Elemente de Teoria Grafurilor

8/3/2019 Elemente de Teoria Grafurilor

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ELEMENTE DE TEORIA GRAFURILOR

1. Noiuni generale

n general, pentru situaiile care necesit la rezolvare unoarecare efort mintal (i un caz tipic este cel al celor din economie),se caut, n primul rnd, o metod de reprezentare a lor care spermit receptarea ntregii probleme dintr-o privire (pe ct posibil) iprin care s se evidenieze ct mai clar toate aspectele acesteia.

n acest scop se folosesc imagini grafice gen diagrame, schi e,grafice etc. O reprezentare dintre cele mai utilizate este cea pringrafuri. Acestea sunt utilizate n special pentru vizualizareasistemelor i situaiilor complexe. n general, vom reprezentacomponentele acestora prin puncte n plan iar relaiile (legturile,

dependenele, influenele etc) dintre componente prin arce de curbcu extremitile n punctele corespunztoare. ntre dou puncte potexista unul sau mai multe segmente (n funcie de cte relaii dintreacestea, care ne intereseaz, exist) iar segmentelor li se pot asociasau nu orientri (dup cum se influeneaz cele dou componente ntre ele), numere care s exprime intensitatea relaiilor dintrecomponente etc.

Este evident, totui, c aceast metod are limite, att dinpunct de vedere uman (prea multe puncte i segmente vor facedesenul att de complicat nct se va pierde chiar scopul pentrucare a fost creat claritatea i simplitatea reprezentrii, aceasta

devenind neinteligibil) ct i din punct de vedere al tehnicii decalcul (un calculator nu poate "privi" un desen ca un om).

Din acest motiv, alturi de expunerea naiv-intuitiv a ceea ceeste un graf, dat mai sus, se impune att o definiie riguroas ct ialte modaliti de reprezentare a acestora, adecvate n generalrezolvrilor matematice.

Definiia 1 Se numete multigraf un triplet G = (X, A, f) n careX i A sunt dou mulimi iar f este o funcie, definit pe produsul

vectorial al lui X cu el nsui (X2 = XX), care ia valori n mulimea

prilor mulimii A (notatP(A))

Mulimea X se numete mulimea nodurilor multigrafului ielementele sale se numesc noduri (sau vrfuri) ale multigrafului, iarA reprezint mulimea relaiilor (legturilor) posibile ntre dou noduriale multigrafului.

Definiia 2 Se numete graf orientat un multigraf n caremulimea A are un singur element: A = {a}.

n acest caz mulimea prilor mulimii A are doar douelemente: mulimea vid i ntreaga mulime A. Dac unei perechi


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orientate (xi, xj) din X2 i se asociaz prin funcia f mulimea A atunci

spunem ca exist arc de la nodul xi la nodul xj iar perechea (xi,xj) seva numi arcul (xi,xj). Nodul xi se numete nod iniial sau extremitateiniial a arcului (x

i,xj) iar nodul x

jse numete nod final sau

extremitate final a arcului (xi,xj). Arcul (xi,xj) este incident spreinterior vrfului xji incident spre exterior vrfului xi. Dac pentru unarc nodul iniial coincide cu nodul final atunci acesta se numetebucl. Nodurile xi i xj se vor numi adiacente dac exist cel puinunul din arcele (xi,xj) i (xj,xi).

Dac unei perechi orientate (xi, xj) din X2 i se asociaz prin

funcia f mulimea vid atunci spunem c nu exist arc de la nodulxi la nodul xj.

Este evident c a cunoate un graf orientat este echivalent cua cunoate vrfurile i arcele sale. Din acest motiv putem defini ungraf orientat prin perechea (X,U), unde X este mulimea vrfurilorsale iar U mulimea arcelor sale.

De asemenea, putem cunoate un graf orientat cunoscndmulimea nodurilor i, pentru fiecare nod, mulimea arcelor incidentespre exterior. Din acest motiv putem defini un graf orientat ca opereche (X,) unde X este perechea nodurilor iar este o func iedefinit pe X cu valori n mulimea prilor lui X, valoarea acesteiantr-un nod xi, (xi) X fiind mulimea nodurilor adiacente nodului

xi, prin arce pentru care xi este extremitatea iniial.

Definiia 3 Se numete graf neorientat un multigraf n caremulimea A are un singur element iar funcia f are proprietatea:

f[(xi,xj)] = f[(xj,xi)] , oricare ar fi nodurile xii xj din X

n aceste condiii, dac f[(xi,xj)] = f[(xj,xi)] = A atunciperechea neorientat {xi,xj} este o muchie iar dac f[(xi,xj)] =f[(xj,xi)] = spunem c nu exist muchie ntre vrfurile xii xj.

Deoarece, n cele mai multe din cazurile practice care vor fi

analizate n acest capitol, situaia este modelat matematic printr-ungraf orientat, vom folosi, pentru simplificarea expunerii, denumireade graf n locul celei de graf orientat iar n cazul n care graful esteneorientat vom specifica acest fapt la momentul respectiv.

2. Moduri de reprezentare ale unui grafA. O prim modalitate de reprezentare este listarea efectiv a

tuturor nodurilor i a arcelor sale.B. Putem reprezenta graful dnd pentru fiecare nod mulimea

nodurilor cu care formeaz arce n care el este pe prima

poziie.C. Putem reprezenta geometric graful, printr-un desen n plan,


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reprezentnd fiecare nod printr-un punct(cercule) i fiecarearc printr-un segment de curb care are ca extremitinodurile arcului i pe care este trecut o sgeat orientatde la nodul iniial spre cel final.

D. Putem folosi o reprezentare geometric n care nodurile

sunt reprezentate de dou ori, n douiruri paralele, de lafiecare nod din unul din iruri plecnd sgei spre nodurilecu care formeaz arce n care el este pe prima poziie, depe al doilea ir (reprezentarea prin coresponden).

E. Un graf poate fi reprezentat printr-o matrice ptraticboolean, de dimensiune egal cu numrul de noduri, ncare o poziie aij va fi 1 dac exist arcul (xi,xj) i 0 n cazcontrar, numit matricea adiacenelor directe.

F. Un graf poate fi reprezentat printr-o matrice ptratic latin,de dimensiune egal cu numrul de noduri, n care pe o

poziie aij va fi xixj dac exist arcul (xi,xj) i 0 n cazcontrar.

Exemplu: Dac n reprezentarea A avem graful G = (X,U),unde X = {x1, x2, x3, x4, x5, x6} i U = {(x1,x1), (x1,x2), (x1,x4),(x1,x5), (x2,x3), (x2,x4), (x2,x6), (x3,x1), (x3,x2), (x4,x5), (x5,x2),(x6,x4)}, atunci n celelalte reprezentri vom avea:

B x1 {x1, x2, x4, x5} C

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

02c400000088767565640000034c0000008676696577000003d4000000246c756d69000003f8000000146d6561730000040c0000002474656368000004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c0000080c7465787400000000436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e31000000000000000000000000000000000000000000000000000000

x2 {x3, x4, x6}x3

{x1

, x2}

x4 {x5}


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x5 {x2}x6 {x4}

D (reprezentarea prin coresponden)x1 x2 x3 x4 x5 x6



x1 x2 x3 x4 x5 x6

x3

x4

x5

x6

x2

x1

E F


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3. Concepte de baz n teoria grafurilor1. semigraf interior al unui nod xk: este mulimea arcelor =

{(xffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d485020200000000000000000000000000000000000000000000

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

65776c6574742d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e31000000000000000000000000000000000000000000000000000000j,xk)/ (xj,xk) U}care sunt incidente spre interior nodului xk;

2. semigraf exterior al unui nod xk: este mulimea arcelor ={(xffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f0210000

06d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d4850202000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001163707274000001500000003364657363000001840000006c77747074000001f000000014626b707400000204000000147258595a00000218000000146758595a0000022c000000146258595a0000024000000014646d6e640000025400000070646d6464000002c4000000887675656400000

34c0000008676696577000003d4000000246c756d69000003f8000000146d6561730000040c0000002474656368000


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004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c0000080c7465787400000000436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d70616e7900006465736300000000000000127352474220494543363139

36362d322e31000000000000000000000012735247422049454336313936362d322e31000000000000000000000000000000000000000000000000000000k,xi)/ (xk,xi) U}care sunt incidente spre exterior nodului xk;

3. semigradul interior al unui nod xk: este numrul arcelorcare sunt incidente spre interior nodului xk = cardinalul luii se noteaz cu ;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

03f8000000146d6561730000040c0000002474656368000004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c0000080c7465787400000000436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e31000000000000000000000000000000000000000000000000000000ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f0210000

06d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d4850202000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001163707274000001500000003364657363000001840000006c77747074000001f000000014626b707400000204000000147258595a00000218000000146758595a0000022c000000146258595a0000024000000014646d6e640000025400000070646d6464000002c4000000887675656400000

34c0000008676696577000003d4000000246c756d69000003f8000000146d6561730000040c0000002474656368000


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004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c0000080c7465787400000000436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d70616e7900006465736300000000000000127352474220494543363139

36362d322e31000000000000000000000012735247422049454336313936362d322e31000000000000000000000000000000000000000000000000000000

4. semigradul exterior al unui nod xk: este numrul arcelorcare sunt incidente spre exterior nodului xk = cardinalul luii se noteaz cu ;ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d53465400000000494543207352474200000

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

004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c0000080c746578740


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0000000436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e31000000000000000000000000

0000000000000000000000000000005. gradul unui nod xk: este suma semigradelor nodului xk: = +

;ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d48502020000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

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

65776c6574742d5061636b61726420436f6d70616e7900006465736300000000000000127352474220494543363139


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36362d322e31000000000000000000000012735247422049454336313936362d322e31000000000000000000000000000000000000000000000000000000ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f0210000

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

6. nod izolat: este un nod cu gradul 0;

7. nod suspendat: este un nod cu gradul 1;8. arce adiacente: arce care au o extremitate comun;9. drum ntr-un graf: o mulime ordonat de noduri ale grafului:

(x1, x2, ..., xk), cu proprietatea c exist n graf toate arcelede forma (xi,xi+1) i = 1,...,k-1;

10. lungimea unui drum: este numrul arcelor care l formeaz;11. drum elementar: un drum n care fiecare nod apare o

singur dat;12. drum simplu: un drum n care fiecare arc apare o singur

dat;

13. putere de atingere a unui nod xi X n graful G: numrulde noduri la care se poate ajunge din xi. Puterea deatingere se noteaz cu p(xi), 1 i n i evident p(xi) .ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d48502020000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

00001163707274000001500000003364657363000001840000006c77747074000001f000000014626b707400000204


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000000147258595a00000218000000146758595a0000022c000000146258595a0000024000000014646d6e640000025400000070646d6464000002c400000088767565640000034c0000008676696577000003d4000000246c756d69000003f8000000146d6561730000040c0000002474656368000

004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c0000080c7465787400000000436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e31000000000000000000000000000000000000000000000000000000

14. drum hamiltonian: un drum elementar care trece printoate nodurile grafului;

15. drum eulerian: un drum simplu care conine toate arcelegrafului;

16. lan: un drum n care arcele nu au neaprat acelai sens deparcurgere;

17. circuit: un drum n care nodul iniial coincide cu cel final;18. circuit elementar: un drum n care fiecare nod apare o

singur dat, cu excepia celui final, care coincide cu celiniial;

19. circuit simplu: un drum n care fiecare arc apare o singurdat;

20. circuit hamiltonian: un circuit care trece prin toate

nodurile grafului;21. ciclu: este un circuit n care arcele nu au neaprat acelaisens de parcurgere;

22. ciclu elementar: un ciclu n care fiecare nod apare osingur dat, cu excepia celui final, care coincide cu celiniial;

23. ciclu simplu: un ciclu n care fiecare arc apare o singur dat;

Observaie: ntr-un graf neorientat noiunile de drum i lansunt echivalente i de asemenea cele de circuit i ciclu.

24. graf parial al unui graf G = (X,U): este un graf G'(X,U') cuU' U;

25. subgraf al unui graf G = (X,): este un graf G'(X',') undeX' X i '(xi) = (xi) X' pentru orice xi X';

26. graf redus al unui graf G = (X,U): este un graf G*(X*,U*)

unde X* este format din mulimile unei partiii de muliminevide ale lui X, iar () exist doar dac i j i exist cel puinun arc n U, de la un nod din la un nod din

ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f0210000


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06d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d48502020000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

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


65776c6574742d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e31000000000000000000000000000000000000000000000000000000ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d4850

2020000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000


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00001163707274000001500000003364657363000001840000006c77747074000001f000000014626b707400000204000000147258595a00000218000000146758595a0000022c000000146258595a0000024000000014646d6e640000025400000070646d6464000002c4000000887675656400000

34c0000008676696577000003d4000000246c756d69000003f8000000146d6561730000040c0000002474656368000004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c0000080c7465787400000000436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e31000000000000000000000000000000000000000000000000000000ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d4850202000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001163707274000001500000003364657363000001840000006c77747074000001f000000014626b707400000204000000147258595a00000218000000146758595a0000022c000000146258595a0000024000000014646d6e64000002

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

27. graf tare conex: este un graf n care ntre oricare dounoduri exist cel puin un drum;

28. graf simplu conex: este un graf n care ntre oricare dounoduri exist cel puin un lan;

x1 x2 x3 x4 x5 x6x1 1 1 0 1 1 0x2 0 0 1 1 0 1x3 1 1 0 0 0 0

x4 0 0 0 0 1 0


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x5 0 1 0 0 0 0x6 0 0 0 1 0 0

x1 x2 x3 x4 x5 x6x1 x1

x1

x1x2

0x1x4

x1x5

0

x20 0

x2x3

x2x4

0x2x6

x3 x3x1

x3x2

0 0 0 0

x40 0 0 0

x4x5

0

x50

x5x2

0 0 0 0

x60 0 0

x6x4

0 0

Observaie: Pentru grafuri neorientat noiunile de tare conex isimplu conex sunt echivalente, graful numindu-se doar conex;29. component tare conex a unui graf G = (X,U): este un

subgraf al lui G care este tare conex i nu este subgrafulnici unui alt subgraf tare conex al lui G (altfel spus, ntreoricare dou noduri din component exist cel puin un

drum

i nu mai exist

nici un nod n afara componenteilegat printr-un drum de un nod al componentei).4. Gsirea drumurilor ntr-un graf orientat

Dac privim graful ca imagine a unui sistem, nodurilereprezentnd componentele sistemu-lui, atunci o interpretareimediat a unui arc (xi,xj) este c, componenta xi influeneaz directcomponenta xj. Dac nodurile au semnificaia de stri posibile aleunui sistem atunci un arc (x

i,xj) semnific faptul c sistemul poate

trece direct din starea xi n starea xj. n ambele cazuri se vede cavem de-a face doar cu informaii despre legturi directe; totui,chiar dac o component xi nu influeneaz direct componenta xj eao poate influena prin intermediul altor componente, existnd un irde componente intermediare: x1 x2 ,..., xk, fiecare influennd-odirect pe urmtoarea i xi direct pe x1 iar xk direct pe xj. Astfel, dacdintr-o stare xi nu se poate trece direct ntr-o stare xj s-ar puteatotui n mai multe etape, prin alte stri intermediare. Deoarecegsirea acestor influene sau treceri posibile este de obicei foarteimportant iar pentru un sistem cu mii sau zeci de mii decomponente acest lucru nu mai poate fi fcut "din ochi", este


14/151

necesar formalizarea noiunii de "influene" i "treceri" posibile, nuneaprat directe. Acest lucru a i fost fcut mai sus, deoarece esteevident c "xi influeneaz xj" sau "din starea xi se poate trece nstarea xj" este echivalent cu existena n graf a unui drum de la

nodul xi la nodul xj.n continuare vom da un algoritm prin care putem gsi toatedrumurile dintr-un graf orientat cu un numr finit de noduri.

Pasul 1. Se construiete matricea boolean a adiacenelor directecorespunztoare grafului, notat cu A. n aceasta se afl,evident, toate drumurile de lungime 1.

Este interesant de vzut ce legtur exist ntre aceastmatrice i drumurile de lungime 2. Fie dou noduri xii xj oarecare

din graf. Existena unui drum de lungime 2 ntre ele presupuneexistena unui nod xk, din graf, cu proprietatea c exist att arcul(xi,xk) ct i arcul (xi,xk). Pentru a vedea dac acesta exist, lum pernd fiecare nod al grafului i verificm dac exist sau nu ambelearce ((xi,xk) i (xi,xk)). Aceasta este echivalent cu a verifica dac, nmatricea boolean a adiacene-lor directe, exist vreun indice kastfel nct elementul k al liniei i i elementul k al coloanei j s fieambele egale cu 1. Dac folosim operaiile algebrei booleene deadunare i nmulire:

ffd8ffe000104a4649460001

020100c800c80000ffe20c584943

0 1 ffd8ffe000104a4649460001

020100c800c80000ffe20c584943

0 1


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435f50524f46

494c4500010100000c484c696e6f021000006d6e747

25247422058595a2007ce0

00200090006003100006

1637

435f50524f46

494c4500010100000c484c696e6f021000006d6e747

25247422058595a2007ce0

00200090006003100006

1637


16/151

3704d53465

400000000494543207352474200000000000000000

00000000000f6d600010000

0000d32d485020200000

0000

3704d53465

400000000494543207352474200000000000000000

00000000000f6d600010000

0000d32d485020200000

0000


17/151

0000000000

000000000000000000000000000000000000000000

0000000000000000000000

00000000000011637072

7400

0000000000

000000000000000000000000000000000000000000

0000000000000000000000

00000000000011637072

7400


18/151

0001500000

003364657363000001840000006c77747074000001f

000000014626b707400000

204000000147258595a0

0000

0001500000

003364657363000001840000006c77747074000001f

000000014626b707400000

204000000147258595a0

0000


19/151

2180000001

46758595a0000022c000000146258595a000002400

0000014646d6e640000025

400000070646d6464000

002c

2180000001

46758595a0000022c000000146258595a000002400

0000014646d6e640000025

400000070646d6464000

002c


20/151

4000000887

67565640000034c0000008676696577000003d4000

000246c756d69000003f800

0000146d656173000004

0c00

4000000887

67565640000034c0000008676696577000003d4000

000246c756d69000003f800

0000146d656173000004

0c00


21/151

0000247465

6368000004300000000c725452430000043c000008

0c675452430000043c0000

080c625452430000043c

0000

0000247465

6368000004300000000c725452430000043c000008

0c675452430000043c0000

080c625452430000043c

0000


22/151

080c746578

7400000000436f70797269676874202863292031393

938204865776c6574742d5

061636b61726420436f6d

7061

080c746578

7400000000436f70797269676874202863292031393

938204865776c6574742d5

061636b61726420436f6d

7061


23/151

6e79000064

657363000000000000001273524742204945433631

3936362d322e3100000000

00000000000000127352

4742

6e79000064

657363000000000000001273524742204945433631

3936362d322e3100000000

00000000000000127352

4742


24/151

2049454336

313936362d322e3100000000000000000000000000

0000000000000000000000

000000

2049454336

313936362d322e3100000000000000000000000000

0000000000000000000000

000000

0 0 1 0 0 01 1 1 1 0 1

atunci verificrile de mai sus sunt echivalente cu a verifica dac

elementul de pe poziia (i,j) din A2 este egal cu 1. Valoarea 1 spunedoar c exist cel puin un drum de lungime 2 de la x

ila x

j. Dac

dorim s vedem i cte sunt, vom folosi regulile de nmulire i


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adunare obinuit.De asemenea, se poate observa c existena unui drum de

lungime 3 de la xi la xj presupune existena unui nod xk astfel ncts existe un drum de lungime 2 de la xi la xki un arc de la xk la xj,

care este echivalent cu a verifica dac exist vreun indice k astfelnct elementul k al liniei i din matricea A2i elementul k al coloaneij din A sunt ambele egale cu 1 sau, mai simplu, dac elementul (i,j)

din A3 este 1.Din cele de mai sus se observ c existena drumurilor de

lungime k este dat de valorile matricei Ak, dac s-au folosit regulile

algebrei booleene i numrul lor este dat de Ak, dac s-au folositregulile obinuite.

Pasul 2. Vom calcula succesiv puterile lui A pn la puterea An-1

Dac ntre nodurile xi i xj exist un drum de lungime natunci el va conine un numr de noduri mai mare sau egal nu n+1i, cum n graf sunt doar n vrfuri, este clar c cel puin unul, szicem xk, va aprea de dou ori. Vom avea n acest caz un drum dela xi pn la prima apariie a lui xk, i un drum de la ultima apariie alui xk la xj. Eliminnd toate nodurile dintre prima apariie a lui xki

ultima apariie a sa vom obine un drum de la xi la xj, n care xkapare o singur dat. Aplicnd acest procedeu pentru toate nodurilecare apar de mai multe ori pe drum, vom obine un drum de la xi laxj, n care fiecare nod apare o singur dat (deci un drumelementar), care are evident cel mult n-1 arce. n concluzie, dacexist vreun drum de la xi la xj atunci existi un drum elementar i,

deci, va exista o putere a lui A, ntre A1i An-1, n care poziia (i,j)este diferit de 0. Pentru deciderea existenei unui drum ntreoricare dou noduri este suficient, deci, calcularea doar a primelor

n-1 puteri ale lui A.

Pasul 3. Se calculeaz matricea D = A + A2 + ... + An-1

Dac ne intereseaz doar existena drumurilor dintre noduri,nu i numrul lor, vom folosi nmulirea i adunarea boolean iconform observaiei de mai sus:

dij =

ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e747252474


26/151

22058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d485020200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000116370727400000150000000336465736300000184000

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

00000000000000000000000000000000000000000000

n acest caz, observnd c:

A(A + I)n2 = CA 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

2d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e310000000000

00000000000000000000000000000000000000000000ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d485020200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000116370727400000150000000336465736300000184000

0006c77747074000001f000000014626b707400000204000000147258595a00000218000000146758595a0000022c000000146258595


27/151

a0000024000000014646d6e640000025400000070646d6464000002c400000088767565640000034c0000008676696577000003d4000000246c756d69000003f8000000146d6561730000040c0000002474656368000004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c0000080c74657874000000

00436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e310000000000

000000000000000000000000000000000000000000002 + Affd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d485020200000000000000000000000000000000


000000000000000000000000000000000000000000003 + ... + CAffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d485020200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000116370727400000150000000336465736300000184000

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

0000000012735247422049454336313936362d322e31000000000000000000000000000000000000000000000000000000n1 = A +


28/151

A2 + A3 + ... + An-1 = D

rezult c e suficient s calculm doar puterea n-2 a matricei A + I iapoi s-o nmulim cu A. Avantajul acestei metode, n ceea ce privete

economia de timp, este susinut i de urmtoarea observaie: dac Dconine toate perechile de arce ntre care exist drum atunci:

D = (A + A2 + ... + An-1) + An + An+1 + ... + An+k = D oricare arfi k 0

A(A + I)n2+k = (A + A2 + ... + An-1) + An + An+1 + ... +

An+k-1 = D = A(A + I)n2

A(A + I)n2+k = A(A + I)n2 oricare ar fi k 0

deci de la puterea k = n-2 toate matricile Ak sunt egale. Putem,deci, calcula direct orice putere a lui A+I mai mare sau egal cu n-1

(de exemplu calculnd (A+I)2, (A+I)4, (A+I)8, ..., , r fiind primaputere a lui 2 pentru care 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

000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e310000000000

00000000000000000000000000000000000000000000r n-2).Procedeul de mai sus nu asigur dect aflarea faptului dac

exist sau nu drum ntre dou noduri, eventual ce lungime are icte sunt de aceast lungime. Totui, n problemele practice cel maiimportant este s tim care sunt efectiv aceste drumuri. Deoarecetoate drumurile pot fi descompuse n drumuri elementare i nproblemele practice n general acestea sunt cele care intereseaz,paii urmtori ai algoritmului vor fi dedicai gsirii lor. Pentru gsirea

acestora se folosete reprezentarea grafului prin matricea latin dela cazul F.


29/151

Pasul 4. Construim matricea latin L asociat grafului, unde:

lij =



00000000000000000000000000000000000000000000i matriceaffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50

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 , definitprin:

ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247

422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6


30/151

000100000000d32d4850202000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001163707274000001500000003364657363000001840000006c77747074000001f000000014626b707400000204000000147258595a00000218000000146758595a0000022c00000014625859

5a0000024000000014646d6e640000025400000070646d6464000002c400000088767565640000034c0000008676696577000003d4000000246c756d69000003f8000000146d6561730000040c0000002474656368000004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c0000080c7465787400000000436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e31000000000

000000000000000000000000000000000000000000000 =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

000246c756d69000003f8000000146d6561730000040c0000002474656368000004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c0000080c7465787400000000436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e310000000000

00000000000000000000000000000000000000000000numit matricea latin redus.

Gsirea unui drum de lungime 2 de la xi la xj presupunegsirea unui nod cu proprietatea c exist arcele (x

i,x

k) i (x

k,xj) i

memorarea vectorului (xi, xk, xj). Aceasta este echivalent cu a gsiun indice k astfel nct elementul de pe poziia k a liniei i, dinmatricea L, s fie xi,xki elementul de pe poziia k al coloanei j, dinmatriceaffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d485020200000000000000000000000000000000

000000000000000000000000000000000000000000000000000000000000000116370727400000150000000336465736300000184000


31/151

0006c77747074000001f000000014626b707400000204000000147258595a00000218000000146758595a0000022c000000146258595a0000024000000014646d6e640000025400000070646d6464000002c400000088767565640000034c0000008676696577000003d4000000246c756d69000003f8000000146d6561730000040c0000002474

656368000004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c0000080c7465787400000000436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e31000000000000000000000000000000000000000000000000000000 , s fie xj.Vom nmuli deci matricea L cu matriceaffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e747252474

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

0000043c0000080c625452430000043c0000080c7465787400000000436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e31000000000000000000000000000000000000000000000000000000 ~, folosindns nite reguli de calcul speciale, numite nmulire i adunare latin.

Definiia 1: Se numete alfabet o mulime de semne numitesimboluri sau litere {si/iI} unde I este o mulime

oarecare de indici, finit sau nu.Definiia 2: Se numete cuvnt un ir finit de simboluri notat s.ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d4850202000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001163707274000001500000003364657363000001840000006c77747074000001f000000014626b707400000204000000147258595a00000218000000146758595a0000022c000000146258595

a0000024000000014646d6e640000025400000070646d6464000002c400000088767565640000034c0000008676696577000003d4000


32/151

000246c756d69000003f8000000146d6561730000040c0000002474656368000004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c0000080c7465787400000000436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d70616e790000646573630000000000

000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e31000000000000000000000000000000000000000000000000000000

Definiia 3: Se numete nmulire latin o operaie definit pemulimea cuvintelor unui alfabet, notat "", astfel: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


ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f02

1000006d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d4850202000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001163707274000001500000003364657363000001840000006c77747074000001f000000014626b707400000204000000147258595a00000218000000146758595a0000022c000000146258595a000002400000

0014646d6e640000025400000070646d6464000002c400000088767565640000034c00000086766965


33/151

77000003d4000000246c756d69000003f8000000146d6561730000040c0000002474656368000004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c0000080c7465787400000000436f70797269676874202863292

031393938204865776c6574742d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e310000000000000000000000000000

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

000000146d6561730000040c0000002474656368000004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c0000080c7465787400000000436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e31000000000000000000000

000000000000000000000000000000000ffd8ffe000104a4649460001020100c800c80000ffe2

0c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d4850202000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001163707274000001500000003364657363000001840000006c77747074000001f000000014626b707400000204000000147258595a000002180000

00146758595a0000022c000000146258595a0000024000000014646d6e640000025400000070646d64


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64000002c400000088767565640000034c0000008676696577000003d4000000246c756d69000003f8000000146d6561730000040c0000002474656368000004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c000

0080c7465787400000000436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e31000000000000000000000

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

676696577000003d4000000246c756d69000003f8000000146d6561730000040c0000002474656368000004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c0000080c7465787400000000436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e31000000000000000000000

000000000000000000000000000000000

(produsul a dou cuvinte se obine prin concatenarea lor)nmulirea latin este asociativ, are ca elementneutru cuvntul vid, nu e comutativi un elementeste inversabil doar dac este cuvntul vid.

Definiia 3: Se numete adunare latin o funcie definit pemulimea cuvintelor unui alfabet cu valori nmulimea parilor mulimi cuvintelor, notat "" astfel:ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce

00020009000600310000616373704d5346540000000049454320735247420000000000000000000000


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000000f6d6000100000000d32d4850202000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001163707274000001500000003364657363000001840000006c77747074000001f000000014626

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

ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d4850202000000000000000

000000000000000000000000000000000000000000000000000000000000000000000000000000001163707274000001500000003364657363000001840000006c77747074000001f000000014626b707400000204000000147258595a00000218000000146758595a0000022c000000146258595a0000024000000014646d6e640000025400000070646d6464000002c400000088767565640000034c0000008676696577000003d4000000246c756d69000003f8000000146d6561730000040c0000002474656368000004300000000c725452430000043c0000080c675452430

000043c0000080c625452430000043c0000080c7465787400000000436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e310000000000000000000000000000

00000000000000000000000000ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c

696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d53465400000


36/151

00049454320735247420000000000000000000000000000f6d6000100000000d32d485020200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000116370727400000150000000336465736300


000000000000000000000000000000000ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000

000000f6d6000100000000d32d4850202000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001163707274000001500000003364657363000001840000006c77747074000001f000000014626b707400000204000000147258595a00000218000000146758595a0000022c000000146258595a0000024000000014646d6e640000025400000070646d6464000002c400000088767565640000034c0000008676696577000003d4000000246c756d69000003f8000000146d6561730000040c00000024746563680


000000000000000000000000000000000 =ffd8ffe000104a4649460001020100c800c80000ffe2

0c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce


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00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d4850202000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000


000000000000000000000000000000000(suma a dou cuvinte este mulimea format din cele dou

cuvinte)

Pasul 5. Se calculeaz succesiv matricile:

L2 = Lffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d4850202000000000000000000000000000000000000000000000000000000000000000000000


3938204865776c6574742d5061636b61726420436f6d70616e79000064657363000000000000001273524742


38/151

2049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e31000000000000000000000000000000000000000000000000000000ffd8ffe000104a4649460001020100c800c80000ffe20c5

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

00000000000000000000000000000000000000000000000000 , L3 = L2


0000003364657363000001840000006c77747074000001f000000014626b707400000204000000147258595a00000218000000146758595a0000022c000000146258595a0000024000000014646d6e640000025400000070646d6464000002c400000088767565640000034c0000008676696577000003d4000000246c756d69000003f8000000146d6561730000040c0000002474656368000004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c0000080c7465787400000000436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d

70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000


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000012735247422049454336313936362d322e31000000000000000000000000000000000000000000000000000000ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f

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

000000 , ... ,Lk+1 = Lkffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6000100000000d32d4850202000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001163707274000001500000003364657363000001840000006c777470740000

01f000000014626b707400000204000000147258595a00000218000000146758595a0000022c000000146258595a0000024000000014646d6e640000025400000070646d6464000002c400000088767565640000034c0000008676696577000003d4000000246c756d69000003f8000000146d6561730000040c0000002474656368000004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c0000080c7465787400000000436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d70616e79000064657363000000000000001273524742

2049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e310000


40/151

00000000000000000000000000000000000000000000000000ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce00020009

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

folosind operaiile de nmulire i adunare latin, alfabetul fiindmulimea nodurilor grafului, unde operaia de nmulire este uormodificat, produsul dintre dou elemente ale matricilor fiind 0, dacunul dintre ele este 0 sau au un nod comun i este produsul latin allor, n caz contrar.

Din felul cum a fost construit, matricea Lk va conine toatedrumurile elementare de lungime k. Cum un drum elementar poateavea cel mult n noduri (cte are graful cu totul) rezult c:

primele n-1 puteri ale lui L conin toate drumurile

elementare din graf; puterile lui L mai mari sau egale cu n au toate elementeleegale cu 0;

matricea Ln-1 conine toate drumurile hamiltoniene din graf(dac exist).

Observaie: Deoarece obinerea matricii D prin metoda de maisus presupune un volum foarte mare de calcule (de exemplu, dacgraful are 100 de noduri, ridicarea unei matrici de 100100 laputerea 100) pentru obinerea acesteia se poate aplica i urmtorulalgoritm:


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Pas 1. Se construiete matricea de adiacen A;Pas 2. Pentru fiecare linie i se adun boolean la aceasta toate

liniile j pentru care aij = 1.Pas 3. Se reia pasul 2 pn cnd, dup o aplicare a acestuia,

matricea rmne aceeai (nu mai apare nici un 1)Ultima matrice obinut este matricea drumurilor D numiti

matricea conexiunilor totale.Aceast metod, dei mai simpl nu spune nsi care sunt

aceste drumuri, pentru gsirea lor aplicndu-se, de exemplu,nmulirea latin

5. ARBORI. Problema arborelui de valoare optim

n acest subcapitol grafurile vor fi considerate neorientate.

5.1. Noiunea de arboreUn arbore este un graf neorientat, finit, conex i fr cicluri.

Grafurile din fig. 4.1. sunt arbori.




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Studiul arborilor este justificat de existena n practic a unui

numr mare de probleme care pot fi modelate prin arbori. Dintreacestea amintim:

1. construirea unor reele de aprovizionare cu ap potabil (saucu energie electric sau termic etc) a unor puncte deconsum, de la un punct central;

2. construirea unor ci de acces ntre mai multe puncteizolate;

3. desfurarea unui joc strategic;4. luarea deciziilor n mai multe etape (arbori decizionali);5. evoluii posibile ale unui sistem pornind de la o stare iniial;6. construirea unei reele telefonice radiale, a unei reele de

relee electrice;7. legarea ntr-o reea a unui numr mare de calculatoare;8. organigramele ntreprinderilor;9. studiul circuitelor electrice n electrotehnic (grafe de fluen

etc);10. schemele bloc ale programelor pentru calculatoare etc.

n toate problemele de mai sus se dorete ca, dintre muchiile

unui graf neorientat, s se extrag arborele optim din mulimeatuturor arborilor care pot fi extrai din graful dat.Deoarece definiia arborelui este dificil de aplicat pentru

deciderea faptului c un graf este arbore sau nu (i n special suntgreu de verificat conexitatea i mai ales existena ciclurilor) existmai multe caracterizri posibile ale unui arbore, acestea fiind datede teorema de mai jos:

Teorem. Dac H este un graf neorientat finit, atunciurmtoarele afirmaii sunt echivalente:

H este arbore;H nu conine cicluri i, dac se unesc printr-o muchie dou

noduri neadiacente, se formeaz un ciclu (i numai unul).Arborele este, deci, pentru o mulime de noduri dat, grafulcu numrul maxim de arce astfel nct s se pstrezeproprietatea c nu are cicluri);

H este conex i dac i se suprim o muchie se creeaz doucomponente conexe (arborele este graful conex cu numrulminim de arce);

H este conex i are n-1 muchii;

H este fr cicluri i are n-1 muchii;Orice pereche de noduri este legat printr-un lani numai unul.


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x1

a)

x1

x1

x1

x1

x1

x1

b

)

x1

x1

x1

x1

c)

Fi

gu


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ra4.

1

Demonstraie :

1) 2). ntre cele dou noduri adiacente noii muchii introduse existadeja un drum n fostul graf. Acest drum, mpreun cu noul arcva forma evident un ciclu i afirmaia 2) a fost demonstrat.

2)3). Pentru oricare dou vrfuri neunite printr-o muchie,adugnd muchia dintre cele dou vrfuri s-ar crea, conformipotezei, un ciclu care conine aceast muchie, deci doudrumuri ntre cele dou noduri, din care unul nu conine nouamuchie, adic n graful iniial exista un drum ntre cele dounoduri. Dac nu exist cicluri nseamn c ntre oricare dounoduri exist un singur drum. Pentru dou noduri unite printr-o muchie, aceasta este chiar drumul corespunztor celordou noduri. Dac suprimm aceast muchie ntre cele dounoduri nu va mai exista nici un drum, formndu-se doucomponente conexe.

3)4). Demonstraia se face prin inducie dup n = numrul denoduri ale grafului. Pentru n=2 este evident. Presupunem

afirmaia adevrat pentru toate grafurile cu cel mult nnoduri. Dac graful are n+1 noduri, prin suprimarea uneimuchii se formeaz dou componente conexe fiecare avndcel mult n noduri (n1 n, n2 n i n1 + n2 = n+1) i deci aun1 1 respectiv n2 1 muchii. n concluzie graful iniial aavut (n1 1) + (n2 1) +1 = n1 + n2 1= (n+1)-1 muchii,ceea ce era de demonstrat.

4)5). Dac ar avea un ciclu atunci prin suprimarea unui arc alacestuia ar rmne de asemenea conex. Eliminm acest arcapoi repetm procedeul pentru graful parial rmas i tot aa

pn cnd nu mai rmne nici un ciclu. n acest momentgraful rmas este conex i nu are cicluri deci este arbore ideci are n-1 arce, n contradicie cu faptul c el avea n-1 arcenainte de a ncepe suprimarea arcelor;

5)6). Dac ntre dou noduri ar exista dou drumuri atunci acesteaar forma la un loc un ciclu. Deci ntre 2 noduri este cel multun drum. Dac ntre dou noduri nu ar exista nici un drum arfi cel puin dou componente conexe n graf, fiecare fiindarbore (pentru c nu exist cicluri) i deci fiecare ar avea unnumr de arce cu 1 mai mic dect numrul de noduri. Fcndadunarea, ar rezulta c n graf sunt strict mai puin de n-1arce.


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6)1). Dac H ar avea un ciclu, ntre dou noduri ale acestuia arexista dou lanuri, n contradicie cu ipoteza.

Presupunem c avem un graf pentru care am verificat dejadac este conex. Dac nu este atunci acesta, evident, nu are nici un

graf parial care s fie arbore.Presupunem de asemenea c fiecrei muchii i este asociat o

valoare real.

5.2. Algoritmi pentru gsirea arborelui de valoare optim

Vom da mai jos trei algoritmi pentru determinarea unui grafparial al grafului, care s fie arbore i pentru care suma valorilorarcelor sale s fie minim (sau maxim).

Toi algoritmii descrii n continuare extrag arborele princolectarea una cte una a muchiilor acestuia.

A. Algoritmul lui Kruskal

Pasul 1. Dintre toate muchiile grafului se alege muchia de valoareminim (maxim). Dac minimul este multiplu se alege lantmplare una din muchiile respective. Deoarece acest "lantmplare" trebuie cumva tradus n limbajul calculatorului,n cazul implementrii unui program bazat pe acest algoritm,vom perturba din start valorile muchiilor, la k muchii cuaceiai valoare V adunnd respectiv valorile , 2, ... , k,

unde este foarte mic (n orice caz, k mai mic dectdiferena dintre valoarea acestor arce si valoarea imediatsuperioar a unui arc), pozitiv.

Pasul 2. Dintre toate muchiile rmase, se alege cea de valoareminim (maxim);

Pasul 3. Dintre toate muchiile rmase, se alege cea de valoareminim (maxim), astfel nct s nu se formeze cicluri cu celedeja alese;

Pasul 4. Se reia algoritmul de la pasul 3 pn se colecteaz n-1muchii.

Dei s-a demonstrat c algoritmul gsete ntotdeauna arboreleoptim, el are dezavantajul c este foarte laborios (de fiecare dattrebuie calculat minimul unei mulimi mari sau foarte mari existsituaii n practic n care graful are sute de mii de arce) i, n plus,trebuie aplicat un algoritm special ca s respectm condiia de a nuse forma cicluri, la alegerea unui nou arc.

O metod posibil este ca, dup adugarea fiecrui arc, s sempart graful n componente conexe i s alegem apoi un arc carenu are ambele extremitile n aceeai component conex.

De asemenea este clar c, n cazul existenei arcelor de valoriegale, deoarece se alege la ntmplare, exist mai multe variante de


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evoluie a alegerii arcelor. Totui, cu toate c pot fi mai multe grafurila care se poate ajunge prin acest algoritm, ele vor avea toateaceeai valoare (minima (sau maxima) posibil).

B. Algoritmul lui Sollin

Pasul 1. Pentru fiecare nod se alege muchia adiacent de valoareminim (maxim).

Pasul 2. Se evideniaz componentele conexe, existente n grafulparial format din arcele alese pn n acest moment.

Pasul 3. Pentru fiecare component conex se alege muchiaadiacent de valoare minim (maxim). Prin muchieadiacent unei componente conexe nelegem o muchie careare o singur extremitate printre nodurile componenteirespective.

Pasul 4. Se reia algoritmul de la pasul 2 pn rmne o singurcomponent conex. Aceasta este arborele optim cutat.

Acest algoritm asigur de asemenea gsirea arborelui optim,necesit mult mai puine calcule (la fiecare alegere se calculeazminimul doar pentru muchiile adiacente unui singur nod), evitautomat formarea ciclurilor, dar, pentru grafuri foarte mari, la unmoment dat pot exista att de multe componente conexe caretrebuie memorate succesiv, nct calculul devine greoi sau, pe

calculator, depete posibilitile de memorare ale calculatorului.

C. O variant a algoritmului lui Kruskal

Pasul 1. Dintre toate muchiile grafului se alege cea de valoareminim (maxim);

Pasul 2. Dintre toate muchiile adiacente componentei conexeformat din arcele alese pn n acest moment, se alege ceade valoare minim (maxim);

Pasul 3. Se reia pasul 2 pn se colecioneaz n-1 muchii.

Algoritmul are toate avantajele algoritmului lui Sollin i, nplus, lucreaz cu o singur component conex, fiind mult mai uorde implementat pe calculator i mult mai rapid n execuie.

Exemplu: Administraia unei localiti montane a hotrtconstruirea unor linii de teleferic care s lege oraul de cele 8 puncteturistice importante din jurul acestuia. n urma unui studiu au fost

puse n evidena toate posibilitile i costurile de conectare aobiectivele turistice ntre ele i cu oraul, acestea fiind prezentate n


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figura 4.2.Se cere gsirea variantei de construcie de cost minim, care s

asigure accesul din ora la oricare din obiectivele turistice.

ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f5

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

Rezolvare

Condiia de cost minim implic dou obiective:1. S se construiasc minimul de arce necesare;2. S se construiasc cele mai ieftine legturi.

Referitor la numrul de arce necesar, facem observaia c,dac din ora se va putea ajunge la orice obiectiv turistic, atunci seva putea ajunge i de la orice staiune la oricare alta (trecnd prin

ora), deci trebuie ca arcele alese pentru construcie s formeze laun loc un graf conex.


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n concluzie, cutm un graf parial conex cu un numr minimde arce, adic un arbore. n plus, suma costurilor arcelor sale trebuies fie minim. Vom aplica pe rnd cei trei algoritmi pentru gsireaacestuia:

A. Kruskal

La primul pas poate fi ales unul din arcele OP3 sau OP7, eleavnd valoarea minim 2. Putem alege oricum primul arc dintre celedou pentru c la al doilea pas va fi ales cellalt.

La pasul trei poate fi ales unul din arcele OP5, OP6 sau P1P6care au valoarea minim 3. Nici n acest caz nu are vre-o importan ordinea alegerii, deoarece pot fi alese succesiv toate trei fr a seforma nici un ciclu.

Al aselea arc poate fi ales dintre arcele P4P5i P1P2, care au

valoarea minim 4. Nici n acest caz nu are vre-o importan ordineaalegerii, deoarece pot fi alese succesiv ambele, fr a se forma niciun ciclu.

Urmtoarea valoare disponibil a unui arc este 5, dar arcul optnu poate fi ales dintre arcele OP1, P6P7, dei au valoarea minim 5.Arcul OP1 nu poate fi ales deoarece s-ar forma ciclul OP1P6, iarP6P7 ar duce la ciclul OP6P7. Urmtoarea valoare minim este 6,pentru arcul P5P7 dar nu poate fi ales deoarece se formeaz ciclulOP5P7.

Valoarea urmtoare, 7, o au arcele OP4, P2P3i P5P8. OP4 nupoate fi ales deoarece s-ar forma ciclul OP5P4. Arcul P2P3 nu poatefi ales deoarece s-ar forma ciclul OP6P1P2P3. Arcul P5P8 nuformeaz nici un ciclu i el va fi al optulea arc ales. n acest caz,deoarece s-au adunat 8 arce ntr-un graf cu 9 noduri, am obinutgraful cutat.

Acest arbore este reprezentat n figura 4.3.

O

P2

P1

P6

P3


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P7

P4

P5

P8

9

7

8

8

4

5

3

3

2

2

5

7

3

4

8

7

6

8

8


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Figu

ra4.2

9

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0000246c756d69000003f8000000146d6561730000040c0000002474656368000004300000000c725452430000043c0000080c675452430000043c0000080c625452430000043c0000080c7465787400000000436f70797269676874202863292031393938204865776c6574742d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e31000000000000000000000000000000000000000000000000000000


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B. Sollin

Vomalege:

pentrunodul O

arculOP3

pentrunodulP1

arculP1P6

pentrunodulP2

arculP1P2

pentrunodulP3

arculOP3

pentrunodulP4

arculP4P5

pentrunodulP5

arculOP5

pentrunodulP6

arculP1P6

pentrunodul

P7

arculOP7

pentrunodulP8

arculP5P8

Rezult graful parial:

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

30000043c0000080c625452430000043c0000080c7465787400000000436f70797269676874202863292031393938204865776c65747


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42d5061636b61726420436f6d70616e790000646573630000000000000012735247422049454336313936362d322e31000000000000000000000012735247422049454336313936362d322e31000000000000000000000000000000000000000000000000000000

Dup cum se vede, s-au format dou componente conexe: C1= {P1,P2,P6}

C2 ={O,P3,P4,P5,P7,P8}.

Vom alege: pentru C1 arcul OP6pentru C2 arcul OP6

i obinem o singur component conex, care este arborele cutat.

O

P2

P1

P6

P3

P7

P4

P


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5

P8

4

3

3

2

2

3

4

Figura4.3

7

O

P2

P1

P6

P3


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P7

P4

P5

P8

4

3

22

3

4

7

F

igura4.4

C. Varianta algoritmului lui KruskalSuccesiunea alegerii arcelor va fi:

1 OP3

2 OP7

3 OP6

4 OP5


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5 P1P6

6 P1P2

7 P4P5

8 P5P8

6. Cuplajul a dou mulimi disjuncte. Probleme de

afectare (de repartiie)n practica economic sunt foarte des ntlnite probleme n

care se dorete asocierea optim a elementelor unei mulimi X ={x1, x2, ... , xn} cu elementele unei alte mulimi Y = {y1, y2, ... ,ym} n condiiile unor limitri existente (i cunoscute) aleposibilitilor de asociere.

n general, fiecare asociere posibil xi yj aduce un anumitefect aij (profit, cost etc) care poate fi calculat i vom presupune ceste cunoscut.

Limitrile asupra asocierilor se traduc de obicei prin faptul c:

1. Un element xi poate fi asociat doar cu anumite elementedin Y i reciproc;

2. La sfrit, fiecrui element din X i s-a asociat cel mult unelement din Y i reciproc.

Asocierea optim presupune, de obicei, dou obiective:

1. S se fac maximul de asocieri;2. Suma efectelor asocierilor s fie maxim (sau minim, nfuncie de semnificaia acestora).

Reprezentarea geometric a situaiei de mai sus este un grafde forma:

ffd8ffe000104a4649460001020100c800c80000ffe20c584943435f50524f46494c4500010100000c484c696e6f021000006d6e74725247422058595a2007ce00020009000600310000616373704d5346540000000049454320735247420000000000000000000000000000f6d6


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Elemente de Teoria Grafurilor

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