Filtering Random Networks to Synthesize Interconnection Networks

0 downloads 0 Views 356KB Size Report
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   

 

      "