Interconnection Networks with Multiple Objectives ... Projects Agency' the Air Force Research Laboratory' or the US Gov- ernment. ... gestion. Two measures used to describe the node-pair dis- .... was well under 5% for graph sizes greater than 32. It is also ..... miniscule fraction of the multitude of interconnection net-.
Æ !
! ! "
!
#
!
Ì
!
" #$% #&% #'% ( Æ ) ! " "" # $ %& ' % ($ ) '*+,*- $ Æ .
&/ ' % ($ 0 )
)
* Æ
( (
* #+% ,
)
) - *
) ) ) ) )
. / 0 1 /2 #3% - #4% #5%
#6% #7% #$8% #$$%
$ *
9 :
) * ( ) Æ ;<
Æ )
)
#$&%
=
#$'% *
0 >
9?:
9:
Æ
*
#$+% = , @ ,
Æ ,
#$3%
Æ = ) @ @
/ #$4% 9 :
?9 : 9 :
* * 9 :
A B =
) ) . *
CD -
)
) 0
@ )
) 9 3E: 0
!
" # $ %
! !
D 0 ) F
) =
@ , )
) , @ ,
!
!
/ .
A 9 : )
A H &8 H + 7636583&& #$5% @
9 : )
= ) #$6% "2I¾ #$7% #&8% 9 : Æ H 9 : @ #&$% #&&%
/ @ = #$6%
9 :
= H $ &
, -
A ! "
388( * &34(" - $ $888 * J
&8+6
) 3E * '&
=
J KL #&'% #&+% 73E 0.030
Average Runtime (sec)
* D Æ @ G
degree 6
0.025
degree 5
0.020 degree 4
0.015 0.010
degree 3
0.005 0
0
50
100 150 200 Number of nodes
250
300
'
& !
) ) ) ) 0
) )
" )
) )
) ) )
)
) )
@
)
)
) - ,
)
)
. - ) =
)
) ) - & )
) Æ
) ) )
)
)
) J ) ) Output from graph generation phase
, ' + = ' 0 ! #&3%
9: #&4% ' #&5% (
#&6% 9(@: ' #&7% +
+ #'8%
. ' &
*
#'$% = #'&%
* F ) * 6 &34 ' 4 - * $888
- ' *
) 10
degree 3
9 8 Diameter ( ∆ )
"
degree 4
7
degree 5
6 5 degree 6
4 3 2
E1 C1
0 E2
E1
Filter 2
E2 C2 U
E3 C3 E C
100
150 200 Number of nodes
250
300
' " C=C1 C2 C 3
Filter 3
50
E3
U
Filter 1
Evaluation part Checking part
' + 01 2
- + '
) - - +
(@ * (@ Æ
. (@ ) * 9 $: +
J *
0 " $ 3
10
14
1.0 Min. Diameter
0.8
8 Diameter ( ∆ )
10 8 6
Shuffle Exchange Network Cube Connected Cycles Chordal Ring of degree 3 Moebius Graphs Random Networks of degree 3 Multi Tree Structures
4 2 0 0
20
40
60
80 100 120 Number of Nodes
140
160
180
' - 4 "
*
* 9 :
9 ?: $ #'+% J 9 ?: ,
, )
) , - 3
)
9 - ': ;<
)
( &
( Æ
- - Æ D F
D
J @ I / 9/: F #'3% 5 67 $ " 6 $ $ $ 89 $ $ 56 7 $
6
Min. Diameter count
0.6
4
0.4
2
0.2
0
0
20
40
60 80 Number of nodes
100
120
Fraction of the generated graphs
12 Diameter ( ∆ )
#
0 140
' : '1
"
, ,
JM@ 9J M @ :
/ / Æ
#'4% @
. (
J> #'5% D ) , ) )
>
) " ) ) ! H 9 : 9 : ! H 9 : 9 : - 0 9 : 9 : 9 : 9 :
$
;(
( ( N @ --
¾ 9: 9: H
¾ 9: !
!
9: 9:
, ) #'4% #'6% @
) H ½ O ¾ O ¿ O O
=
0 ) JM@
) = - =
$ "
* = ( , )
)
; < ) ) F $888 )
"
-
/
' +
@ * ,
@ J>
#'7% =
4+ 6 6
#+8% - #+$% # & &
- ?9 : 9 : 9 : * 9 :
* 4+ ' +
* )
) - 4 ' + + ' - 5 ' +
%
;(
? $ 4
30
= . 9JD=: #+&% . *
9:
#+'% ) #++%
)
;) < * A #+3% ) * + , -
- #+4% &34
GP@
.
.
P
- 9 :
88 8& * 4+ ' / ; @A " 3 $888
( " "
'4 9 - $8: 6 6 )
) 4 $& J
+
, , -
140
132 meshtorus random
120 100 Diameter
#+4% .
* =
A
- 7 )
=
@
- = ) * =
99
80 60 40 2118
20 0
Inter−module network
'
2
4
1614
1413
1212
6 8 10 Number of module pins
1211
12
' $
Pins
... ..... .
... ..... .
Intra−module network
... ..... .
...
... ..... .
Modules
' , 0 + "
- ) + +
= 4+ * $8&+ D )
J )
+ + * $4 +
- $8 * , -
& + &$
D , * * - $&6 * 6 - $$ + ) '& '&
'& && & '& $ '& ) &
$ ) @ F ) * #+5%
- #'+%
35
32
meshtorus random
30 26 24
Diameter
25
22
20
22 19
17
15
21 18
17
14
10
1615
14 8
8
5 0
1
4
8 16 32 64 Number of modules
128
256
1024
' $ # * ,
0 - @ ) .
) ,
) , ) , @ * A $888 Æ = ) , P
Q R 89 8+9 89 8-9 8:9 8*9
'
% ' " D 0 @& A +, *E= ,> B 0 C ? 0 F @ ! "A -: :*E*- '$ ,,* 0 ; " ; F @
"A > :::E:*: ,>, & G %H @? A !" : =E* ,* ; ;$ # $ & ( ,>:
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
&/A !MM / M
8*9 ; D " % ( @ $ A " !" ,,- *>+E*>: 8=9 C ; 0. @ $$ " B& A " ./ " ,=, ->E->, 8>9 0 I D
" " 0 & % & 1 , D .$" " F & ( ++
() ' & ,>- $
D
"