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C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
Algoritmica grafurilor - Cursul 1
2 octombrie 2020
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C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
Cuprins
1 Descrierea cursuluiInformaµii legate de curs
2 Aplicaµii ale teoriei grafurilor
3 Vocabularul teoriei grafurilorDe�niµia grafului
4 Exerciµii pentru seminarul din s pt mâna urm toare
5 Exerciµii rezolvate (parµial)
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C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
Informaµii legate de curs
Pagina cursului:http://profs.info.uaic.ro/~olariu/curent/AG/AG.html
Obiective:Cursurile acoper noµiunile de baz în Teoria algoritmic a grafurilor.Cuno³tinµele acumulate vor � aplicate în proiectarea unor algoritmi e�-cienµi pentru rezolvarea problemelor de optimizare combinatorial .
Cursuri: E. F. Olariu (semianii A ³i E), C. Fr sinaru (semianul B).
Prezentarea online a notelor de curs, care vor � postate ca �³iere.pdf înaintea �ec rui curs.
Aceste �³iere vor conµine ³i exerciµiile pentru seminarii.
Cursurile ³i seminariile vor � organizate folosind Cisco Webex Meet-ings (aplicaµie pe care trebuie s v-o instalaµi).
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C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
Informaµii legate de curs
Seminarii: E. F. Olariu, C. Fr sinaru, P. Diac, A. Ioniµ
Fiecare seminar va conµine un scurt test asupra cuno³tinµelor dincursurile ³i seminariile anterioare.
În timpul seminariilor vor � discutate soluµii la exerciµiile propuse.
Fiecare seminar este dedicat unui num r de trei-patru exerciµii(postate în avans în notele de curs) cu scopul de a aprofunda con-ceptele introduse la curs.
Studenµii sunt încurajaµi s propun soluµii originale (prezentate pesuport electronic: pdf, jpg, png etc).
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Informaµii legate de curs
Notarea:
Seminar: teste (max. 2� 12 = 24p), activitatea la seminara (max.18pb) - max. 42 puncte.
Teme pentru acas : trei seturi de exerciµii, max. 16p �ecare - max.48 puncte.
Examen �nal scris (obligatoriu): ³ase exerciµii/probleme (�ecare acâte 10p) +10p din o�ciu - max. 70 puncte.
Pentru a susµine teza scris în sesiune sunt necesare cel puµin 30 depuncte din cele maximum 90p din activitatea de seminar ³i temele pentruacas .Din maximum 160p limita de promovare este de 80 puncte.
aDou r spunsuri corecte ³i la obiect = 1 punct.bCu aproximaµie.
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Informaµii legate de curs
Notarea:
Seminar. Testele sunt susµinute online în Webex Meetings subforma unor întreb ri la care trebuie dat un r spuns nu prea lung,dar justi�cat.
Teme pentru acas . Soluµiile pot � elaborate în echipe de pân ladoi studenµi. Soluµiile la aceste teme vor � scrise în LaTeX sauWord (Write) ³i trimise electronic (atât sursa cât ³i .pdf-ul). Bonus2 puncte pentru �ecare tem scris în LaTeX. Aceste soluµii vor �veri�cate cu un instrument anti-plagiat iar studenµii care copie vor� penalizaµi (pentru �ecare problem copiat punctajul va � de �2puncte).
Examen �nal scris (obligatoriu). Studenµii nu au voie s consultemateriale în timpul examenului.
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Informaµii legate de curs
Cuprinsul cursului:
Vocabularul teoriei grafurilor.
Probleme de drum: parcurgerea sistematic a grafurilor, drumuride cost minim, conexiune.
Arbori parµiali de cost minim: union-�nd, complexitateamortizat .
Teoria cuplajelor.
Fluxuri în reµele.
Reduceri polinomiale între probleme de decizie pe grafuri.
Abord ri ale problemelor NP-di�cile.
Grafuri planare.
Tree-decomposition.
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Aplicaµii ale teoriei grafurilor
Figure: Facebook/Twitter
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Aplicaµii ale teoriei grafurilor
Figure: A Facebook network of a few users
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Aplicaµii ale teoriei grafurilor
Grafurile sunt folosite pentru analiza reµelelor sociale/de ³tiri (precumFacebook sau Twitter) pentru a determina caracteristici cum ar �:
nivelul de conexiune, densitatea;
in�uenµa utilizatorilor asupra reµelei (centralitatea, potenµialul reµeleisociale (social networking potential);
nivelul de segmentare: m sura clusteriz rii;
robusteµea sau stabilitatea structural a reµelei.
Analiza reµelelor este folosit pentru
data mining ³i agregare, marketing;
analiza comportamentului ³i predicµie în reµea;
colectarea de informaµii ³i analiza securit µii (supravegherea teror-ismului ³i a crimei organizate) etc.
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C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
Aplicaµii ale teoriei grafurilor
În afara studiului reµelelor sociale (un subiect la mod ast zi) exist multe alte aplicaµii ale teoriei grafurilor:
Arbori parµiali de cost minim: pentru conectarea e�cient a unorpuncte de comunicaµie (e. g. în IT&C);
Drumuri ³i circuite Euleriene: Problema po³tasului chinez - s sedetermine un drum/circuit de cost minim care trece prin �ecaremuchie a unui graf o singur dat (pentru salubrizarea str zilor,expedierea po³tei sau a unor servicii, colectarea de³eurilor etc.);
Drumuri ³i circuite Hamiltoniene: vizitarea e�cient a unui num rde puncte (dintr-un ora³, dintr-o µar etc); Problema comis-voiajorului, Problema rut rii vehiculelor;
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Aplicaµii ale teoriei grafurilor
Colorarea grafurilor: colorarea h rµilor (a feµelor unui graf pla-nar), plani�carea cursurilor/seminariilor (problema orarului), plan-i�carea unei sesiuni, alocarea frecvenµelor radio mobile, alocarearegi³trilor de memorie.
Figure: Regiunile Franµei începând cu 2016
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Aplicaµii ale teoriei grafurilor
Cuplaje: probleme de asignare, în studii de chimie computaµional ³i de chimie matematic asupra materialelor organice.
Figure: Gale Shapley algorithm
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Aplicaµii ale teoriei grafurilor
Cuplaje stabile: repartizarea rezidenµilor în spitale, proceduri de ad-mitere în înv µ mântul superior, identi�carea repartiz rii optime aorganelor donate pentru transplant (e. g. rinichi), problema aloc riiproiectelor c tre studenµi etc.
Fluxuri de valoare maxim : determinarea nivelului de înc rcare înreµelele de transport pentru îmbun t µirea condiµiilor de tra�c, re-construcµia imaginilor din proiecµia razelor X (în tomogra�e), plan-i�carea procesoarelor paralele etc.
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Aplicaµii ale teoriei grafurilor
Aplicaµii în informatic
În managementul bazelor de date, bazele de date de tip graf folosescstructurile de date tip graf pentru stocare ³i interogare.
Graph rewriting systems folosite în veri�carea sistemelor software.
Quantum computation.
Modelarea documentelor web drept grafuri ³i clusterizarea acestora.
Aproximarea ³i compresia datelor.
Modelarea reµelelor de senzori cu grafuri (folosind de exemplu dia-grame Voronoi).
�i multe altele.
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De�niµia grafului
Notaµii
Pentru o mulµime �nit X :
jX j = card(X ) 2 N este cardinalul lui X ;
Dac jX j = k , atunci X este o k -mulµime;
2X = P(X ) este mulµimea p rµilor lui X : 2X = fY : Y � X g,j2X j = 2jX j;ÇX
k
å= fY : Y � X ; jY j = kg,
∣∣∣∣∣ÇX
k
å∣∣∣∣∣ = ÇjX jk å.Algoritmica grafurilor - Cursul 1 2 octombrie 2020 16 / 54
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De�niµia grafului
De�niµia 1
Un graf este o pereche G = (V ;E), unde:
V = V (G) o mulµime �nit , nevid ; este mulµimea nodurilor (vâr-furilor) lui G ;
E = E(G) este o submulµime a lui
ÇV (G)
2
å; este mulµimea muchi-
ilor lui G .
jGj = jV (G)j este ordinul grafului G , iar jE(G)j este dimensiunea sa.
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De�niµia grafului
De�niµia 2
Dac G = (V ;E) ³i e = uv = vu = fu ; vg 2 E este o muchie a lui G ,
spunem c
e conecteaz (sau leag ) nodurile u ³i v ;
nodurile u ³i v sunt adiacente sau u ³i v sunt vecine;
e este incident cu u ³i v ;
u ³i v sunt capetele (extremit µile) lui e .
Vecin tatea nodului u este NG(u) = fv 2 V (G) : uv 2 E(G)g.
Dou muchii e ³i f sunt adiacente dac au un cap t în comun: je\f j = 1.
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De�niµia grafului
Fie G = (V ;E) un graf ³i v 2 V .
Gradul unui nod v : dG(v) = jNG(u)j.
v este un nod izolat dac dG(v) = 0 ³i pendant (sau frunz ) dac dG(v) = 1.
o proprietate util : X
v2V
dG(v) = 2jE j:
Un graf poate � reprezentat în plan ca o �gur constând dintr-o mulµimede puncte (forme geometrice mici: puncte, cercuri, p trate etc) core-spunzând vârfurilor sale ³i curbe care conecteaz vârfurile corespunz -toare muchiilor din graf.
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C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
De�niµia grafului
Un exemplu:
b b
bb
b
b
bb
b
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 20 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
De�niµia grafului
Acela³i graf din nou:b b
bb
b
b
bb
b
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 21 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
De�niµia grafului
Putem ad uga etichete (nume, numere etc) ³i culori nodurilor ³i muchi-ilor obµinând reprezent ri vizuale mai bune.Mai jos sunt trei reprezent ri vizuale ale aceluia³i graf:
G = (f1; 2; 3; 4g; f12; 13; 14; 23; 24; 34g)
bc bc
bc bc
1
2 3
4 rsrs
rs rs
a b
c d
u
u
u
u
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 22 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
De�niµia grafului
De�niµia 3
Mulµime stabil (sau mulµime independent de noduri) în G = (V ;E):
S � V astfel încât
ÇS
2
å\ E = ?.
Cu alte cuvinte S � V este o mulµime stabil a lui G dac nu exist nicio muchie între nodurile sale.
Notaµie
Cardinalul maxim al unei mulµimi stabile a lui G este num rul de sta-bilitate (sau num rul de independenµ ) al lui G ³i se noteaz cu �(G).
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 23 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
De�niµia grafului
Exemplu
În urm torul graf (al lui Petersen) avem dou mulµimi stabile de cardinalmaxim (de ce?):
b b
b
b
b
b
b
b
b
b
b
b
b b
b
b bb
b b
b
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 24 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
De�niµia grafului
Mulµimi stabile: Problem de optimizare
P1 Input: G graf:Output: �(G) ³i o mulµime stabil S cu jS j = �(G):
Mulµimi stabile: Problem de decizie
SM Instanµ : G graf; k 2 N:Întrebare: Exist o mulµime stabil S în G , astfel încâtjS j > k?
NP-complet (Karp, 1972).
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 25 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
De�niµia grafului
De�niµia 4
Cuplaj (mulµime independent de muchii) înG = (V ;E): M � E astfel
încât pentru orice e ; f 2M , dac e 6= f , atunci e \ f = ?.
Cu alte cuvinte, M � E este un cuplaj dac oricare dou muchii ale salenu noduri în comun.
Notaµie
Cardinalul maxim al unui cuplaj în G este numit num rul de cuplaj(num rul de muchie-independenµ ) al lui G ³i se noteaz cu �(G).
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 26 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
De�niµia grafului
Exemplu
Pentr urm torul graf sunt marcate dou cuplaje cu ro³u ³i verde (aldoilea �ind de cardinal maxim - de ce?):
b
b
b
b
b
b
b
b
b
b
b
b
b
b
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 27 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
De�niµia grafului
Cuplaj maxim: Problem de optimizare
P2 Input: G graf:Output: �(G) ³i un cuplaj M cu jM j = �(G):
Edmonds (1965) a ar tat c P2 2 P.
Not
Problemele P1 ³i P2 sunt similare: în amândou se cere s se determineun membru de cardinal maxim al unei familii de mulµimi relativ la ungraf dat. Ce face ca ele s �e diferite?
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 28 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
De�niµia grafului
De�niµia 5
Pentru p 2 N, o p-colorare a grafului G = (V ;E) este o funcµie c :
V ! f1; : : : pg astfel încât c(u) 6= c(v) pentru �ecare uv 2 E .
Merit notat c mulµimea tuturor nodurilor cu aceea³i culoare este omulµime stabil (se mai nume³te clas de colorare). Deoarece noduriadiacente au culori diferite, o p-colorare corespunde unei partiµii a luiV cu cel mult p mulµimi stabile (sau clase de colorare).
Notaµie
Num rul cromatic al grafului G este cea mai mic valoare a lui p astfelîncât G are o p-colorare. Acest parametru se noteaz cu �(G). (�(G) 6
jG j - de ce?)
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 29 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
De�niµia grafului
Exemplu
Pentru urm torul graf avem marcate dou color ri (�(G) = 3!)
b
b
b
b
b
b
b b
b
b
bbbb
bbb
b
b
b
b
bb
bb
b
b
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 30 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
De�niµia grafului
Colorarea nodurilor: Problem de optimizare
P3 Input: G graf:Ouput: �(G) ³i o �(G)-colorare:
Colorarea nodurilor: Problem de decizie
COL Instanµ : G graf; p 2 N:Întrebare: Exist o p-colorare a lui G?
NP-complet pentru p > 3 (Karp, 1972).
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 31 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
De�niµia grafului
De�niµia 6
Pentru p 2 N, a p-muchie colorare a grafului G = (V ;E) este o funcµie
c : E ! f1; : : : pg astfel încât c(e) 6= c(f ) pentru orice e ; f 2 E cu
je \ f j = 1.
Se poate observa c , dat o p-muchie colorare, o mulµime de muchii cuaceea³i culoare este un cuplaj. Astfel, o p-muchie colorare corespundeunei partiµii a lui E cu cel mult p cuplaje.
Notaµie
Indexul cromatic al grafului G : cea mai mic valoare a lui p astfelîncât G are o p-muchie colorare. Acest parametru este notat cu �0(G).(�0(G) 6 jE(G)j - de ce?)
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 32 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
De�niµia grafului
Exemplu
Pentru urm torul graf am marcat o colorare a muchiilor (�0(G) = 3!)
bb
b
b
bb
b
b
b b
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 33 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
De�niµia grafului
Colorarea muchiilor: Problem de optimizare
P3 Input: G graf:Ouput: �0(G) ³i o �0(G)-colorare:
Colorarea muchiilor: Problem de decizie
COL Instanµ : G graf; p 2 N:Întrebare: Exist o p-muchie colorare of G?
NP-complet pentru p > 3 (Holyer, 1984).
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 34 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
De�niµia grafului
De�niµia 7
Dou grafuri G1 = (V1;E1) ³i G2 = (V2;E2) sunt izomorfe dac exist
o bijecµie ' : V1 ! V2 astfel încât pentru orice dou noduri u1; v1 2 V1,
u1 ³i v1 sunt adiacente în G1 (i. e., u1v1 2 E1) dac ³i numai dac '(u1)
³i '(v2) sunt adiacente in G2 (i. e., '(u1)'(u2) 2 E2).
Cu alte cuvinte, dou grafuri sunt izomorfe dac exist o bijecµie întremulµimile lor de noduri care induce o bijecµie între mulµimile lor demuchii.
Notaµie
G1�= G2.
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 35 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
De�niµia grafului
Exemplu
Grafurile de mai jos sunt izomorfe.
b b
b
b bb
b b
b b
a b
c d
e
1
2
3 4
5
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 36 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
De�niµia grafului
Testarea izomor�smului: Problem de optimizare
ISO Input: grafurile G1 ³i G2:
Ouput: Sunt G1 ³i G2 izomorfe?
Nu se ³tie dac este în P sau dac este NP-complet .Exist un algoritm cu timp de execuµie quasipolinomial (i. e., 2O((log n)c)
pentru un c > 0, Babai, 2015).
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 37 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
De�niµia grafului
Exemplu
Urm toarele dou grafuri au acelea³i ordine, dimensiuni ³i secvenµe alegradelor, dar nu sunt izomorfe (de ce?).
b
b
b
b b
b
b b
b b
b
b
b
b b
b b
b b
b
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 38 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
Exerciµii pentru seminarul din s pt mâna urm toare
Exerciµiul 1. Grafurile de mai jos sunt izomorfe?
Exerciµiul 1'. Determinaµi num rul minim de culori cu care poate �colorate nodurile grafului de mai jos:
Algoritmica grafurilor - Cursul 1 2 octombrie 2020 39 / 54
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Exerciµii pentru seminarul din s pt mâna urm toare
Exerciµiul 1�. Determinaµi num rul de stabilitate al grafului de mai jos:
Exerciµiul 2.Fie P(n) =" În orice graf cu cel puµin n noduri exist trei noduri dis-tincte care sunt dou câte dou adiacente sau dou câte dou neadia-cente." Ar taµi c 6 este cea mai mic valoare a lui n 2 N pentru careP(n) este adev rat .
Exerciµiul 3.
(a) Exist un graf cu gradele 3; 3; 3; 3; 5; 6; 6; 6; 6; 6; 6?
(b) Exist un graf bipartit cu gradele 3; 3; 3; 3; 3; 5; 6; 6; 6; 6; 6; 6; 6; 6?
(c) Exist un graf cu gradele 1; 1; 3; 3; 3; 3; 5; 6; 8; 9?Algoritmica grafurilor - Cursul 1 2 octombrie 2020 40 / 54
C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms * C. Croitoru - Graph Algorithms *
Exerciµii pentru seminarul din s pt mâna urm toare
Exerciµiul 4.Doi studenµi, L(azy) ³i T(hinky), trebuie s g seasc un drum particularîntre dou noduri �xate într-un graf rar G : jE(G)j = O(jV (G)j). Lconsider c , deoarece graful este rar, num rul de drumuri dintre celedou noduri trebuie s �e mic, ³i o soluµie lazy este s genereze (cubacktracking) toate aceste drumuri ³i apoi s reµin drumul dorit. T nueste de acord ³i d urm torul exemplu: �e H = K2 � Pn�1 (n > 3);ad ug m la H dou noduri noi x ³i y �ecare unite prin câte dou muchiicu câte una dintre cele dou perechi de noduri adiacente de grad 2 dinH .Graful obµinut, G , este rar dar num rul de drumuri dintre x ³i y estefoarte mare. Explicaµi lui L acest exemplu desenând G, ar tând c esterar ³i determinând num rul de drumuri dintre x ³i y .
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Exerciµii pentru seminarul din s pt mâna urm toare
Exerciµiul 5. O sesiune de examene trebuie plani�cat folosind urm -toarele speci�caµii: mulµimea examenelor este cunoscut ; �ecare studenttrimite o list a examenelor la care dore³te s �e examinat; �ecare exa-men are loc cu toµi studenµii înscri³i la examen (care este scris); �ecarestudent poate participa la cel mult un examen într-o aceea³i zi.Construiµi un graf cu ajutorul c ruia s raspundeµi la urm toarele între-b ri (prin determinarea unor parametri corespunz tori):
(a) Care este num rul maxim de examene care pot � organizate într-oaceea³i zi?
(b) Care este num rul minim de zile necesare organiz rii unei sesiuni?
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Exerciµii pentru seminarul din s pt mâna urm toare
Exerciµiul 6. (exerciµiul 5 - continuare)Un programator isteµ ³i îndemânatic se întreab dac cele dou prob-leme NP-hard din exerciµiul anterior nu ar putea � rezolvate în timppolinomial deoarece graful construit pare s aparµin unei clase specialede grafuri.
(a) Ar taµi c pentru orice graf dat, G , exist un input pentru problemade plani�care anterioar astfel încât graful construit (ca mai sus) s �e tocmai G .
Programatorul sugereaz urm toarea �abordare greedy� pentru ar spunde la a doua întrebare din exerciµiul 4: începând cu prima zi, seplani�c zilnic un num r maxim de examene (din mulµimea examenelorneplani�cate înc ), pân când toate examenele vor � plani�cate.
b) Ar taµi c aceast abordare greedy este gre³it printr-un contraex-emplu.
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Exerciµii pentru seminarul din s pt mâna urm toare
Exerciµiul 7. Un exemplu de optimizare a unui compilator este tehnicaaloc rii regi³trilor de memorie:cele mai folosite variabile sunt µinute înregistrii cu acces rapid ai procesorului pentru a le avea la îndemân înmomentul când compilatorul are nevoie de ele (pentru anumite operaµiiCPU).Construiµi un graf cu ajutorul c ruia s raspundeµi la urm toarele între-b ri (prin determinarea unor parametri corespunz tori):
(a) Care este num rul maxim de variabile de care nu este nevoie înacela³i timp?
(b) Care este num rul minim de regi³tri necesari acestor variabile?
Indicaµie: avem dou tipuri de obiecte: variabilele (cu valorile lor) ³ioperaµiile CPU (sau operatorii) care folosesc una sau mai multe variabile.
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Exerciµii pentru seminarul din s pt mâna urm toare
Exerciµiul 8. (exerciµiul 7 - continuare) Un student se întreab dac laîntreb rile de mai sus (care corespond unor probleme NP-hard) nu s-arputea da un r spuns în timp polinomial deoarece graful construit pares aparµin unei clase speciale de grafuri.
(a) Ar taµi c pentru orice graf dat, G , exist un input pentru problemade alocare a regi³trilor astfel încât graful construit (ca mai sus) s �e tocmai G .
Studentul sugereaz urm toarea �abordare greedy� pentru a r spundela a doua întrebare din exerciµiul 6: începând cu primul registru, sealoc �ec rui registru nefolosit un num r maxim de variabile (dintrecele nealocate înc ), pân când toate variabilele vor � alocate.
b) Ar taµi c aceast abordare greedy este gre³it printr-un contraex-emplu.
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Exerciµii pentru seminarul din s pt mâna urm toare
Exerciµiul 9. G este numit graf autocomplementar dac G ³i comple-mentul s u, G, sunt izomorfe (G �= G).
(a) Ar taµi c un graf autocomplementar este conex ³i c jG j � 0 sau1 mod 4.
(b) Determinaµi toate grafurile autocomplementare cu cel mult 7noduri.
(c) Ar taµi c pentru orice graf G exist un graf autocomplementar Hastfel încât G este subgraf indus al lui H .
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Exerciµii rezolvate (parµial)
Exerciµiul 1. Grafurile de mai jos sunt izomorfe?
Indicaµie: Dac grafurile sunt izomorfe atunci ele trebuie s aib acela³inum r de: noduri, muchii, componente conexe, circuite (induse) de oanumit lungime �xat etc.Soluµia 1.
Nodurile de grad 3 din graful G1 formeaz (induc) în G1 un circuitde lungime patru (f r corzi).
Ce formeaz nodurile de grad 3 din graful G2?
Soluµia 2.
Câte circuite de lungime patru f r corzi exist în graful G1 ³i câteîn graful G2?
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Exerciµii rezolvate (parµial)
Exerciµiul 1'. S se determine num rul cromatic al grafului de mai jos:
Soluµie (³i metod ).
Se caut în graf un subgraf complet (o clic ) de cardinal maxim (încazul nostru fy ; v ; z ; tg).
Se coloreaz nodurile acestui subgraf cu atâtea culori câte noduriconµine (c(y) = 1, c(v() = 2, c(z ) = 3, c(t) = 4).
Apoi se încearc extinderea acestei color ri ad ugând culori numaidac este neap rat nevoie: c(x ) poate � 2 etc
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Exerciµii rezolvate (parµial)
Exerciµiul 1�. S se determine num rul de stabilitate al grafului de maijos:
Soluµie (³i metod ).
Exist în graf mulµimi stabile de cardinal 2? Da: fu ; yg deci �(G) >
2.
Exist în graf mulµimi stabile de cardinal 3? ... dac r spunsul estea�rmativ, atunci �(G) > 3.
Etc.
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Exerciµii rezolvate (parµial)
Exerciµiul 2.Fie P(n) =" În orice graf cu cel puµin n noduri exist trei noduri dis-tincte care sunt dou câte dou adiacente sau dou câte dou neadia-cente." Ar taµi c 6 este cea mai mic valoare a lui n 2 N pentru careP(n) este adev rat .Soluµie.
Pentru 3 6 n 6 5 avem contraexemplele: P3 (drumul cu 3 noduri,f r corzi) C4 (circuitul cu 4 noduri, f r corzi) ³i, respectiv, C5
(circuitul cu 5 noduri, f r corzi).
Pentru n > 6, �e v0 2 V (G); jNG(v0)j > 3 or jNG(v0)j > 3.
Putem presupune c v0 are cel puµin vecini în G (de ce?), v1, v2,v3, atunci ace³ti vecini sunt �e doi câte doi ne-adiacenµi (formeaz o mulµime stabil ) sau exist o muchie vivj 2 E(G) (v0; vi ; vj va �o clic ).
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Exerciµii rezolvate (parµial)
Exerciµiul 3.
(c) Exist un graf cu gradele 1; 1; 3; 3; 3; 3; 5; 6; 8; 9?
Soluµie.
Se veri�c mai întâi dac suma gradelor este par (de ce?). (Sumaeste par .)
Dac ar exista un graf cu aceste grade ar avea 10 noduri, deci nodulde grad 9 este adiacent cu toate celelalte.
Putem ³terge din (ipoteticul) graf nodul de grad 9 ³i subgraful r masar avea gradele 0; 0; 2; 2; 2; 2; 4; 5; 7.
Nodurile de grad 0 pot � ignorate (de ce?). Exist un graf cu gradele2; 2; 2; 2; 4; 5; 7?
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Exerciµii rezolvate (parµial)
Exerciµiul 4. Soluµie.
H este un graf de ordin (2n � 2) (ladder graph - vezi �gura 5) ³ijG j = 2n ³i jE(G)j = 3n � 1, deci G este un graf rar: jE(G)j =
O(jG j).
Se poate demonstra prin inducµie dup n c num rul de xy-drumuriîn G este 2n (Cum?).
Acest num r este exponenµial în ordinul luiG (metoda backtrackingnu este prea e�cient în acest caz).
u1 un−1un−2u3u2
v1
y
vn−1vn−2v3v2
x
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Exerciµii rezolvate (parµial)
Exercise 5. Solution
Foe G = (V ;E) un graf, unde V e mulµimea examenelor. Pentru�ecare examen, v , se construieµe lita tuturor studenµilor care susµinacest examen; atunci uv 2 E dac ³i numai dac listele lui u ³i vau studenµi în comun.
Presupunem c S � V este mulµimea examenelor dintr-o anumit zi; pentru orice u 6= v 2 S nu putem avea uv 2 E , deci S eo mulµime stabil (independent ) de noduri din G . R spunsul laîntrebarea (a) este ...
În acest fel, o sesiune corespunde unei partiµii a mulµimii V înmulµimi stabile. Num rul minim de clase într-o asemenea partiµieeste r spunsul la cea de-a doua întrebareacest num r este ...
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Exerciµii rezolvate (parµial)
Exercise 6. Solution
(a) Fie G = (V ;E) un graf; de�nim V ca �ind mulµimea exameneloriar E mulµimea studenµilor: studentul uv 2 E va � "înscris" doarla examenelei u ³i v .
(b) Fie G = P6, V (G) = f1; 2; : : : ; 6g cu muchiile (i ; i + 1) (mod 6) -drumul cu 6 noduri f r corzi, �(P6) = 3, �(P6) = 2 - este un grafbipartit.
Dac primei zile îi asociem mulµimea stabil f1; 4; 6g, atunci G seva colora cu 3 culori.
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