MC ES

ibm10 0.922 0.724 0.975

ibm11 0.955 0.805 0.551

ibm12 0.900 0.655 0.329

ibm13 0.923 0.495 0.901

ibm14 0.747 0.866 0.823

ibm15 0.946 0.777 0.368

ibm16 0.947 0.828 0.01

ibm17 0.941 0.645 0.575

ibm18 0.938 0.836 0.487

Average 0.914 0.743 0.579

|correlation coefficients|MC is mutual connectivity

ES is edge separability

Calculate correlation coefficients between ISPL and wirelength

For comparison, calculate correlation coefficient between:

•Mutual Contraction (UCSB) and HPWL

•Edge Separability (UCLA) and HPWL

Validation: 2. Effect of Pin Count

0

2

4

6

8

10

12

14

1 2 3 4 5 6 7 8 9

Pin Count (k)

Ave

rage

ISP

L

0

200

400

600

800

1000

1200

1400

1 2 3 4 5 6 7 8 9

Pin Count (k)

Ave

rage

HP

WL

Average length of k-pin nets

Average ISPL of k-pin nets

Objective: Test the effect of pincount on both average wirelength and ISPL

• For every k (2…) on ibm01:

• Calculate the average ISPL of all k-pin net

• Run a placer and calculate the average placed wirelength of all k-pin nets

Correlation coefficient of 0.95 between average HPWL and average ISPL (typical result)

Validation: 3. Average ISPL and Total Wirelength

Objective: Is the average ISPL correlated with the total wirelength?

Circuit Rentparameter

Wirelength Av. ISPL

GNL50 0.50 53788 6.924

GNL55 0.55 56632 6.991

GNL60 0.60 65948 7.244

GNL65 0.65 73502 7.399

GNL70 0.70 77365 7.515

• Synthesize netlists (10k nodes/nets) with varying Rent parameter with GNL• A higher rent parameter more global communication larger wirelength• Calculate the average ISPL of each netlist• Place the netlists using mPL and measure the HPWL

Perfect correlation between average ISPL and total wirelength

Validation: 4. Individual Net Length Prediction

Objective: Given two arbitrary nets i and j with the same number of pins, can we a priori decide which net will be longer?

Predictor

Oracle

ij

Success?

{, <}

{, <}

Yes/No

• Dragon gives the best placement and will be used as an Oracle• Performance of the Predictor:

• Lower bound is 50%• Upper bound is the performance of any other placer

• What is performance if we use ISPL for the predictor?

Validation: 4. Individual Net Length Prediction

bench Capo9 mPL4 FS2.6 MC ISPL

ibm10 71.91 71.84 72.07 52.71 61.88

ibm11 70.57 70.75 71.34 52.20 59.77

ibm12 71.70 72.71 72.72 51.55 60.86

ibm13 71.15 71.76 71.56 52.44 60.22

ibm14 69.45 69.53 70.34 52.25 59.48

ibm15 71.56 72.31 72.23 52.26 59.96

ibm16 70.03 70.69 70.84 52.71 60.65

ibm17 73.08 73.5 73.06 51.41 61.14

ibm18 69.72 70.41 70.43 52.45 60.89

Average 70.49 70.84 71.18 52.44 59.67

MC: Mutual ContractionISPL: Intrinsic Shortest Path Length

The success of prediction in percentage

Validation: 5. ISPL and Net Length Distribution

00.20.40.60.8

11.21.41.61.8

2

1 1397 2793 4189 5585 6981 8377 9773 11169 12565 13961 15357 16753 18149

Net Number

Nor

mal

ized

HP

WL

and

ISP

L• Sort all nets according to their ISPL and their HPWL• Plot all sorted HPWL normalized to the maximum HPWL value• Plot all sorted ISPL normalized to the maximum ISPL value ISPL and HPWL have roughly similar profiles

HPWL ISPL

• Objective: Examine the relationship between ISPL and HPWL profiles








Outline

Applications: 1. A Priori Wirelength Total Estimation

• Devise an analytical model between ISPL and HPWL

• Using empirical data, we find an exponential relationship between ISPL and wirelength

sgkk

keasw )(

)(swk

symbol Meaning

k Number of pins on a net

s ISPL

ak constant of netlist and k

gk constant of netlist and k

HPWL of k-pin net with ISPL=s

0

5

10

15

20

25

30

35

40

0 2 4 6 8 10 12 14 16 18

ISPLH

PWL

actu

al a

nd

fittin

g

Exponential fittingActual results

Applications: 1. A Priori Total Wirelength Estimation

k ak gk

2 1.56 0.12

… … …

20 … …

• How to determine ak and gk?

• Ideal modeling (not a priori): based on the netlist characteristics from the placement (only useful for model validation and calibration)

• Static modeling (a priori): fixed values for all netlists based on values typically encountered

sgkk

keasw )(


sgeasw22

2

22

2 )(

sgmm m

easw 2

2 2)(

sgaswy 22

22

22

2 ln)(ln sgeasw12

2

12

1 )( sgaswy 12

12

12

1 ln)(ln

sgaswy mmmm222 ln)(ln

An estimate function that minimizes the total square error

dscy sgdscy eaeeesw 222 )(

Objective: Given m ideal models, how to calculate an approximate static model

sgeasw 222 )( ?

m ideal exponential fits (from typical netlists)

linearize

calculate exp model


circuit static ideal circuit static ideal

ibm09 18.04% -3.52% ibm14 -15.09% 0.64%

ibm10 7.36% 9.39% ibm15 -26.23% 3.14%

ibm11 -2.82% -1.45% ibm16 -15.55% 0.44%

ibm12 -13.55% -1.56% ibm17 -30.94% -0.34%

ibm13 -8.26% 0.22% ibm18 -10.65% -1.11%

• Calculate the total wirelength of the IBM (version 1) benchmarks (unit size cells) using ideal model

• Calculate typical values and use it for a priori static modeling.

On the average, ideal modeling is 3.61% accurate compared to actual HPWL. Static modeling is 16.60% accurate

Applications: 2. A Priori Global Interconnect Prediction

Global interconnects hurt performance and are typically buffered

• Objective: Can we a priori decide which nets are going to be “long” before placement?

Definition: a net is global (long) if it is in the top 5% of the longest nets in the final placement

Given a netlist: 1. Calculate the ISPL of all nets 2. Sort all nets based on their ISPL 3. Plot net count vs ISPL

0

20

40

60

80

100

120

0 5 10 15 20 25 30 35

ISPL

Inte

rcon

nect

If we declare all nets with ISPL 15 then we declare 10% of all nets global, actually capturing the 60% of the future global interconnects All nets Global nets








Outline

Relationship to Rent Parameter

We develop a characterization, Range Parameter, of VLSI netlists

|)(|

)()( )(

uincident

eISPLuR uincidente

Definition 1: The range of a node u is the average ISPL of all nets incident to it.

Definition 2: The Range of a netlist is the average range of all nodes V.

||

)(

V

uRR Vu

The larger a node’s range , the more wirelength it needs to communicate with its neighbor

A large Range parameter predicts that a netlist would require a large amount of global communication.

Relationship to Rent Parameter

Intuitive connection to Rent parameter: a netlist with large Rent parameter requires more global communication in any good placement

0

2

4

6

8

10

12

14

Circuit

Ran

ge a

nd R

ent

Range Rent Correlation coefficient of 0.701

Rent Parameter Range Parameter

• Calculated in top-down fashion • Calculated in bottom-up fashion

• Useful for complete netlist characterization

• Useful for complete netlist characterization

• Useless for individual net prediction

• Useful for individual net prediction

• Unstable value • Stable value (same topology)

bench Rent Range bench Rent Range

ibm09 0.24 0.27 ibm14 1.00 0.61

ibm10 0.34 0.77 ibm15 1.05 1.18

ibm11 0.63 0.41 ibm16 1.25 0.56

ibm12 0.60 0.86 ibm17 1.85 1.92

ibm13 0.50 0.54 ibm18 1.51 0.89

Runtime normalized with respect to FengShui

• Hypergraph to graph transformation

• works on graphs or hypergraphs

Conclusions

• Developed the new concept of Intrinsic Shortest Path Length (ISPL)

• Demonstrated strong correlation between ISPL and HPWL

• Used it for individual net length predictor

• Correlated average ISPL with total wirelength

• Studied the relationship between ISPL and HPWL distributions

• Developed a characterization to VLSI netlists and studied its relation to Rent parameter

• Used ISPL for two practical applications:Total wirelength estimationGlobal interconnect prediction

Future Work

• Runtime improvement

• Studying the effect of different net weights on ISPL performance

• Better wirelength models

• Synthetic benchmark generation based on ISPL

• Analytical relationship between Range and Rent parameters

• Fixed blocks/white space effects

• Deducing wirelength distribution, pin-effect count from the analytical models

• Estimating RSMT by using weighting coefficients

Thank you

Intrinsic Shortest Path Length: A New, Accurate A Priori Wirelength Estimator Andrew B. KahngSherief...

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