SLIP 2000 April 8, 2000 --1-- Efficient Representation of Efficient Representation of Interconnection Length Interconnection Length Distributions Using Generating Distributions Using Generating Polynomials 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
18
Embed
Efficient Representation of Interconnection Length Distributions Using Generating Polynomials
Efficient Representation of Interconnection Length Distributions Using Generating Polynomials. D. Stroobandt (Ghent University) H. Van Marck (Flanders Language Valley) - PowerPoint PPT Presentation
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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
• Simple Manhattan grids: not so difficult– just start counting– more clever: use convolution
• But what with...?– anisotropic grids– partial grids
SLIP 2000April 8, 2000 --5--
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
SLIP 2000April 8, 2000 --6--
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
SLIP 2000April 8, 2000 --7--
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
SLIP 2000April 8, 2000 --8--
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
SLIP 2000April 8, 2000 --9--
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
SLIP 2000April 8, 2000 --10--
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
SLIP 2000April 8, 2000 --11--
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
SLIP 2000April 8, 2000 --12--
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
SLIP 2000April 8, 2000 --13--
B
Examples (cont.)
• Complicated architectures
Ak
n
|| 2 X
C
n
Bk
SLIP 2000April 8, 2000 --14--
Examples (cont.)
C
n
Bk
k
||
C
n
E
C
n
F
+
SLIP 2000April 8, 2000 --15--
C
n
Ek
C
n
F
+
Examples (cont.)
||
n
n
k
*
n*
kx
*k
SLIP 2000April 8, 2000 --16--
Examples (cont.)
||
C
n
Ek
C
n
F
+
*k
1
k+1x
SLIP 2000April 8, 2000 --17--
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
SLIP 2000April 8, 2000 --18--
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