Ministerul Educaţiei şi Tineretului din Republica Moldova Universitatea Universitatea de Stat din Moldova
Lucrare de laborator
Tema: Mulţimi stabile în grafuri neorientate. Algoritmul Malgrange
Student: Student: Profesor: Novac L.
Chişinău 2009
I. Noţiuni preliminare: Fie M o matrice binară, fnită, de dimensiune m×n, cu mulţimea de linii L={l 1,l2,…,lm} şi mulimea de coloane ={c1,c2,…,cm}! "om nota #=$%,&' matricea (ormată din elementele de la intersecia liniilor %) şi a coloanelor & *! Fie acum # 1 şi # 2 două submatrice ale matricei M, determinate de +erecile de mulimi de linii şi coloane $%i,&i' i=1,2 $# 1=$%1,&1' şi # 2=$%2,&2''! -acă %1%2 , &1&2 $# 1 # 2', matricea # 1 se numeşte submatrice a matricei # 2 ! -acă toate elementele din # sunt e.ale cu 1, submatricea # a matricei M se numeşte completă -acă submatricea # este com+letă şi /n M nu e0istă o altă submatrice com+letă # as3el /nc/t # #, se numeşte principală -acă orice element e.al cu 1 din M a+arine cel +uin unei submatrici din (amilia ={# 1,# 2, …,# +}, această (amilie se numeşte acoperire a matricei ! ardinalul mulimii stabile interior ma0ime a .ra(ului 4 se notea5ă +rin !0"#$ şi se numeşte număr de stabilitate internă
II. escrierea algoritmului Fie % matricea de adiacenţă a unui .ra( neorientat 4=$678'9
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iar ; este matricea com+lementară a acesteia $elementele < i ale matricei ; se calculea5ă /n ba5a
elementelor a i ale matricei % du+a (ormula &i'()* ai''9 a
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) Scopul lucrării+ -e a construi toate submatricile +rinci+ale +ătra>ce ale lui ;, /n ba5a cărora se +ot determina toate mulţimile ma0imale stabile interior ale ale .ra(ului 4, +rin u>li5area al.oritmului Mal.ran.e! ,asul ) onstruim o aco+erire arbitrară : a matricei ;! ?n calitate de aco+erirea : se ia (amilia tuturor submatricelor com+lete din ; de (orma # i=$%i,&i', unde @%@=1, iar & i este (ormată din coloanele matricei ;, ce conţin unitatea /n linia % i! -coperirea ini.ială C0 a matricei ; este9 C0 ( { $a,ac', $b,b', $c,ace(', $d,d', $e,ce', $(,c(' }! ,asul 2 Aentru p=0, construim (amilia X p={ƒ i ƒi } B (amilia tuturor i astel, încî ƒj C p , ƒ j ƒ submatricelor com+lete din +, care care se conin /n alte submatrice ale lui +!
?n acest ca5 /0( ,asul onstruim (amilia de submatrice 1"Cp/p$, care se obine +rin a+licarea o+eraiilor ⋂ şi ⋃ asu+ra tuturor +erecilor +osibile de matrice # i, # din Cp/p, cu condiia ca aceste elemente noi să nu le conină +e submatricele din Cp/p! C0/0( C0 1"C0/0$ ( 1"C0$9
$a, ac' ⋃ $c,ace(' = $a ⋃ c, ac ⋂ acef ' = "ac3ac$ $a, ac' ⋃ $e,ce' = $a ⋃ e, ac ⋂ ce' = "ae3c$ $a, ac' ⋃ $(,c(' = $a ⋃ (, ac ⋂ cf ' = "a%3c$ $c, ace(' ⋃ $e,ce' = $c ⋃ e, ace( ⋂ ce' = "ce3ce$ $c, ace(' ⋃ $(,c(' = $c ⋃ (, ace( ⋂ cf ' = "c%3c%$ $e,ce' ⋃ $(,c(' = $e ⋃ (, ce ⋂ cf ' = "e%3c$ 1"C0/0$ ={ $ac,ac', $ae,c', $a(,c', $ce,ce', $c(,c(', $e(,c' }! ,asul 4 Formăm aco+erirea de matrice9 Cp5) Cp/p$ C
"1"Cp/p$$
C) C0/0$ "1"C0/0$$ = { "a3ac$, $b,b', $c,ace(', $d,d', "e3ce$, $(,c(', $ac,ac', $ae,c', $a(,c', $ce,ce', "c%3c%$ , $e(,c' }! ,asul 6 -acă Cp5) C) 70
Cp, atunci considerăm p(p5) Di ne re/ntoarcem la ,asul 2
,asul 2 onstruim (amilia /p5)(/)+
2 "a3ac$ "ac3ac$ "e3ce$ "ce3ce$ "%3c%$ "c%3c%$ /) ={ $a,ac', $e,ce', $(,c(' }! ,asul onstruim (amilia 1"Cp5)/p5)$ = 1"C)/)$+ 1E61= { $b,b', $c,ace(', $d,d', $(,c(', $ac,ac', $ae,c', $a(,c', $ce,ce', $e(,c' }!
$c,ace(' ⋃ $(,c(' = $c ⋃ (, ace( ⋂ c(' = $c(,c(' $c,ace(' ⋃ $ac,ac' = $c ⋃ ac, ace( ⋂ ac' = $ac,ac' $c,ace(' ⋃ $ae,c' = $c ⋃ ae, ace( ⋂ c' = $ace,c' $c,ace(' ⋃ $a(,c' = $c ⋃ a(, ace( ⋂ c' = $ac(,c' $c,ace(' ⋃ $ce,ce' = $c ⋃ ce, ace( ⋂ ce' = $ce,ce' $c,ace(' ⋃ $e(,c' = $c ⋃ e(, ace( ⋂ c' = $ce(,c' $(,c(' ⋃ $ac,ac' = $( ⋃ ac, c( ⋂ ac' = $ac(,c' $(,c(' ⋃ $a(,c' = $( ⋃ a(, c( ⋂ c' = $a(,c' $(,c(' ⋃ $ce,ce' = $( ⋃ ce, c( ⋂ ce' = $ce(,c' $(,c(' ⋃ $e(,c' = $( ⋃ e(, c( ⋂ c' = $e(,c' $ac,ac' ⋃ $ce,ce' = $ac ⋃ ce, ac ⋂ ce' = $ace,c' $ae,c' ⋃ $a(,c' = $ae ⋃ a(, c ⋂ c' = $ae(,c' $ae,c' ⋃ $ce,ce' = $ae ⋃ ce, c ⋂ ce' = $ace,c' $ae,c' ⋃ $e(,c' = $ae ⋃ e(, c ⋂ c' = $ae(,c'
$a(,c' ⋃ $ce,ce' = $a( ⋃ ce, c ⋂ ce' = $ace(,c' $a(,c' ⋃ $e(,c' = $a( ⋃ e(, c ⋂ c' = $ae(,c' $ce,ce' ⋃ $e(,c' = $ce ⋃ e(, ce ⋂ c' = $ce(,c'
1"C)/)$ = { $c(,c(', $ac,ac', $ace,c', $ac(,c', $ce,ce', $ce(,c', $a(,c', $e(,c', $ae(,c', $ace(,c' } ,asul 4 C2 C)/)$ "1"C)/)$$ = { $b,b', $c,ace(', $d,d', "%3c%$ , $ac,ac', "ae3c$, "a%3c$, $ce,ce', "e%3c$, $c(,c(', "ace3c$, "ac%3c$, "ce%3c$, "ae%3c$, $ace(,c' } ,asul 6 C2
7)
,asul 2 onstruim (amilia /p5)(/2+ "%3c%$ "c%3c%$ "ae3c$ "ace3c$3 "ae%3c$ şi "ace%3c$ "a%3c$ "ac%3c$3 "ae%3c$ şi "ace%3c$ "e%3c$ "ce%3c$ şi "ace%3c$
"ace3c$ "ace%3c$ "ac%3c$ "ace%3c$ "ce%3c$ "ace%3c$ "ae%3c$ "ace%3c$ /2 = {$(,c(', $ae,c', $a(,c', $e(,c', $ace,c', $ac(,c', $ce(,c', $ae(,c'} ,asul onstruim (amilia 1"Cp5)/p5)$ = 1"C2/2$+ C2/2= { $b,b', $c,ace(', $d,d', $ac,ac', $ce,ce', $c(,c(', $ace(,c'}
$c,ace( '⋃ $ac,ac' = $c ⋃ ac, acef ⋂ ac ' = "ac3ac$ $c,ace( '⋃ $ce,ce' = $c ⋃ ce, acef ⋂ ce ' = "ce3ce$ $c,ace(' ⋃ $c(,c(' = $c ⋃ cf, acef ⋂ cf ' = "c%3c%$ $c,ace( ' ⋃ $ace(,c' = $c ⋃ acef, acef ⋂ c ' = "ace%3c$ $ac,ac' ⋃ $ce,ce' = $ac ⋃ ce, ac ⋂ ce ' = "ace3c$ $ac,ac' ⋃ $c(,c(' = $ac ⋃ cf, ac ⋂ cf ' = "ac%3c$ $ac,ac' ⋃ $ace(,c' = $ac ⋃ acef, ac ⋂ c ' = $ace(,c' $ce,ce' ⋃ $c(,c(' = $ce ⋃ cf, ce ⋂ cf ' = "ce%3c$ $ce,ce' ⋃ $ace(,c' = $ce ⋃ acef, ce ⋂ c ' = $ace(,c' $c(,c(' ⋃ $ace(,c' = $c( ⋃ acef, cf ⋂ c ' = $ace(,c' 1"C2/2$ = { $ac,ac', $ce,ce', $c(,c(', $ace(,c', $ace,c', $ac(,c', $ce(,c' } ,asul 4 C C2/2$ "ce%3c$ }
"1"C2/2$$ = { $b,b', $c,ace(', $d,d', $ac,ac', $ce,ce', $c(,c(', $ace(,c', "ace3c$, "ac%3c$,
G
,asul 6 C
72
,asul 2 onstruim (amilia /p5)(/+ "ace3c$ "ac%3c$ "ce%3c$
"ace% c$ "ace%3c$ "ace%3c$
/ ={ $ace,c', $ac(,c', $ce(,c' } ,asul onstruim (amilia 1"Cp5)/p5)$ = 1"C/$+ C/ ={ $b,b', $c,ace(', $d,d', $ac,ac', $ce,ce', $c(,c(', $ace(,c' } 1"C/$ = { $ac,ac', $ce,ce', $c(,c(', $ace(,c', $ace,c', $ac(,c', $ce(,c' }
4 ,asul 4 C4 C/$
"1"C/$$ ( C
,asul 6 C4 ( 7 3 re8ultă că C conţine toate submatricele principale ale matricei ,asul :onstruim o (amilie nouă F /n care includem submatricele +ătra>cema0imale ale submatricelor +rinci+ale din C3 res+ecHnd condiia ca fecare dintre acestea să nu le conină altă submatrice +ătra>că din F ! C { $b,b', $c,ace(', $d,d', $ac,ac', $ce,ce', $c(,c(', $ace(,c', $ace,c', $ac(,c', $ce(,c' } F
= { $b,b', $d,d', $ac,ac', $ce,ce', $c(,c(' }
"/r(urile ce cores+und liniilor $coloanelor' matricelor (amiliei F (ormea5ă o mulime stabilă interior! %s3el, numărul tuturor mulimilor stabil interior este e.al cu @ F @ $@F @=G'! S = { {b}, {d}, {a, c}, {c, e}, {c, (} } $ SI(amilia tuturor mulimilor stabile interior a .ra(ului 4'!
Mulimile stabile interior ma0im sunt ;a3 c<3 ;c3 e< şi ;c3 %< Arin urmare !0"#$ ( 2
,ro=ramul Jinclude Kstrin.! Jinclude Kstdio! Jinclude Kconio! struct multsime{ struct submul>me{ car %1:N,&1:N7 }sm1::N7 }2N,61N7 car nit1:N,nru1:N7 unsi.ned cm+$car O, car '7
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unsi.ned c2$car O, carO'7 Qoid it$carO , carO'7 Qoid ru$carO , carO'7 Qoid sort$car O'7 Qoid del$unsi.ned , unsi.ned'7 unsi.ned cecR$car O,car O,unsi.ned n'7 unsi.ned eSl$unsi.ned ,unsi.ned '7
int main$'{ TTclrscr$'7 multsime U7 unsi.ned i, , n, com1:N1:N7 TTcom I com+lementara car Ostd=V%&-WF4XV, lit2N={YE:Y}7 int bPNPN={ :,1,:,1,1,1,TT1 1,:,1,1,1,1,TT2 :,1,:,1,:,:,TTC 1,1,1,:,1,1,TT 1,1,:,1,:,1,TTG 1,1,:,1,1,:}7TTP n=P7 TO +rin3$Vintrodu nr de Qir(uri9V'7 scan($VZiV, [Q'7 +rin3$Vintrodu matricea de adiacenta9V'7 (or$i=:7 iKQ7 i\\' (or$=:7 KQ7 \\'{ +rin3$VEn bZcNZcN=V,stdiN,stdN'7 scan($VZiV, [biNN'7 }OT +rin3$Vcom+lementara9EnV'7 (or$i=:7 iKn7 i\\' { (or$=:7 Kn7 \\' { comiNN=1IbiNN7 +rin3$ V Zi V,comiNN'7} +rin3$VEnV'7 } (or$i=:7 iKn7 i\\' { TT trans(ormarea in litere lit:N=O$std\i'7lit1N=YE:Y7 strcat$:N!smiN!%,lit'7 (or$=:7 Kn7 \\' i($comiNN==1' { lit:N=O$std\'7 lit1N=YE:Y7 strcat$:N!smiN!&,lit'7} } unsi.ned m=:,R=:,Q=n,l7 TTIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII do{ 1N=:N7 m=n7 n=Q7 TTafsarea : submultsimelor %,& Aasul 2 +rin3$V%fsarea lui T Aasul 2EnV'7 (or $i=:7iKm7i\\' +rin3$V$Zs@Zs'EnV, 1N!smiN!%, 1N!smiN!&'7 TTsubmultsimele din N (or $i=:7iKm7i\\' (or $=:7Km7\\' i( $$i]=' [[ $c2$:N!smiN!%, :N!smN!%'' [[ $c2$:N!smiN!&, :N!smN!&'''{ strc+^$6:N!smRN!%,:N!smN!%'7 strc+^$6:N!smRN!&,:N!smN!&'7 R\\7 } TTafsarea 6N +rin3$VEnafsam multsimea 6EnV'7 (or $i=:7iKR7i\\'{
_
i( $strlen$6:N!smiN!%'' +rin3$V6$Zs@Zs'EnV, 6:N!smiN!%, 6:N!smiN!&'7} .etc$'7 TT$NT6N' (or$i=:7iKn7i\\' (or$=:7KR\17\\' i($ ]strcm+$:N!smiN!%, 6:N!smN!%' [[ ]strcm+$:N!smiN!&, 6:N!smN!&'' {del$i,n'7nII7}
TTN reuniunea si intersec>a cautarea in N it si ru +rin3$VEnAasul CEnV'7 (or$i=:7iKnI17i\\' (or$=i\17Kn7\\'{ ru$:N!smiN!%, :N!smN!%'7 it$:N!smiN!&, :N!smN!&'7 +rin3$V$Zs,Zs'8$Zs,Zs'=$Zs,Zs'EnV,:N!smiN!%,:N!smiN!&,:N!smN!%,:N!smN!&,nru,strlen$nit':`nit9V"idV'7 i( $strlen$nit' [[ strlen$nru' [[ cecR$nru,nit,Q''{ strc+^$:N!smQN!%, nru'7 strc+^$:N!smQN!&, nit'7 Q\\7 } } .etc$'7 +uts$V:NV'7 (or $l=:7lKm7l\\' i($strlen$1N!smlN!%'' +rin3$VEtZ!2d!1N$Zs@Zs'EnV,l, 1N!smlN!%, 1N!smlN!&'7 +uts$V1NV'7 (or $l=:7lKQ7l\\' i($strlen$:N!smlN!%'' +rin3$VEtZ!2d!:N$Zs@Zs'EnV,l, :N!smlN!%, :N!smlN!&'7 .etc$'7 n=Q7 +rin3$V6=ZdV,R'7 +rin3$VEnicluEnEnV'7 }ile$eSl$m,n''7 +rin3$VEndone]EnV'7 TTIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII car Qt1:N1:N7 +uts$Ve5ultat9V'7 +rin3$VF=V'7 (or $l=:7lKQ7l\\' i( $$strcm+$:N!smlN!%,:N!smlN!&'==:' [[ $strlen$:N!smlN!&''' +rin3$V$Zs@Zs'7V,:N!smlN!%, :N!smlN!&'7 +rin3$VEnU={V'7 (or $l=:7lKQ7l\\' i( $$strcm+$:N!smlN!%,:N!smlN!&'==:' [[ $strlen$:N!smlN!&'''{ +rin3$V{V'7 (or $i=:7iKstrlen$:N!smlN!%'7i\\' +rin3$VZsZcV,i`V,V9VV,:N!smlN!%iN'7 +rin3$V}V'7 } +rin3$V}V'7
.etc$'7 return 17 } TTcontrolea5a da ca intrun sie se contsine caracterul Qariabila unsi.ned cm+$car O+,car c'{ TTreturn : dace este return 1 daca nui unsi.ned i=:7 ile $O$+\i'' i($O$+\i\\'==c' return :7 return 17 }7 unsi.ned c2$car Oa, car Ob'{ i( $$strlen$a'=strlen$b'''{ ile $Ob' i( $cm+$a,O$b'\\'' return :7 return 17} return :7 }
TTintersectsia a 2 siruri Qoid it$car Oa,car Ob'{ car inter1:N={YE:Y}7 unsi.ned i=:7 i($strlen$a' [[ strlen$b'' ile $O$b\i'' i($]cm+$a,O$b\i\\''' O$inter\strlen$inter''=O$b\iI1'7 sort$inter'7 strc+^$nit,inter'7 } TTreunuinea a 2 siruri Qoid ru$carO a, carO b'{ int i=:7 strc+^$nru,a'7 car lit2N={YE:Y}7 ile $O$b\i'' i( $ cm+$nru,bi\\N' ' {lit:N=O$b\iI1'7 strcat$nru,lit'7} sort$nru'7 } TTsortarea crescator a sirului $metoda bulelor' Qoid sort$car O+'{ car tm+7 unsi.ned i,,0=strlen$+'7 (or$i=17 iK07 i\\' (or$=:7 K0Ii7 \\' i( $+N +\1N ' {tm+=+N7 +N=+\1N7 +\1N=tm+7} } TTster.erea din N Qoid del$unsi.ned i,unsi.ned n'{ (or $7iKn7i\\'{ :N!smiN=:N!smi\1N7 } } TTdaca )n N nu se contsine a si b return 1 unsi.ned cecR$car Oa,car Ob,unsi.ned n'{ (or$int i=:7iKn7i\\' i($ strcm+$:N!smiN!%,a'==: ' i($strcm+$:N!smiN!&,b'==:' { return :7} return 17 } unsi.ned eSl$unsi.ned n,unsi.ned m'{ unsi.ned e.ale,,i7 i( $n]=m' return 17 (or$i=:7iKn7i\\'{ e.ale=:7 (or$=:7Km7\\' i($$strcm+$:N!smiN!%,1N!smN!%'==:' @@ $strcm+$:N!smiN!&,1N!smN!&'==:''{ e.ale=17breaR7} } i($e.ale' return :7 else return 17 }
Re8ultatul complementara+ ) 0 ) 0 0 0 0 ) 0 0 0 0 ) 0 ) 0 ) )
0 0 0 ) 0 0 0 0 ) 0 ) 0 0 0 ) 0 0 ) ->sharea lui C? ,asul 2
C"-@-C$ C"A@A$ C"C@-CEB$ C"@$
C"E@CE$ C"B@CB$
C"-E@C$ C"-B@C$ C"CE@CE$ C"CB@CB$ C"EB@C$ C"@$ C"@$ C"@$ C"-CE@C$ C"-CB@C$ C"CEB@C$ C"-CEB@C$ C"-EB@C$
a>sham multsimea / /"-@-C$ /"E@CE$ /"B@CB$
"A3A$U"-B3C$("-AB3Did$ "A3A$U"CE3CE$("ACE3Did$ "A3A$U"CB3CB$("ACB3Did$ "A3A$U"EB3C$("AEB3Did$ "C3-CEB$U"3$("C3Did$ "C3-CEB$U"-C3-C$("-C3-C$ "C3-CEB$U"-E3C$("-CE3C$ "C3-CEB$U"-B3C$("-CB3C$ "C3-CEB$U"CE3CE$("CE3CE$ "C3-CEB$U"CB3CB$("CB3CB$ "C3-CEB$U"EB3C$("CEB3C$ "3$U"-C3-C$("-C3Did$ "3$U"-E3C$("-E3Did$ "3$U"-B3C$("-B3Did$ "3$U"CE3CE$("CE3Did$ "3$U"CB3CB$("CB3Did$ "3$U"EB3C$("EB3Did$ "-C3-C$U"-E3C$("-CE3C$ "-C3-C$U"-B3C$("-CB3C$ "-C3-C$U"CE3CE$("-CE3C$ "-C3-C$U"CB3CB$("-CB3C$ "-C3-C$U"EB3C$("-CEB3C$ "-E3C$U"-B3C$("-EB3C$ "-E3C$U"CE3CE$("-CE3C$ "-E3C$U"CB3CB$("-CEB3C$ "-E3C$U"EB3C$("-EB3C$ "-B3C$U"CE3CE$("-CEB3C$ "-B3C$U"CB3CB$("-CB3C$ "-B3C$U"EB3C$("-EB3C$ "CE3CE$U"CB3CB$("CEB3C$ "CE3CE$U"EB3C$("CEB3C$ "CB3CB$U"EB3C$("CEB3C$ C0F 00C)F"-@-C$ 0)C)F"A@A$ 02C)F"C@-CEB$ 0C)F"@$ 04C)F"E@CE$ 06C)F"B@CB$ 0:C)F"-C@-C$ 0GC)F"-E@C$ 0HC)F"-B@C$ 09C)F"CE@CE$ )0C)F"CB@CB$ ))C)F"EB@C$ C)F 00C0F"A@A$ 0)C0F"C@-CEB$ 02C0F"@$ 0C0F"-C@-C$ 04C0F"-E@C$ 06C0F"-B@C$ 0:C0F"CE@CE$ 0GC0F"CB@CB$ 0HC0F"EB@C$ )2C0F"-CE@C$ )C0F"-CB@C$ )4C0F"CEB@C$ )6C0F"-CEB@C$ ):C0F"-EB@C$ /( Ciclu
,asul "A3A$U"C3-CEB$("AC3Did$ "A3A$U"3$("A3Did$ "A3A$U"-C3-C$("-AC3Did$ "A3A$U"-E3C$("-AE3Did$
->sharea lui C? ,asul 2 C"A@A$ C"C@-CEB$ C"@$ C"-C@-C$
doneI Re8ultat+ B("A@A$J"@$J"-C@-C$J"CE@CE$J"CB@CB$J
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