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" ,3;aBasic Switch ModelbSome definitionseFinding a maximum size match$/How do we find the maximum size (weight) match?00f$Network flows and bipartite matching%%$~Finding a maximum size bipartite matching is equivalent to solving a network flow problem with capacities and flows of size 1.: Network Flows;,A maximum network flow example By inspection -$ <A maximum network flow example$>)Ford-Fulkerson method of augmenting paths**$fSet f(v,w) = -f(w,v) on all edges. Define a Residual Graph, R, in which res(v,w) = cap(v,w)  f(v,w) Find paths from s to t for which there is positive residue. Increase the flow along the paths to augment them by the minimum residue along the path. Keep augmenting paths until there are no more to augment.B4 " Z( H?Example of Residual Graph@Example of Residual GraphA#Complexity of network flow problemsIn general, it is possible to find a solution by considering at most VE paths, by picking shortest augmenting path first. There are many variations, such as picking most augmenting path first. Best Algorithm based on work by Dinic. Complexity: O(n m) E!*,Finding a maximum size match$/How do we find the maximum size (weight) match?00,$Network flows and bipartite matching%%$~Finding a maximum size bipartite matching is equivalent to solving a network flow problem with capacities and flows of size 1.:Network flows and bipartite matching Ford-Fulkerson method ;%$ $Network flows and bipartite matching%%$  $Network flows and bipartite matching%%$ !$Network flows and bipartite matching%%$ "$Network flows and bipartite matching%%$ #$Network flows and bipartite matching%%$CComplexity of Maximum Matchings Maximum Size Matchings: Algorithm by Dinic (~1970) O(n1/2m) Maximum Weight Matchings (assignment problem) Algorithm by Kuhn (1955) O(N3) In general: Hard to implement in hardware Slooooow.$. (   ( P ! _+Another algorithm: Longest Port First (LPF),,$ /LPF0Properties of LPFqLPF is stable for all admissible Bernoulli traffic. An LPF match is a maximum size match. How can this be stable? ` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@(?vKd@ " d@ ` n?" dd@   @@``PR    @ ` ` p>>    (    6b "P b T Click to edit Master title style! !$  0db " b RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0`b "`` b *   0!b "`  b ,   00&b "@ b B* H  0޽h ? ̙33 $Blank Presentation 0 zr  (     0/V M(  V P*    0V  ( V R*  d  c $ ?4h  V  0(V  ?c V RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  6V M  V P*    6V   V R*  H  0Bhy ? ̙3380___PPT10.  8(    0V M(  V >*   0V  ( V @*   6V M  V >*   6,V   V @* H  0Bhy ? ̙3380___PPT10.   @ 8 0" (  ~  s * _bP  b r ! 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Queue occupancies: RB  s *Do0 RB  s *Do,FB  S DE FB  S D FB   S D||FB   S DU< < FB   S DU, , FB   S D  FB   S D FB  S D FB  S D5FB  S D FB  S DE lE FB  S D \ FB  S DLFB  S DU l UFB  S DU \ UFB  S D L FB  S D  FB  S D  FB  S D55FB  S D  RB  s *Do \   6c>  A Occupancy    6   6xc, `LNN(n)> H  0޽h ? ̙33   0(//(  x  c $0c  c x  c $c   c d  <Gu^ LB  c $DH MLB  c $DS& sLB  c $DM LB  c $DH sLB   c $Dm sLB   c $Ds LB   c $D LB   c $D LB   c $D LB  c $D LB  c $D LB  c $D LB  c $DX LB  c $Ds LB  c $D LB  c $D LB  c $D  F ` >   `>   <c` 5A  <c 5B  <Xc 5C  <c 5D  <J  5E  <DJ >  5FZ2  s *3! Z2  s *3A Z2  s *3a Z2   s *3 Z2 ! s *3  Z2 " s *3 ! F  `/ *  # `  *  $ <X JH `  51 % < JF /  52 & <JD -  53 ' <JB +  54 ( <h J@ )   55 ) <J> ' *  56Z2 * s *3 ` Z2 + s *3 @` Z2 , s *3 `` Z2 - s *3 ` Z2 . s *3 ` Z2 / s *3 ` H  0޽h ? 33y___PPT10Y+D=' = @B +   2*==(  x  c $H*J  J x  c $+J  J d  <Gv   < -J` 5A  <h0J`H   51X2  0 3  <4JzpZ HSource s$ 3X2   0iv   <9J@$ F FSink t$3RB   s *DPFRB   s *DzFRB   s *DFRB  s *DFRB  s *DPb i RB  s *DsK i RB  s *D] i RB  s *D @ iRB  s *DHRB  s *DH RB  s *D 3 hRB  s *D&8 m LB  c $DH" MLB  c $DS> sLB  c $DM" LB  c $DH( sLB  c $Dm" sLB  c $Ds3 LB  c $D( LB  c $D8 LB  c $D8 LB   c $D" LB ! c $D3 LB " c $D8 LB # c $DX LB $ c $Ds" LB % c $D4 LB & c $D LB ' c $D (  ( <  5F - <HYJF /  52 . <]JD -  53 / <`JB +  54 0 <`dJ@ )   55 1 <8hJ > ' *  56R2 2 s *3! R2 3 s *3A R2 4 s *3a R2 5 s *3 R2 6 s *3  R2 7 s *3  ! R2 8 s *3  ` R2 9 s *3@ ` R2 : s *3` ` R2 ; s *3 ` R2 < s *3 ` R2 = s *3 ` H  0޽h ? 33y___PPT10Y+D=' = @B +  !!y(  x  c $ {JP  J X2  00@RB  s *D`X2  00-RB  s *D] RB  s *D`RB  s *D] RB   s *D] RB   s *D] RB   s *D] -`RB   s *D`] -   <J HSource s$ 3  <܁J FSink t$3  <̉J  1a  <+B#style.visibility<*%(+%  !!>>\!(  x  c $J   J X2  00@RB  s *D`X2  00-RB  s *D] RB  s *D`RB  s *D] RB   s *D] RB   s *D] -`RB   s *D`] -   <J HSource s$ 3   <J FSink t$3  <J  1a  <J  1c  <LJ  1b  <J  1dz 0Y@  Y0@,$D 0  <J0= 810  <J0=w 810  <DJ Y @ 810  <L pE W 71  <8L i P 71  <L Y @ 71  <l L0 `=G 810  <<L}  810  <L*  8Step 1:  R2  s *3] @R2  s *3] @R2  s *3] @R2  s *3] @RB   s *D] z    !  ,$D 0`2 " 0 C0 `B # 0D> `2 $ 00 0 `B % 0D> ` ZB & s *D ZB ' s *D ` ZB ( s *D ` `B ) 0D>` 0 ZB * s *D` 0  + <<L * 0  HSource s$ 3 , <,L* 0  FSink t$3 - <$L  1a . <(L  1c / <d&L`  1b 0 <.L `  1d 1 <1L7I 0 0  L10, 10*3 2 <H6L30 @  810 3 <d4L  L10, 10*3 4 <8L  H  71 5 <0AL  71 6 <FL  71 7 <DL3 @  810 8 <LL g  L10, 10*3 9 <DL}  DFlow is of size 10Z2 : s *3` ZB ; s *D ` Z2 < s *3 ` Z2 = s *3 ` Z2 > s *3` H  0޽h ? 33~___PPT10^+A1DB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(+.*  P&H&CC %(   x   c $ZL`  L X2   0pXB   0D>]X2   0p]XB   0D>]- ]XB   0D>PXB   0D>P- PRB   s *D]- PRB   s *D]- PXB   0D>]- XB   0D>- P   <^L HSource s$ 3   <`L FSink t$3   <hL0i P 1a   <4gL0 P 1c   <jLm  1b   < sL  1d   <\wL L10, 10*3   <zL K10, 1*3   <L  L10, 10*3   <L`   71   <8L u  71   <xL 6  J1, 1*3   <ĎLM -4 K10, 1*3   <ؓL   L10, 10*3   <L# 8Step 2:     <L$p UFlow is of size 10+1 = 11 LB   c $DR2   s *3 - R2   s *3 - R2   s *3 - R2 !  s *3 - gz   "   ,$D 0`2 #  0 @C `B $  0D>  `2 %  0- C `B &  0D> ] `B '  0D>  `B (  0D> ] `B )  0D> ] `B *  0D>] - `B +  0D>]  -  ,  <L= C  HSource s$ 3 -  < L= C  FSink t$3 .  <L  1a /  <L  1c 0  <4Ls  1b 1  <DL s  1d 2  <lL4\ - C  L10, 10*3 3  <пL0C *  K10, 2*3 4  <XL  K10, 9*3 5  <LP  n  I1,1*3 6  <L     I1,1*3 7  < L f  J1, 1*3 8  < L t  K10, 2*3 9  <4L z  L10, 10*3 :  <HL  = Maximum flow: ;  <\L!  UFlow is of size 10+2 = 12 L p  <  p ,$D 0 =  <`L  > Not obvious   >  s BC DE(FԔ PP P 0  @   p Z2 ?  s *3 ] Z2 @  s *3] `B A  0D> ] Z2 B  s *3 ] Z2 C  s *3] H   0޽h ? 33~___PPT10^+DB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*" %(+  F(  x  c $L  L   c $ N0 L "P@08XH  0޽h ? 33y___PPT10Y+D=' = @B +#  F">"??!(  x  c $b   N X2  0pCXB  0D>] X2  0p0XB  0D>] ` ]RB  s *D PRB  s *DP ` PRB   s *D] ` PRB   s *D] ` PXB   0D>]` 0RB   s *D` 0P   <b 3s  <N~L 3t  <N0 P 1a  <0N0 P 1c  <N  1b  < b  1d  <N70  L10, 10*3  <$N3@ 810  <(N  L10, 10*3  <,N H  71  </N  71  <3N  71  <d6N3 @ 810  <9N   L10, 10*3  <H"N} z DFlow is of size 10X2  0 C XB  0D1  X2  0 0 XB  0D1 ` ` XB   0DԔ  XB ! 0DԔ ` RB " s *D ` RB # s *D ` XB $ 0D1 0 XB % 0DԔ ` 0 & < EN` ~L  3t ' <|HN   1a ( <KN   1c ) <4ONP  p 1b * <RNP  p 1d + <UNp -W  810 , <HYN 3@ 810 - <\N    810 . <0aN H  71 / <cN  71 0 < gN  71 1 <|jN 3 @  810 2 <mNp c pW  810 3 <|qN`   3s 4 <DuN 0C  8:res(v,w) = cap(v,w)  f(v,w) 33333 5 <NFK  eResidual Graph, R 6 6 <N  AAugmenting pathLB 7 c $D} pR2 8 s *3  ` R2 9 s *3 ` R2 : s *3  ` R2 ; s *3 ` R2 < s *3p ` R2 = s *3p ` R2 > s *3p ` R2 ? s *3p` H  0޽h ? 33y___PPT10Y+D=' = @B +,  (( IIv((  x  c $ВN`  N X2  0pXB  0D>]X2  0p]XB  0D>]- ]XB  0D>PXB  0D>P- PRB   s *D]- PRB   s *D]- PXB   0D>]- XB   0D>- P   <ėN B s$3  <NK B t$3  <N0i P 1a  <N0 P 1c  <Nm  1b  <N  1d  <N L10, 10*3  <N K10, 1*3  <N  L10, 10*3  <N`   71  <,N u  71  <DN 6  J1, 1*3  <NM -4 K10, 1*3  <N   L10, 10*3  <N# 8Step 2:    <pN$p UFlow is of size 10+1 = 11 X2  0 `i XB  0D1 ;( 9 X2  0 Xi XB   0D19 ;( XB ! 0D1 ( XB " 0D1 X9 XB # 0D19 X  $ <Nc T%i  B s$3 % <Nc ti  B t$3 & <TN   1a ' <N B   1c ( <N   1b ) <N B '  1d * <XN -  H10*3 + <Ri @P  G1*3 , <R2 + 8   H10*3 - <`R6 { 0   G1*3 . <NB  )  G1*3 / <t R2 H  G1*3 0 <4R0 e  G1*3 1 <R H U  H10*3 2 <HR`   >Residual GraphLB 3 c $DNF 0 @ 4  0@`B 5 0D1 `B 6 0D1( `B 7 0D1(  8  BCDE(F <x` @   0` 0 8 9  BCDE(F ,X( pH$ @    ` @ : <#RY 0@ G9*3 ; <(R l G9*3z 0 @ <  0@,$D 0`B = 0DԔ `B > 0DԔ( `B ? 0DԔ(  @ C BCDE(FԔ <x` @   0` 0 8 A C BCDE(FԔ ,X( pH$ @    ` @R2 B s *3 - R2 C s *3 - R2 D s *3 - R2 E s *3 - R2 F s *3 (  R2 G s *3 ( R2 H s *3  R2 I s *3 H  0޽h ? 33~___PPT10^+jDB' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(+y  0 0(   x   c $3R   R x   c $4R R H   0޽h ? 33y___PPT10Y+D=' = @B +   P/?(  x  c $:R  R x  c $t;R   R d  <Gu^ LB  c $DH MLB  c $DS& sLB  c $DM LB  c $DH sLB  c $Dm sLB  c $Ds LB  c $D LB  c $D LB  c $D LB   c $D LB ! c $D LB " c $D LB # c $DX LB $ c $Ds LB % c $D LB & c $D LB ' c $D  8 ` >  >`>   <<@R` 5A ( <CR 5B ) <FR 5C * <JR 5D + <\MR  5E , <PR >  5FZ2 2 s *3! Z2 3 s *3A Z2 4 s *3a Z2 5 s *3 Z2 6 s *3  Z2 7 s *3 ! 8  `/ *  ?`  *   <URH `  51 - <@YRF /  52 . <\RD -  53 / <_RB +  54 0 <cR@ )   55 1 <`fR> ' *  56Z2 8 s *3 ` Z2 9 s *3 @` Z2 : s *3 `` Z2 ; s *3 ` Z2 < s *3 ` Z2 = s *3 ` H  0޽h ? 33y___PPT10Y+D=' = @B +  2*`==(  x  c $vR  R x  c $vR  R d  <Gv   <xR` 5A  <0|R`H   51X2  0 3  <RzpZ HSource s$ 3X2   0iv   <TR@$ F FSink t$3RB   s *DPFRB   s *DzFRB   s *DFRB  s *DFRB  s *DPb i RB  s *DsK i RB  s *D] i RB  s *D @ iRB  s *DHRB  s *DH RB  s *D 3 hRB  s *D&8 m LB  c $DH" MLB  c $DS> sLB  c $DM" LB  c $DH( sLB  c $Dm" sLB  c $Ds3 LB  c $D( LB  c $D8 LB  c $D8 LB   c $D" LB ! c $D3 LB " c $D8 LB # c $DX LB $ c $Ds" LB % c $D4 LB & c $D LB ' c $D (  ( <R 5B ) <R 5C * <`R 5D + <R  5E , <R >  5F - <lRF /  52 . <ܩRD -  53 / <RB +  54 0 <R@ )   55 1 <\R > ' *  56R2 2 s *3! R2 3 s *3A R2 4 s *3a R2 5 s *3 R2 6 s *3  R2 7 s *3  ! R2 8 s *3  ` R2 9 s *3@ ` R2 : s *3` ` R2 ; s *3 ` R2 < s *3 ` R2 = s *3 ` H  0޽h ? 33J  p==(  x  c $R@  R d  <  6   <R\\  5A  <`R  51X2  0_     <R   4 sX2  0U V    <RH  B t$3XB   0DԔ @ XB   0DԔ @ XB   0DԔ ' RB   s *D 4  XB  0DԔ _ XB  0DԔ _ XB  0DԔ _ RB  s *D_  RB  s *D 3 RB  s *D 3 ' RB  s *D_  RB  s *De ' XB  0DԔ RB  s *De RB  s *D` RB  s *DZ XB  0DԔ RB  s *DZ ! XB  0DԔ RB  s *DU RB  s *Dk  RB  s *D e  RB   s *D q  RB ! s *D k , RB " s *Dq RB # s *Dv RB $ s *DZ ! LB % c $D e 2 RB & s *D, Z 2  ' <xR\I  5B ( <R\D  5C ) <0R \\ 1  5D * <R7 \H W  5E + <R] \E }  5F , <DR  52 - <R   53 . <R %  54 / <DS' G  55 0 <SI i  56R2 1 s *3`- R2 2 s *3- R2 3 s *3- R2 4 s *3 - R2 5 s *3 - @ R2 6 s *3 - ` R2 7 s *3_ R2 8 s *3 R2 9 s *3 R2 : s *3  R2 ; s *3 ? R2 < s *3 _  = <R0 P V&Residual Graph for first three paths: ''H  0޽h ? 33k  ==(  x  c $S0P  S d  <  6   <S\\  5A  <S  51X2  0_     <S   4 sX2  0U V    <SH  B t$3XB   0DԔ @ XB   0DԔ @ XB   0DԔ ' RB   s *D 4  XB  0DԔ _ XB  0DԔ _ XB  0DԔ _ RB  s *D_  XB  0DԔ 3 XB  0DԔ 3 ' XB  0DԔ_  XB  0DԔe ' XB  0DԔ RB  s *De RB  s *D` RB  s *DZ XB  0DԔ RB  s *DZ ! XB  0DԔ RB  s *DU RB  s *Dk  RB  s *D e  XB   0DԔ q  RB ! s *D k , RB " s *Dq RB # s *Dv RB $ s *DZ ! LB % c $D e 2 XB & 0DԔ, Z 2  ' <h-S\I  5B ( <x1S\D  5C ) <P5S \\ 1  5D * <9S7 \H W  5E + < <D3Ԕip ndB ? <D3Ԕ H  0޽h ? 33  e]>>(  x  c $S  S d  <  6   <S\\  5A  <LS  51X2  0_     <S   4 sX2  0U V    <SH  B t$3XB   0DԔ @ XB   0DԔ @ XB   0DԔ ' XB   0DԔ 4  XB  0DԔ _ XB  0DԔ _ XB  0DԔ _ XB  0DԔ_  XB  0DԔ 3 XB  0DԔ 3 ' XB  0DԔ_  XB  0DԔe ' RB  s *D RB  s *De XB  0DԔ` RB  s *DZ XB  0DԔ RB  s *DZ ! RB  s *D XB  0DԔU RB  s *Dk  RB  s *D e  XB   0DԔ q  RB ! s *D k , XB " 0DԔq RB # s *Dv RB $ s *DZ ! LB % c $D e 2 XB & 0DԔ, Z 2  ' <S\I  5B ( <\S\D  5C ) <S \\ 1  5D * <S7 \H W  5E + <pS] \E }  5F , <S  52 - <(S   53 . <S %  54 / <S' G  55 0 <PSI i  56R2 1 s *3`- R2 2 s *3- R2 3 s *3- R2 4 s *3 - R2 5 s *3 - @ R2 6 s *3 - ` R2 7 s *3_ R2 8 s *3 R2 9 s *3 R2 : s *3  R2 ; s *3 ? R2 < s *3 _  = <SW? Y)Residual Graph for last augmenting path: *** > <lS 5J^ pNote that the path augments the match: no input and output is removed from the match during the augmenting step.2qVH  0޽h ? 33  F>22(  x  c $$S0  S d  <  6   <S\\  5A  <D[  51X2  0_     <[   4 sX2  0U V    <( [H  B t$3XB   0DԔ @ XB   0DԔ @ XB   0DԔ ' XB   0DԔ 4  XB  0DԔ _ XB  0DԔ _ XB  0DԔ _ XB  0DԔ_  XB  0DԔ 3 XB  0DԔ 3 ' XB  0DԔ_  XB  0DԔe ' XB  0DԔ` XB  0DԔ XB  0DԔU XB  0DԔ q  XB  0DԔq XB  0DԔ, Z 2   <([\I  5B  <[\D  5C  <[ \\ 1  5D  <<[7 \H W  5E   <[] \E }  5F ! < [  52 " <P$[   53 # <'[ %  54 $ <+[' G  55 % <d.[I i  56R2 & s *3`- R2 ' s *3- R2 ( s *3- R2 ) s *3 - R2 * s *3 - @ R2 + s *3 - ` R2 , s *3_ R2 - s *3 R2 . s *3 R2 / s *3  R2 0 s *3 ? R2 1 s *3 _  2 <5[Wi  DMaximum flow graph: H  0޽h ? 33u  %#%(  x  c $L>[  [ d  <  6   <A[\\  5A  <8D[  51XB  0DԔ` XB  0DԔ XB  0DԔU XB   0DԔ q  XB   0DԔq XB   0DԔ, Z 2    <LI[\I  5B   <$M[\D  5C  <O[ \\ 1  5D  <8S[7 \H W  5E  <V[] \E }  5F  <Y[  52  <L][   53  <`[ %  54  <d[' G  55  <`g[I i  56R2  s *3`- R2  s *3- R2  s *3- R2  s *3 - R2  s *3 - @ R2  s *3 - ` R2  s *3_ R2  s *3 R2  s *3 R2  s *3  R2   s *3 ? R2 ! s *3 _  " <n[W  GMaximum Size Matching: r % S r[ [ H  0޽h ? 33y  (0(  (x ( c $4y[P  [ x ( c $z[ [ H ( 0޽h ? 33y___PPT10Y+D=' = @B +[  z;(  r  S T[P  [ `  c $A F??! 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