1er Cours : Introduction MPRI 2010 2011 › ~habib › Documents › 1ercours.pdf · The great...

1er Cours : Introduction MPRI 2010–2011

1er Cours : IntroductionMPRI 2010–2011

Michel [email protected]

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

Chevaleret, septembre 2010

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


Schedule

Introduction

Which algorithms ?

Structuration and decomposition

Schedule of the course

Problems and Exercises

Graph Representations

Chordal graphsParcours en largeur lexicographique LexBFSOrdre d’elimination simplicial


Introduction

First question

Why such a course ?

Fagin’s theorems in structural complexity

Characterizations of P and NP using graphs and logics fragments.

NP

The class of all graph-theoretic properties expressible in existentialsecond-order logic is precisely NP.

P

The class of all graph-theoretic properties expressible in Hornexistential second-order logic with successor is precisely P.


Introduction

Practical issues

Many applications involve graph algorithms, in particular manyfacets of computer science ! ! !


Introduction

The great importance of the right model

It is crucial to use all the characteristics of the problem you wantto solve. Two examples from Peter Winkler’s nice book onMathematical Puzzles :

1. Soldiers in a field

2. A problem on rectangles


Introduction

Graph invariants

A graph invariant is a function G → f (G ) ∈ N, which is invariantunder isomorphism (i.e. if G and H are isomorphic thenf (G ) = f (H)).


Which algorithms ?

Which algorithms ?

◮ Optimal

◮ Linear

◮ Efficient

◮ Simples

◮ Easy to program


Which algorithms ?

In our work on optimisation problems on graphs, we will metseveral complexity barriers :

◮ Polynomial versus NP hardIn this case an algorithm running in O(n75.m252) could be anice result !

◮ Linear versus Boolean matrix multiplicationAlgorithms running in O(n.m), hard to find a lower bound !

◮ Practical issues need linear algorithms in order to be applied◮ on huge graphs such as Web graph . . ..◮ many times such as inheritance in object oriented

programming.◮ Need for heuristics !



Our Claims or thesis

An efficient algorithm running on a discrete structure is alwaysbased on a combinatorial decomposition of this discrete structure.For some other cases a geometric representation of the structureprovides the algorithm.

Examples

◮ Chordal graph recognition and maximal clique trees (particular case of treewidth).

◮ Transitive orientation and modular decompostion.

◮ Max Flow and decomposing a flow in a sum of positivecircuits.

◮ Minimum spanning trees and matroids

◮ . . .



Uses of geometric reprsentations

◮ Computing a maximum clique (of maximum size) or anefficient representation of an interval graph.

◮ Numerous algorithms on planar graphs use a representation ofthe dual.



Academic year : 2010-2011

◮ 2 members of the group : Algorithmique et Combinatoire duLIAFA :

◮ Michel Habib (Pr Univ. Paris Diderot)

◮ Nicolas Trotignon (CR1 CNRS)

◮ contact : [email protected]



Examples of algorithmic problems

◮ shortest paths, reachability (existence of a path)

◮ Center and diameter computation

◮ graphe encoding, distance labeling

◮ Routing

◮ Sandwich (or completion) problems

◮ Colorations



Decompositions used here

◮ modular decomposition

◮ split decomposition

◮ Largeur de clique (cliquewidth)

◮ Largeur d’arborescence (treewidth), largeur de chaıne(pathwidth)

◮ Largeur de branche (branchwidth)



Graph classes used here

◮ Tournaments (Tournois), ensembles ordonnes, graphes bipartis

◮ Cographs

◮ Interval graphs

◮ Graphes triangules (Chordal)

◮ Graphes de comparabilite

◮ Graphes sans triplet asteroıdal

◮ Graphes distance-hereditaires

◮ Graphes de permutation

◮ Graphes parfaits

◮ Graphes convexes, bi-convexes, . . .



Typical questions

◮ Algorithmes de reconnaissance, si possible lineaires etrobustes.

◮ Algorithmes de decomposition efficaces

◮ Codages compacts et representations

◮ Enumeration et generation aleatoire

◮ Calculs de diametre, routages

◮ Optimisation d’invariants sur la classe (par exemple cliquemaximum). Soit un algorithme polynomial, soit une reductionde NP-completude.



Application domains

Themes◮ Biocomputing

◮ Networks (P-2-P)

◮ Distributed systems

◮ Analysis of huge graphs

GANG

Une equipe-projet INRIA–LIAFA sur les graphes et les reseaux,dirigee par L. Viennot



Extensions

◮ Algorithmes et structures donnees dynamiques

◮ Algorithmique probabiliste ou randomisee

◮ Applications aux tres grands graphes

◮ Algorithmique sous-lineaire, decision sans avoir consideretoute la donnee.



Notation

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



Some exercises

1. A linear algorithm for isomorphism on trees

2. Computation of the diameter of a tree

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



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

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

◮ 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.

◮ 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 |2

4

Graphes planaires

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



Research Problems

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

◮ 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



◮ Such an ordering always exists

◮ 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.

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



◮ Implicit hypothesis : the memory words have k bits withk > ⌈log(|V |)⌉

◮ To be sure, consider the bit encoding level



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

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

◮ 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

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

◮ 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 ?



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



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

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



◮ Find a minimum sized representation

◮ If G is undirected a minimum is unique.



What is an elementary operation for a graph ?

◮ Traversing an edge or Visiting the neighbourhood ?

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

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


Chordal graphs

Parcours en largeur lexicographique LexBFS

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

fin


Chordal graphs


1

76

5

4

3

2


Chordal graphs


C’est un parcours en largeur avec une regle de tie-break bizarre.


Chordal graphs

Ordre d’elimination simplicial

Graphe triangule

5

1 4 38

6 7 2

Un sommet est simplicial si son voisinage est une clique.


σ = [x1 . . . xi . . . xn] est un ordre d’elimination simplicial si xi estsimplicial dans le sous-graphe Gi = G [{xi . . . xn}]

ca b


Chordal graphs


Caracterisation

Un graphe est triangule ssi il possede un ordre d’eliminationsimplicial.

Caracterisation LexBFS [Rose, Tarjan et Lueker 1976]

Un graphe G est triangule ssi tout ordre LexBFS de G estsimplicial.

1 8

7

6

5

4

32


Chordal graphs


How can we prove such a theorem ?

1. A direct proof, finding the invariants ?

2. Find some structure of chordal graphs

3. Understand how LexBFS explores a chordal graph


Chordal graphs


A characterisation theorem for chordal graphs

Theorem

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

(0) G is chordal (every cycle of length ≥ 4 has a chord) .

(i) G has a simplicial elimination scheme

(ii) Every minimal separator is a clique

1er Cours : Introduction MPRI 2010 2011 › ~habib › Documents › 1ercours.pdf · The great...

Documents

Transcript of 1er Cours : Introduction MPRI 2010 2011 › ~habib › Documents › 1ercours.pdf · The great...