SLIP 2000April 8, 2000 --1--

Efficient Representation of Efficient Representation of Interconnection Length Distributions Interconnection Length Distributions

Using Generating PolynomialsUsing Generating Polynomials

D. Stroobandt (Ghent University)

H. Van Marck (Flanders Language Valley)

Supported by an IUAP research program on optical computing of the

Belgian Government and the Fund for Scientific Research, Flanders

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Outline

• Enumerating interconnection length distributions

• Advantages of generating polynomials• Construction of generating polynomials• Extraction of the distributions• Examples• Conclusions

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Enumerating Interconnection Length Distributions

• Distributions contain two parts:site density function and probability distribution

all possibilities

requires enumerationprobability of occurrence

shorter wires more probable

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Enumerating Interconnection Length Distributions (cont.)

• Simple Manhattan grids: not so difficult– just start counting– more clever: use convolution

• But what with...?– anisotropic grids– partial grids

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Generating Polynomials

• Site function (discrete distribution f(l)) describes, for each length l, the number of pairs between all cells of a set A and a set B, a distance l apart (enumeration problem)

• Two ways of reducing calculation effort:– using generating polynomials– using symmetry in the topology of the architecture

• Generating polynomial: moment-generating polynomial function of f(l) (Z-transform)

0

)()(l

lxlfxVmaxl

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Advantages of Generating Polynomials

• Efficient representation– allows easy switching to path-based enumeration

– compact representation

as rational function– example

l(p)=8

p

n

A=B

p

plxxV )()(

2

21

1

0

1

0

)1(

22)(

)(

x

nxnxxxV

xxV

n

n

i

n

j

ji

otherwise0

)0()(2

)0(

)( nlln

ln

lf

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Advantages (cont.)

• Easy to find relevant properties– total number of paths

– average length (also higher order moments)

• Easy construction of complex polynomials

n

A=B)1()(max

0

Vlfl

l

1

0

0

)(

1)(

)(

)(

max

max

x

l

l

l

l

xVdx

xdV

lf

llf

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Construction of Polynomials

• Composition (adding and subtracting polynomials)

n

B

A n

B

A

__

n

B

A||

|||

A

Bn X

2n

1

1

21n

i

in xx

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Construction of Polynomials (cont.)

• Convolution (multiplication of polynomials)– composing paths from “base” paths

*BA

nn

CA

nn

||

D

nn

C * B

nn

D

21

02

2

1

1

)1(

)1(

x

xxx

x

xx nn

i

in

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Extraction of Distributions

• Construction of polynomials much easier than construction of distributions but… how to extract distributions from polynomials?

• Much simpler than general Z-transform

• Theorem

• Quotient term important, remainder vanishes

• Note: summation bound to be chosen between n-1 and n-i+1 without effect on result

in

li

il

i

n

x

xOx

i

ln

x

x

0

1

)1(

)(

1

1

)1(

1

1

)()!1(

1

1

1 i

j

jlnii

ln

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Extraction of Distributions (cont.)

• Simple substitution of terms by summation of combinatorial functions (with few factors)

• The different ranges of the distribution naturally follow from this!

in

li

il

i

n

x

xOx

i

ln

x

x

0

1

)1(

)(

1

1

)1(

ibl

jk

jj

j

i

lbalf

00 1

1)(

ib

l

ljk

jj

j

xi

lbaxV

00 1

1)(i

k

j

bj

x

xaxV

j

)1()( 0

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A=B

A=B

Examples

• Manhattan grid– convolution of x, y parts– subtract– divide by 2

– extraction = substituting

n

02xn

4

312322

)1(

)(2422)(

x

xOnxxnxxxV

nnnn

110220

3302220

14

112

14

124

14

132

14

1222)(

nlnl

nlnl

lnn

ln

lnn

lnlf

otherwise0

)2(

)0(

)( 3)12)(2)(12(

3)166( 22

nln

nl

lf lnlnln

lnlnl

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B

Examples (cont.)

• Complicated architectures

Ak

n

|| 2 X

C

n

Bk

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Examples (cont.)

C

n

Bk

k

||

C

n

E

C

n

F

+

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C

n

Ek

C

n

F

+

Examples (cont.)

||

n

n

k

*

n*

kx

*k

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Examples (cont.)

||

C

n

Ek

C

n

F

+

*k

1

k+1x

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Examples (cont.)

• Resulting generating polynomial:

• Extraction by simple substitution and calculation of the combinatorial functions:

)34()1(

)1(2)( 1212

4

2

xnxxnxxx

xxxV nnnn

kn

otherwise0

)143()24)(14)(4(

)132()24)(14)(4(

)23)(3()13()13(2

)12())(1)(2(

)2)((5)1(3)1()1())(1)(2(

)10(0

)(

31

31

32

knlknlknlknlkn

knlknlknlknlkn

lknlknlknnlkn

knlknklklkl

lknlknlknnlknknlkklklkl

kl

lf

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Conclusions

• Generating polynomials make enumeration easier– more efficient representation (1 equation, not 5)– easy to obtain characteristic parameters– construction facilitated by using symmetry

(composition, convolution easy with polynomials)– extraction by substitutions of terms, can be

automated by symbolic calculator tools!

• Same technique can be used for calculating cell-to-I/O-pad lengths

• Enumeration viable for complex architectures