1er Cours : Introduction MPRI 2012 2013habib/Documents/Cours_1-2012.pdf1er Cours : Introduction MPRI 2012{2013 Introduction Practical issues Graphs made up with vertices and edges

1er Cours : Introduction MPRI 2012–2013

1er Cours : IntroductionMPRI 2012–2013

Michel [email protected]

http://www.liafa.univ-Paris-Diderot.fr/~habib

Chevaleret, septembre 2012

http://www.liafa.univ-Paris-Diderot.fr/~habib


Schedule

Introduction

Which algorithms ?

Structuration and decomposition

Schedule of the course

Problems and Exercises

Graph Representations


Introduction

Academic year : 2012-2013

I Research team : Distributed algorithms and graphs, LIAFA :

I Part IPractical graph algorithmsMichel Habib (Pr Univ. Paris Diderot)

I Part IITreewithStructure theorems (minors theory and graph decompositionsRobertson and Seymour)Pierre Charbit (MdC, Univ. Paris Diderot)

I contact : [email protected]


Introduction

Google et Facebook


Introduction

Next challenges

I Merging the 2 Webs ?

I Recommender Algorithms (bipartite graph clustering) orreasoning with heterogeneous data.


Introduction

Quelques grosses capitalisations boursières !


Introduction

Programme détaillé de la partie I

I Classes de graphes (cographes, graphes triangulés, graphesd’intervalles ...), Représentations géométriques

I Une théorie des parcours de graphes

I Techniques de partionnement.

I Graphes triangulés et algorithmes gloutons.

I Décompositions de graphes calculables polynomialement :décomposition modulaire, en coupes, 2-join ... Etude desmeilleurs algorithmes disponibles.

I Décompositions de graphes NP-difficiles à calculer : largeurd’arborescence, de rang ou de cliques et invariants de graphes.


Introduction

Programme suite

I Quelques algorithmes sur les ensembles ordonnésI Applications directes de ces notions à :

I Certains calculs dans les réseaux (diamètre, centres, flots),I La théorie de la complexité paramétréeI Des problèmes de phylogénie en Bioinformatique


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

Graphs made up with vertices and edges (or arcs) provide a verypowerful tool to model real-life problems. Weights on vertices oredges can be added.Many applications involve graph algorithms, in particular manyfacets of computer science !Also many new leading economical applications Google PageRankand FaceBook are graph based


Introduction

The great importance of the right model

It is crucial to use all the characteristics of the problem you wantto solve and find a good model. Two examples from PeterWinkler’s nice book on Mathematical Puzzles :All the discrete structures considered her are supposed to be finite.

1. An odd number of soldiers in a field in such a way that allpairwise distances are different. Each soldier is told to keep aneye on the nearest other soldier.Show that at least one soldier is not being watched.

2. A problem on rectangles :A large rectangle of the plane is partitioned into smallerrectangles, each of witch has either integer height or integerwidth (or both). Prove that the large rectangle also has thisproperty.


Which algorithms ?

Which algorithms ?

I Optimal ?

I Linear

I Efficient ?

I Simple ?

I Easy to program


Which algorithms ?

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

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

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

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

programming.I Need for heuristics !


Which algorithms ?

Brute Force algorithms

For some problem

1. Enumerate the set S of solutions

2. Select the optimal one

3. Show that |S | is polynomial in the size of the problem


Which algorithms ?

Variations

1. Find the best way to enumerate S (for example a Gray codein O(|S |) constant time per element)

2. Many variations . . .

3. Sometimes the problem is NP-hard and the the game is tofind the exponential algorithm with the lowest exponent.


Which algorithms ?

Le Parcours en profondeur

DFS(G ) :

for all v ∈ X doFerme(x)← Faux ;

end forfor all v ∈ X doif Ferme(x) = Faux thenExplorer(G , x);

end ifend for


Which algorithms ?

Explorer(G , x) :

Ferme(x)← Vrai ;pre(x);for all xy ∈ U doif Ferme(y) = Faux thenExplorer(G , y) ;

end ifpost(x) ;

end for


Which algorithms ?

Et les deux fonctions suivantes utilisant deux variables : comptpreet comptpost étant initialisées à 1.pre(x) :

pre(x)← comptpre ;comptpre ← comptpre + 1 ;

post(x) :

post(x)← comptpost ;comptpost ← comptpost + 1 ;


Which algorithms ?

L’algorithme de Tarjan pour les composantes fortementconnexes

DFS(G ) :

comptpre ← 1;Resultat ← ∅;for all v ∈ X doFerme(x)← Faux ;

end forfor all v ∈ X doif Ferme(x) = Faux thenExplorer(G , x) ;

end ifend for


Which algorithms ?

Explorer(G , x) :

Empiler(Resultat, x);Pile(x)← Vrai ;Ferme(x)← Vrai ;pre(x)← comptpre; comptpre ← comptpre + 1 ;racine(x)← pre(x);for all xy ∈ U do

if Ferme(y) = Faux thenExplorer(G , y);racine(x)← min{racine(x), racine(y)} ;

else if Pile(y) = Vrai thenracine(x)← min{racine(x), racine(y)} ;

end ifend forif racine(x) = pre(x) then

Dépiler Resultat jusqu’à x ;end if


Which algorithms ?

Fundamental questions about algorithms

1. How can we prove such a nice algorithm ?

2. It has some greedy flavor, why ?


Which algorithms ?

Algorithme de Kosaraju 1978, Sharir 1981

1. Exécuter un parcours en profondeur sur G .

2. Faire un autre parcours en profondeur sur G− avec pour ordreinitial l’ordre postd

3. Les arbres de la forêt recouvrante de G−, sont lescomposantes fortement connexes de G


Which algorithms ?

Goals

I Reach a good level of knowledge on graph algorithms (manystudents FAQ)

I Understand the greedy algorithms

I Try to understand why some simple heuristics works for mostpractical data

I Understand how to use structural results to design algorithms

I Panorama of research



Our Claims or thesis

An efficient algorithm running on a discrete structure is alwaysbased :

I on a theorem describing a combinatorial structure

I a combinatorial decomposition of this discrete structure.

I or in some other cases a geometric representation of thestructure provides the algorithm.

Examples

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

I Transitive orientation and modular decompostion.

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

I Greedy algorithms for minimum spanning trees and matroids

I . . .



Uses of geometric representations

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

I Numerous algorithms on planar graphs use the existence of adual graph. ( Ex : flows transform into paths)




1. Introduction (this course)

2. Chordal and interval graphs and their nice structure

3. Algorithms for modular decomposition and extensions

4. Treewidth and other width parameters (parametrizedcomplexity)

5. Graph Searches, a new approach

6. Diameter Computations



Examples of algorithmic problems

I shortest paths, reachability (existence of a path)

I Center and diameter computations

I Graph encoding, distance labeling

I Routing

I Sandwich (or completion) problems

I Colorations



Decompositions used here

I modular decomposition

I split decomposition

I Cliquewidth)

I Treewidth, pathwidth

I Branchwidth and rankwidth



Graph classes used here

I Tournaments (Tournois), posets, bipartite graphs

I Cographs (P4-free)

I Interval graphs

I Chordal

I Comparability

I Asteroidal triple free

I Distance-hereditary

I Permutation

I Perfect Graphs

I Convex, bi-convex, . . .



Typical questions

I Recognition algorithms, if possible linear-time and robust.

I Efficient decomposition algorithms

I Computations of compact encodings and representation

I Random generation and enumeration

I Routing protocols and diameter estimation

I Computation of some invariant (for example mini coloring ormax clique).



Application domains

ThemesI Bioinformatics (mainly phylogeny and other graph problems)

I Networks and Distributed systems

I Analysis of huge graphs in social sciences (ranking, clustering)

GANG

An INRIA–LIAFA project on graphs an networks, dir. L. Viennot

SAE

A biological group at UPMC with E. Bapteste and P. Lopez

Social sciences

ANR Project AlgoPol, a collaboration with D. Cardon (OrangeLab) and C. Prieur (LIAFA)



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. Propose an algorithm which determines if a clique of size∆ + 1 exists in a graph with maximum degree ∆ ?



Solution de l’exercice

I Propose an algorithm which determines if a clique of size∆ + 1 exists in a graph with maximum degree ∆ ?

I A solutionLet S be the set of vertices with degree ∆ of the graph.Until it exists a ∈ S with a neighbour /∈ S , delete a from S .The remaining connected components of size exactly ∆ + 1are the solutions.

I The algorithm requires O(n + m).



Bound on the number of edges

Triangle free graphs

Show that if G has no triangle then :

|E | ≤ |V |2

4

Planar graphs

Show that if G is a simple planar graph (i.e. without loop andparallel edge) then :|E | ≤ 3|V | − 6



Classes of twin vertices

Definition

x and y are called false twins, (resp. true twins) ifN(x) = N(y) (resp. N(x) ∪ {x} = N(y) ∪ {y}))

Exercise

Propose a good algorithm to compute these classes



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



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.



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 or graphs is Game Theory)



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 .



Exercise

Can the advantages of the 2 previous representations can be mixedin a unique new one ?Adjacency lists : construction in O(n + m)Inicence matrix : cost of the query : xy ∈ E? in O(1)

In other words

Using O(n2) space, but with linear time algorithms on graphs ?



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′



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



Last exercise

Suppose that a graph G = (V ,E ) is given by its adjacency listsand let σ be some total ordering of its vertices.

I How can we sort the adjacency with respect to σ (increasing) ?

I What is the complexity of this operation ?

I Let σ be the total ordering of the vertices with decreasingdegrees, how to compute σ ?