Cours II-2 Cours MPRI 2008--2009habib/Documents/mpri8_2.pdfCours II-2 Cours MPRI 2008–2009...

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Cours II-2 Cours MPRI 2008–2009

Cours II-2Cours MPRI 2008–2009

Michel Habibhabib@liafa.jussieu.fr

http://www.liafa.jussieu.fr/~habib

Rue du chateau des rentiers, septembre 2008

Schedule

Chordal graphsParcours en largeur lexicographique LexBFS

Interval graphs

Sur les algorithmes de graphes

Graph Representations

Treewidth

Schedule

Interval graphs

Treewidth

Schedule

Interval graphs

Treewidth

Schedule

Interval graphs

Treewidth

Schedule

Interval graphs

Treewidth

Chordal graphs

Interval graphs

Treewidth

Chordal graphs

Recognition

Definition

A graph is chordal iff it has no chordless cycle of length ≥ 4.

Characterisation [Dirac 1961 ]

A graph is chordal iff it admits a simplicial elimination scheme.A vertex is simplicial iff its neighbourhood is a clique (completegraph).

LexBFS Characterisation [Rose, Tarjan et Lueker 1976]

A graph is chordal iff any LexBFS ordering of G provides asimplicial elimination scheme.

Chordal graphs

Recognition

Definition

Chordal graphs

Recognition

Definition

Chordal graphs

Minimal Separators

A subset of vertices S is a minimal separator if Gif there exist a, b ∈ G such that a and b are not connected inG − S .and S is minimal for inclusion with this property .

Chordal graphs

Consequences

G has a linear number of maximal cliques.Computing a maximum clique ω(G ) is polynomial.Computing χ(G ) also

Chordal graphs

Maximal Clique trees

A maximal clique tree (sometimes clique tree) is a tree T thatsatisfies the following three conditions :

I Vertices of T are associated with the maximal cliques of G

I Edges of T correspond to minimal separators.

I For any vertex x ∈ G , the cliques containing x yield a subtreeof T .

Chordal graphs

Theorem

Dirac 1961, Fulkerson, Gross 1965, Gavril 1974, Rose, Tarjan,Lueker 1976.

(0) G is chordal.

(i) G has a simplicial elimination scheme

(ii) Every minimal separator is a clique

(iii) G admits a maximal clique tree.

(iv) G is the intersection graph of subtrees in a tree.

(v) Any LexBFS provides a simplicial elimination scheme.

Chordal graphs

Interval graphs

Treewidth

Chordal graphs

Parcours en largeur lexicographique LexBFS

Interval graphs

Treewidth

Chordal graphs

Parcours en largeur lexicographique (LexBFS)

Donnees: Un graphe G = (V ,E ) et un sommet source s

Resultat: Un ordre total σ de VAffecter l’etiquette ∅ a chaque sommetlabel(s)← {n}pour i ← n a 1 faire

Choisir un sommet v d’etiquette lexicographique max.σ(i)← vpour chaque sommet non-numerote w ∈ N(v) faire

label(w)← label(w).{i}fin

Chordal graphs

Maximal clique Graph

Definition

For a chordal graph G we define its Maximal clique graph denotedby C(G) whose vertices are the maximal cliques of G and we putan edge between two maximal cliques C ,C ′ if C ∩ C ′ is a minimalseparator.

I Note that this is not the intersection graph of the maximalcliques of G

I Exercise : Is C(G) chordal ? What is the size of C(G) in termsof n and m ? Same question for a maximal clique tree.

I Any maximal clique tree is a spanning tree of C(G).

Chordal graphs

Definition

Chordal graphs

Definition

Chordal graphs

Definition

Chordal graphs

An example

Chordal graphs

Simplicial elimination schemes

1. Choose a maximal clique tree T

2. While T is not empty doSelect a vertex x ∈ F − S in a leaf F of T ;F ← F − x ;If F = S delete F ;

Chordal graphs

Simplicial elimination schemes

2. While T is not empty doSelect a vertex x ∈ F − S in a leaf F of T ;F ← F − x ;If F = S delete F ;

Chordal graphs

Canonical simplicial elimination scheme

2. While T is not empty doChoose a leaf F of T ;Select successively all vertices in F − Sdelete F ;

Chordal graphs

Canonical simplicial elimination scheme

2. While T is not empty doChoose a leaf F of T ;Select successively all vertices in F − Sdelete F ;

Chordal graphs

Size of a maximal clique tree in a chordal graph

I Let G = (V ,E ) be a chordal graph.

I G admits at most |V | maximal cliques and therefore the treeis also bounded by |V | (vertices and edges).

I But some vertices can be repeated in the cliques. If weconsider a simplicial elimination ordering the size of a givenmaximal clique is bounded by the neighbourhood of the firstvertex of the maximal clique.

I Therefore any maximal clique tree is bounded by |V |+ |E |.

Chordal graphs

Size of CS(G )

Considering a star on n vertices,shows |CS(G )| ∈ O(n2)Not linear in the size of G

Chordal graphs

CS(G ) is not chordal !

Chordal graphs

CS(G ) is not chordal !

Chordal graphs

In fact CS(G ) is dually chordal (almost chordal)and CS(CS(G )) is chordal.

Chordal graphs

Claim 1

CS(G ) is a good tool to study the structure ofchordal graphs.

To prove structural properties of maximal clique trees.

Chordal graphs

Combinatorial structure of CS(G )

Lemma 1 : M.H and C. Paul 95

If C1,C2,C3 is a cycle in CS(G ), with S12,S23 and S23 be theassociated minimal separators then two of these three separatorsare equal and included in the third.

Lemma 2 : M.H. and C. Paul 95

Let C1,C2,C3 be 3 maximal cliques, ifC1 ∩ C2 = S12⊂S23 = C2 ∩ C3 then it yields a triangle in CS(G )

Chordal graphs

Combinatorial structure of CS(G )

Lemma 1 : M.H and C. Paul 95

If C1,C2,C3 is a cycle in CS(G ), with S12,S23 and S23 be theassociated minimal separators then two of these three separatorsare equal and included in the third.

Lemma 2 : M.H. and C. Paul 95

Let C1,C2,C3 be 3 maximal cliques, ifC1 ∩ C2 = S12⊂S23 = C2 ∩ C3 then it yields a triangle in CS(G )

Chordal graphs

Theorem : Blayr and Payton 93

Maximal clique trees are exactly the maximum spanning trees ofCS(G ).

The weight of an edge being the size of the minimal separator itrepresents.

I CS(G ) is the union of all maximal clique trees of G .

I From one maximal clique tree to another there always exists apath of exchanges on triangles.

Chordal graphs

Applications

I Maximal Cardinality Search can be seen as a Prim algorithmfor computing maximal spanning tree of CS(G ).

I LexBFS as an example of Maximal Inclusion Search.

Chordal graphs

Applications

I Maximal Cardinality Search can be seen as a Prim algorithmfor computing maximal spanning tree of CS(G ).

I LexBFS as an example of Maximal Inclusion Search.

Chordal graphs

Others simplicial elimination scheme

1. Maximal cardinality search

2. Minimal inclusion search (MIS)

3. Others ? How to generate all simplicial elimination schemes ?

Chordal graphs

Maximal Cardinality Search : MCS

Resultat: Un ordre total σ de VAffecter l’etiquette 0 a chaque sommetlabel(s)← {n}pour i ← n a 1 faire

Choisir un sommet v d’etiquette maximumσ(i)← vpour chaque sommet non-numerote w ∈ N(v) faire

label(w)← label(w) + 1fin

Chordal graphs

Maximal Inclusion Search : MIS

Resultat: Un ordre total σ de VAffecter l’etiquette ∅ a chaque sommetlabel(s)← {n}pour i ← n a 1 faire

Choisir un sommet v d’etiquette maximaleσ(i)← vpour chaque sommet non-numerote w ∈ N(v) faire

label(w)← label(w) ∪ {i}fin

Chordal graphs

Theorem

G is chordal graph iff any MIS produces a simplicial eliminationscheme.

Chordal graphs

I How to implement MIS search in linear time ?

I 2 known linear implementations : LexBFS, MCS

Chordal graphs

I How to implement MIS search in linear time ?

I 2 known linear implementations : LexBFS, MCS

Interval graphs

Treewidth

Interval graphs

Characterisation Theorem for interval graphs

(0) G = (V ,E ) is interval graph.

(i) G has ...

(ii) It exists a total ordering τ of the vertices of V s.t.∀x , y , z ∈ G with x ≤τ y ≤τ z and xz ∈ E thenxy ∈ E .

(iii) G has a maximal clique path. (A maximal clique pathis just a maximal clique tree T, reduced to a path).

(iv) G is the intersection graph of a family of intervals ofthe real line

Interval graphs

To recognize an interval graph, we just have to compute amaximal clique tree and check if it is a path ?

Interval graphs

Many linear time algorithms already proposed for interval graphrecognition ....using nice algorithmic tools :graph searches, modular decomposition, partition refinement,PQ-trees . . .

Interval graphs

Linear time recognition algorithms for interval graphs

I Booth and Lueker 1976, using PQ-trees.

I Korte and Mohring 1981 using LexBFS and ModifiedPQ-trees.

I Hsu and Ma 1995, using modular decomposition and avariation on Maximal Cardinality Search.

I Corneil, Olariu and Stewart SODA 1998, using a series ofLexBFS.

I M.H, McConnell, Paul and Viennot 2000, using LexBFS andpartition refinement on maximal cliques.

Interval graphs

Treewidth

Notation

For a finite loopless undirected graph G = (V ,E )V set of verticesE edges set|V | = n and |E | = m

Some exercises

1. Algorithme lineaire qui teste l’isomorphisme de deux arbres

2. Calcul du diametre d’un arbre

3. Ecrire un algorithme qui verifie l’existence d’une clique ayant∆ + 1 sommets dans un graphe de degre maximum ∆ ?

Some exercises

Exercices

1. On considere un graphe G dont la clique maximum a pourcardinal 2k, montrer qu’il existe une partition des sommets endeux sous-ensembles A,B tels ω(G (A)) = ω(G (B)) ?Peut-on calculer cette partition en temps polynomial (si onpart d’une clique maximum de G ) ?

2. Quelle est la complexite d’un algorithme de routage ? Parexemple RIP dans Internet.

Exercices

1. On considere un graphe G dont la clique maximum a pourcardinal 2k, montrer qu’il existe une partition des sommets endeux sous-ensembles A,B tels ω(G (A)) = ω(G (B)) ?Peut-on calculer cette partition en temps polynomial (si onpart d’une clique maximum de G ) ?

2. Quelle est la complexite d’un algorithme de routage ? Parexemple RIP dans Internet.

Solution de l’exercice

I Ecrire un algorithme qui verifie l’existence d’une clique ∆ + 1sommets dans un graphe de degre maximum ∆ ?

I Une solutionSoit S l’ensemble des sommets de degre ∆ du graphe.Tant qu’il existe a ∈ S ayant un voisin hors de S , eliminer ade S .Les composantes connexes de taille exactement ∆ + 1 sontdes bonnes reponses.

I L’algorithme est en O(n + m).

Bornes sur le nombre d’aretes

Graphes sans triangles

Montrer que si G n’a pas de triangle alors

|E | ≤ |V |24

Graphes planaires

Montrer que si un graphe est planaire simple (i.e. sans boucle niaretes multiples) alors :|E | ≤ 3|V | − 6

Bornes sur le nombre d’aretes

Graphes sans triangles

Montrer que si G n’a pas de triangle alors

|E | ≤ |V |24

Graphes planaires

Montrer que si un graphe est planaire simple (i.e. sans boucle niaretes multiples) alors :|E | ≤ 3|V | − 6

Research Problems

I Find a linear-time algorithm that minimizes a deterministicautomaton.

I Find a linear-time algorithm which computes a doublylexicographic ordering of a 0-1 matrix, i.e. an ordering of thecolumns and the lines for which lines and columns appear tobe lexicographically ordered.

Research Problems

I Find a linear-time algorithm that minimizes a deterministicautomaton.

I Find a linear-time algorithm which computes a doublylexicographic ordering of a 0-1 matrix, i.e. an ordering of thecolumns and the lines for which lines and columns appear tobe lexicographically ordered.

Example of a doubly lexicographic ordering

C1 C2 C3 C4 C5

L1 1 0 1 0 1

L2 0 1 0 1 1

L3 1 1 0 1 1

L4 0 0 0 0 1

L5 0 1 1 0 0

C3 C1 C4 C5 C2

L4 0 0 0 1 0

L1 1 1 0 1 0

L5 1 0 0 0 1

L2 0 0 1 1 1

L3 0 1 1 1 1

Example of a doubly lexicographic ordering

C1 C2 C3 C4 C5

L1 1 0 1 0 1

L2 0 1 0 1 1

L3 1 1 0 1 1

L4 0 0 0 0 1

L5 0 1 1 0 0

C3 C1 C4 C5 C2

L4 0 0 0 1 0

L1 1 1 0 1 0

L5 1 0 0 0 1

L2 0 0 1 1 1

L3 0 1 1 1 1

I Such an ordering always exists

I Best algorithm to compute one :Paige and Tarjan [1987] proposed an O(LlogL) whereL = n + m + e for a matrix with n lines, m columns and e nonzero values, using partition refinement.

I For undirected graphs, such an ordering of the symmetricincidence matrix, yields an ordering of the vertices which hasnice properties.

Interval graphs

Treewidth

I Implicit hypothesis : the memory words have k bits withk > dlog(|V |)e

I To be sure, consider the bit encoding level

I Implicit hypothesis : the memory words have k bits withk > dlog(|V |)e

I To be sure, consider the bit encoding level

I Adjacency listsO(|V |+ |E |) memory wordsAdjacency test : xy is an arc in O(|N(x)|)

I Adjacency MatrixO(|V |2) memory words (can be compressed)Adjacency test : xy is an arc in O(1)

I Customized representations, a pointer for each arc . . .

For some large graphs, the Adjacency matrix, is not easy to obtainand manipulate.But the neighbourhood of a given vertex can be obtained. (WEBGraph)

Quicksands

I To compute this invariant or this property of a given graph Gone needs to ”see”( or visit) every edge at least once.

I False statement as for example the computation of twins resp.connected components on G knowing G .

Quicksands

I To compute this invariant or this property of a given graph Gone needs to ”see”( or visit) every edge at least once.

I False statement as for example the computation of twins resp.connected components on G knowing G .

Exercice

Comment allier les avantages des deux representations ?liste d’adjacence : construction en O(n + m)Matrices d’adjacence : acces a un arc en O(1)

Espace O(n2) mais en gardant la possibilite d’ecrire desalogrithmes lineaires ?

Exercice

Comment allier les avantages des deux representations ?liste d’adjacence : construction en O(n + m)Matrices d’adjacence : acces a un arc en O(1)

Espace O(n2) mais en gardant la possibilite d’ecrire desalogrithmes lineaires ?

Auto-complemented representations

Initial Matrix

1 2 3 4

1 1 1 1 0

2 0 0 1 0

3 1 0 1 1

4 1 0 0 0

Tagged Matrix

1 2 3 4

1 0 1 0 0

2 1 0 0 0

3 0 0 0 1

4 0 0 1 0

Auto-complemented representations

Initial Matrix

1 2 3 4

1 1 1 1 0

2 0 0 1 0

3 1 0 1 1

4 1 0 0 0

Tagged Matrix

1 2 3 4

1 0 1 0 0

2 1 0 0 0

3 0 0 0 1

4 0 0 1 0

I At most 2n tags (bits).O(n + m′) with m′ << m.Dalhaus, Gustedt, McConnell 2000

I What can be computed using such representations ?Example : strong connected components of G , knowing G ?

I At most 2n tags (bits).O(n + m′) with m′ << m.Dalhaus, Gustedt, McConnell 2000

I What can be computed using such representations ?Example : strong connected components of G , knowing G ?

I Find a minimum sized representation

I If G is undirected a minimum is unique.

I Find a minimum sized representation

I If G is undirected a minimum is unique.

What is an elementary operation for a graph ?

I Traversing an edge or Visiting the neighbourhood ?

I It explains the very few lower bounds known for graphalgorithms on a RAM Machine.

I Our graph algorithms must accept any auto-complementedrepresentation.Partition Refinement

Treewidth

Interval graphs

Treewidth

k-trees

Recursive Definition

A clique with k+1 vertices is a k-tree. If G is a k-tree, then weobtain another k-tree G ′ by adding a new vertex and joining thisvertex to a k-clique of G .

Clearly a k-tree is a chordal graph.1-trees =ordinary trees2-trees = series-parallel networks

Treewidth

Partial k-trees

A partial k-tree is a partial subgraph of a k-tree (deleting edgesand vertices)

Definition

Treewith(G)= least integer k s.t. G is a partial k-tree.

Treewidth

Tree decomposition

G = (V ,E ) has a tree decomposition D = (S ,T )S is a collection of subsets of V , T a tree whose vertices areelements of S such that :

(0) The union of elements in S is V

(i) ∀e ∈ E , ∃i ∈ I with e ∈ G (Si ).

(ii) ∀x ∈ V , the elements of S containing x form asubtree of T .

Definition

Decomp(G ) = MinD(MaxSi∈S{|Si | − 1})

Treewidth

Tree decomposition

(i) ∀e ∈ E , ∃i ∈ I with e ∈ G (Si ).

Definition

Treewidth

Tree decomposition

(i) ∀e ∈ E , ∃i ∈ I with e ∈ G (Si ).

Definition

Treewidth

Tree decomposition

(i) ∀e ∈ E , ∃i ∈ I with e ∈ G (Si ).

Definition

Treewidth

Equivalences

1. ∀G , Treewidth(G ) = Decomp(G )

2. Treewidth(G ) = MinH triangulation of G{ω(H)− 1}3. Computing treewidth is NP-hard.

Treewidth

Equivalences

2. Treewidth(G ) = MinH triangulation of G{ω(H)− 1}

3. Computing treewidth is NP-hard.

Treewidth

Equivalences

2. Treewidth(G ) = MinH triangulation of G{ω(H)− 1}3. Computing treewidth is NP-hard.

Treewidth

Other definitions of trewidth in terms of cop-robber games, usinggraph grammars ....But it turns out that this parameter is a fundamental parameterfor graph theory.