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J# 8 $D 0H D 0޽h ? 33qi___PPT10I.lr+LD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DW' =4@BB BB%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*DD' =1:Bvisible*o3>+B#style.visibility<*D%(D' =-o6Bdissolve*<3<*DD' =%(D' =%(DW' =4@BB BB%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<*DD' =1:Bvisible*o3>+B#style.visibility<*D%(D' =-o6Bdissolve*<3<*DDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* D%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* D%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* D%(+8+0+ D6 +I   H<(  Hr H S Н<qY  < ` H c $A ??B "l P  Hw%w,$D 0` H 0P  H 0(!6Z&  f4Maximum number of links to the network on any switch5 25h H c $A ??  H H 0޽h ? 33E=___PPT10.n`+5yD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-o6Bdissolve*<3<*H+i  0 L(  Lr L S ع<qY  <  L S h)6zY <$ 0 < l @w L@w,$D 0 L 0 +6:@w T"Minimise the maximal switch degree# 2# h L c $A ??@h h L c $A ??}    L c $A  ?? J 8  $D 0  L c $A  ?? J3 8  $D 0H L 0޽h ? 33 ___PPT10.px+53D}' = @B D8' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L6%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-o6Bdissolve*<3<*LD' =%(D' =%(DW' =4@BB BB%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<* LD' =1:Bvisible*o3>+B#style.visibility<* L%(D' =-o6Bdissolve*<3<* LD' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* L%(D' =-o6Bdissolve*<3<* L+8+0+L6 +O  `X` X(  Xr X S p<6qY  6  X S l=6MY<$ 0 6  X 0>6 2Y ,$ 0 bOutput: Number of switches,* ZZ  X c $A  ??v v B8  $D 0 X c $A  ??C i k 8  $D 0l c Y   Xc Y ,$D  0 X 0E6 c Y  ]-Number of links on each switch to the network.Z.h X c $A ??F   l S Y   XS Y ,$D  0h  X c $A ?? S C    X 0I6 S Y  GSize of biggest switch,Z  X <L6f  ,$D  0 v@It is always possible to construct a network with these switchesAA H X 0޽h ? 33___PPT10o.r7iv+D' = @B DV' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(Df' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X6%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(Df' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* X%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* X%(D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* X%(D' =-o6Bdissolve*<3<* X++0+X6 ++0+X6 ++0+ X6 +v;  ,,p\c\,(  \l 3 " c\"3 ,$D 0z `\ s *fC"?3 "R a\ <d6' A Complete Partr \ S ,h6qY  6 l j` P\j`,$D 0N jp@ F  \ X!0 2  \ Zvd @" r `  \ 0z p@ 6 TB  \B c $Dz TB  \B c $D z ZB  \ s *D 6  ZB \ s *D 6 2 \ Zvd @" jo  2 \ Zvd @"  0 F 2 \ Zvd @" s F T jp@ F  \# yj@2 \ Zvd @" r ` \ 0z p@ 6 TB \B c $Dz TB \B c $D z ZB \ s *D 6  ZB \ s *D 6 2 \ Zvd @" jo  2 \ Zvd @"  0 F 2 \ Zvd @" s F T jp@ F  \# J `2 \ Zvd @" r ` \ 0z p@ 6 TB  \B c $Dz TB !\B c $D z ZB "\ s *D 6  ZB #\ s *D 6 2 $\ Zvd @" jo  2 %\ Zvd @"  0 F 2 &\ Zvd @" s F T jp@ F  '\# e 2 (\ Zvd @" r ` )\ 0z p@ 6 TB *\B c $Dz TB +\B c $D z ZB ,\ s *D 6  ZB -\ s *D 6 2 .\ Zvd @" jo  2 /\ Zvd @"  0 F 2 0\ Zvd @" s F  J\ 0Hu6{t 31 2 K\ 0xy6C <  32 2 L\ 0Dx6-&  33 2 M\ 046" 34 2 l  < Q\< ,$D 0@   =\ F 2 2\  fvd @"   f 3\ 6f r `B 4\B 0B.-D E `B 5\B 0B.-D;] ; `B 6\ 0xD  `B 7\ 0xD  2 8\  fvd @"  "I2 9\  fvd @" h S2 :\  fvd @" ;  ZB ;\B s *\ZD _2 <\  `vd @" k T   >\#  <2 ?\  `vd @"   ` @\ 0f r ZB A\B s *B.-D E ZB B\B s *B.-D;] ; ZB C\ s *xD  ZB D\ s *xD  2 E\  `vd @"  "I2 F\  `vd @" h S2 G\  `vd @" ;  ZB H\B s *\ZD _2 I\  `vd @" k  N\ 0t6ZS" 35 2 O\ 0,6Z S  36 2 l 0/+  _\0/+ ,$D 0TB R\ c $D0i  TB S\ c $D TB T\ c $DG`TB U\ c $D TB V\ c $DiT TB W\ c $D TB X\ c $D;'TB Y\ c $DzM TB Z\ c $DZ TB [\ c $D+ + TB \\B c $DD  TB ]\ c $D/}TB ^\ c $D  b\ c $A ??(/8 $D 0H \ 0޽h ? 33___PPT10k.zǶ+@D?' = @B D ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*b\%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*P\%(D' =-o6Bdissolve*<3<*P\D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*Q\%(D' =-o6Bdissolve*<3<*Q\D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*_\%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*c\%(D' =-o6Bdissolve*<3<*c\+  jbt(  tr t S ܝ6qY  6  t S ؞6Y{<$ 0 6  t 0H6 Y ,$ 0 /Exponentially growing number of isomorphic classes for network sizes immediately following a complete network Example: is complete, and has 22 isomorphic classes&nn t 0X6  Y,$ 0 MAlways a complete part of the network Maximum shortest path is of length two MN t c $A ??K8 $D 0 t c $A ??S  8 $@ 0 t c $A ??c W 8 $D 0H t 0޽h ? 33 ___PPT10.8+TrD' = @B D' = @BA?%,( < +O%,( < +D' =%(Du' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t%%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*tn%(DY' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*tn%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t&%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*t&M%(++0+t6 ++0+t6 ++0+t6 +Hr  sk.:d{(  dr d s *fC"?6 dr d S 06qY  6  d Z6v @" g` 11 d Zp6v @"   12 d ZT6v @"  13 d Z6fv @"  z  15  d Z6fv @"   16  d Z<6fv @"  *#  54  d Z6fv @"  ]V  17` d c $A '??i ' d <6v E] A Complete partF d S h` h^b d 6GH)IF d@ S  F d S  F d S 5  F d S 5  F d@ S 5 F d S  F d@ S   F d@ S  F  d S   F !d S D # D ^b "d 6ZGԻHWI  F $d S  z D F %d S  z ] F &d S  ]D  'd <iq,$ 0 ,Saturation point = 63.5% (d <ti ,$ 0 63.5% )d c $o  ,$@ 0 *d c $oh ,$@  0b ,d <GH)Io,$@  0 -d <$iq,$ 0 ,Saturation point = 63.5% .d 0ȩi Z,$ 0 563.5% 2b /d <ZG3HWIo  ,$@ 0 0d c $o z D ,$@ 0 1d c $oh` h,$D 0 2d c $o5  ,$D  0 3d@ c $o5  ,$@  0 4d c $o  ,$@! 0 5d c $oh` ,$@" 0 6d c $o ,$@ 0 7d c $o5  ,$D# 0 9d <4iq,$$ 0 ,Saturation point = 57.2% :d 0i q,$% 0 557.2% 2H d 0޽h ?ddddddd ddddd d dd d ddd ddd ddd ddddd dd d d d!d d d"dd d$d d d%d" d d&d$ dd)d& dd*d*dd,d, d d/d.d d0d0dd1d2d d2d4dd3d6 dd4d8 dd5d: dd6d< dd7d 33}UuU___PPT10UU.`f+p^BDR' = @B DDR' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'d%(D' =-o6Bdissolve*<3<*'dD' =%(Dm' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(d%(D>' =A@BBBB0B%()?)?D' =.M7 BBBBB_M -1.66667E-6 -3.35496E-6 L -0.20295 -0.74652 *3>*B ppt_xB ppt_y=@0BBAApBBнB<<*(dD6' =A@BB BB0B%(D' =-g6B fade*<3<*'dD' =1:Bhidden*o3>+B#style.visibility<*'d%(DH ' =%(D ' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* d%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*)d%(D' =-o6Bdissolve*<3<*)dD' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*d%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*,d%(D' =-o6Bdissolve*<3<*,dD' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*d%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*2d%(D' =%(Do' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<**d%(D' =-o6Bdissolve*<3<**dD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*3d%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*6d%(D' =-o6Bdissolve*<3<*6dD)' =4@BB BB%(D' =-g6B fade*<3<*)dD' =1:Bhidden*o3>+B#style.visibility<*)d%(D)' =4@BB BB%(D' =-g6B fade*<3<*dD' =1:Bhidden*o3>+B#style.visibility<*d%(D)' =4@BB BB%(D' =-g6B fade*<3<*,dD' =1:Bhidden*o3>+B#style.visibility<*,d%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*2d%(D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-d%(D' =-o6Bdissolve*<3<*-dD' =%(Dm' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.d%(D>' =A@BBBB0B%()?)?D' =.M7 BBBBB_M -8.33333E-7 -3.35496E-6 L -0.20469 -0.68883 *3>*B ppt_xB ppt_y=@0BBAApBBѽBf<*.dD6' =A@BB BB0B%(D' =-g6B fade*<3<*-dD' =1:Bhidden*o3>+B#style.visibility<*-d%(D ' =%(D ' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*"d%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/d%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*$d%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0d%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*d%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1d%(D' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*1dD' =1:Bhidden*o3>+B#style.visibility<*1d%(D)' =4@BB BB%(D' =-g6B fade*<3<*/dD' =1:Bhidden*o3>+B#style.visibility<*/d%(D)' =4@BB BB%(D' =-g6B fade*<3<*1dD' =1:Bhidden*o3>+B#style.visibility<*1d%(D)' =4@BB BB%(D' =-g6B fade*<3<*0dD' =1:Bhidden*o3>+B#style.visibility<*0d%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*4d%(D' =-o6Bdissolve*<3<*4dD3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*5d%(D' =-o6Bdissolve*<3<*5dD3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*7d%(D' =-o6Bdissolve*<3<*7dD' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*9d%(D' =-o6Bdissolve*<3<*9dD' =%(Dk' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*:d%(D<' =A@BBBB0B%()?)?D' =.K7 BBBBB]M -5.55556E-7 -3.35496E-6 L -0.2026 -0.62581 *3>*B ppt_xB ppt_y=@0BBAApBBϽBTD<*:dD6' =A@BB BB0B%(D' =-g6B fade*<3<*9dD' =1:Bhidden*o3>+B#style.visibility<*9d%(++0+'di ++0+'di ++0+(di ++0+(di ++0+-di ++0+-di ++0+.di ++0+.di ++0+9di ++0+9di ++0+:di ++0+:di +     l(  lr l S DiqY  i  l VADomain   i`  l c $A ??<bi H l 0޽h ? 33___PPT10i.+D=' = @B +p  x(  xr x S nbqY  <  x S gbY <$ 0 b U x 0db  Yi,$ 0  Further work: Solutions for fixed size switches? 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