du nouveau sur les parcours de graphes...3.using bfs to search an augmenting path provides a...

On the power of graph searching, “Du nouveau sur les parcours de graphes”

On the power of graph searching,“Du nouveau sur les parcours de graphes”

Michel Habib,LIAFA, CNRS et Universite Paris Diderot,

and Gang, Equipe Projet INRIA,[email protected]

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

CPIOR, Nantes, May – june 2012

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


Schedule

Introduction

4 points conditions

Lexicographic Breadth First Search LexBFS

Composition of Searches

Diameter computations and the 4-sweep method


Joint work with D. Corneil (Toronto), P. Crescenzi (Firenze), J.Dusart (PhD), R. Grossi (Pisa), A. Marino (PhD, Pisa)and many others such as : F. de Montgolfier, D. Fortin . . .


Introduction

Introduction

4 points conditions





Introduction

Graph searches are very well known and used

1. Euler (1735) for solving the famous walk problem inKœnisberg

2. Tremaux (1882) and Tarry (1895) using DFS to solve mazeproblems

3. Computer scientists from 1950, in particular in the 70’s, R.E.Tarjan for new applications of DFS....

4. But also : ”Fil d’ariane” in the Greek mythology.


Introduction

Tarry


Introduction

Umberto Eco, ”Il nome della rosa”, Roman, 1980


Introduction

Some definitions

Graph Search

The graph is supposed to be connected so as the set of visitedvertices After choosing an initial vertex, a search of a connectedgraph visits each of the vertices and edges of the graph such that anew vertex is visited only if it is adjacent to some previously visitedvertex.At any point there may be several vertices that may possibly bevisited next ; this generic graph search does not specify whichvertex to choose. The breadth-first search (BFS) and depth-firstsearch (DFS) algorithms are the traditional strategies fordetermining the next vertex to visit.


Introduction

Variations

Graph Traversal

The set of visited vertices is not supposed to be connected (usedfor computing connected components for example)

Cops and robbers games on a graph

The name Graph searching is also used in this context, with aslightly different meaning.


Introduction

Three main aspects for a graph search :

1. its principle or its algorithm(i.e. the description of the tie-break rules for the choice of thenext vertex (edge) to be explored )

2. The study of the underlying tree structure

3. The complexity analysis and its implementation or its program


Introduction

We will focus here on a fourth one :

I Here we consider a graph search as an operator acting on agraph and producing of a total ordering on the set of vertices(the order in which the vertices are visited by the search)

I I will try to convince you that this viewpoint can be helpful


Introduction

I Building bottom up graph algorithms form well-known

I Develop basic theoretic tools for the structural analysis ofgraphs

I No need to store sophisticated data structures, just somelabels on each vertex, can be used to deal with huge graphs

I Le syndrome du petit Poucet or Tom ThumbPutting little stones or breadcrump on the vertices in order tobacktrack


Introduction

Our goals

Fundamental Question

What kind of knowledge can we learn about the structure of agiven graph via graph searching (i.e. with one or a series ofsuccessive graph searches) ?

Practical issues

Build a toolbox of different graph searches and find methods toanalyze them


Introduction

Problem

For an undirected graph G = (V ,E ),explore the vertices of G ”traversing or following” the edges.

ResultI a tree structure rooted at the first visited vertex

I an ordering σ of the vertices

QuestionsI Under which conditions an ordering σ of the vertices

corresponds to a search ?

I What are the properties of these orderings ?


Introduction

Important reference for this today lecture :

These easy questions have been only recently systematicallyconsidered :D.G. Corneil et R. M. Krueger, A unified view of graph searching,SIAM J. Discrete Math, 22, Num 4 (2008) 1259-1276


4 points conditions

Introduction

4 points conditions





4 points conditions

Generic Search

7

1

2

3

4

56Invariant

At each step, an edge between a visited vertex and a unvisited oneis selected


4 points conditions

Generic search

S ← {s}for i ← 1 a n do

Pick an unumbered vertex v of Sσ(i)← vforeach unumbered vertex w ∈ N(v) do

if w /∈ S thenAdd w to S

endend

end


4 points conditions

Generic question ?

Let a, b et c be 3 vertices such that ab /∈ E et ac ∈ E .

a cb

Under which condition could we visit first a then b and last c ?


4 points conditions

Property (Generic)

For an ordering σ on V , if a < b < c and ac ∈ E and ab /∈ E , thenit must exist a vertex d such that d < b et db ∈ E

<

cba

d <

Theorem

For a graph G = (V ,E ), an ordering σ sur V is a generic search ofG iff σ satisfies property (Generic).


4 points conditions

Most of the searches that we will study are refinement of thisgeneric searchi.e. we just add new rules to follow for the choice of the nextvertex to be visitedGraph searches mainly differ by the management of the tie-breakset


4 points conditions

Property (BFS)

For an ordering σ on V , if a < b < c and ac ∈ E and ab /∈ E , thenit must exist a vertex d such that d < a et db ∈ E

d cba

Theorem

For a graph G = (V ,E ), an ordering σ sur V is a BFS of G iff σsatisfies property (BFS).


4 points conditions

Proprerty (DFS)

For an ordering σ on V , if a < b < c and ac ∈ E and ab /∈ E , thenit must exist a vertex d such that a < d < b and db ∈ E .

d cba

Theorem

For a graph G = (V ,E ), an ordering σ sur V is a DFS of G iff σsatisfies property (DFS).


4 points conditions

Applications of BFS

1. Distance computations (unit length), diameter and centers

2. BFS provides a useful layered structure of the graph

3. Using BFS to search an augmenting path provides apolynomial implementation of Ford-Fulkerson maximum flowalgorithm.


4 points conditions

Applications of DFS

Some applications

I Planarity testing.

I Computation of 2-connected (resp. strongly connected)components, 2-SAT solvers

I Computation of a linear extension (topological sorting) for anacyclic digraph, applications to inheritance mechanisms. . . .



Introduction

4 points conditions






Lexicographic Breadth First Search (LexBFS)

Data: a graph G = (V ,E ) and a start vertex s

Result: an ordering σ of V

Assign the label ∅ to all verticeslabel(s)← {n}for i ← n a 1 do

Pick an unumbered vertex v with lexicographically largest labelσ(i)← vforeach unnumbered vertex w adjacent to v do

label(w)← label(w).{i}end

end



1

76

5

4

3

2



It is just a breadth first search with a tie break rule.We are now considering a characterization of theorder in which a LexBFS explores the vertices.



Property (LexB)

For an ordering σ on V , if a < b < c and ac ∈ E and ab /∈ E , thenit must exist a vertex d such that d < a et db ∈ E et dc /∈ E .

d cba

Theorem

For a graph G = (V ,E ), an ordering σ sur V is a LexBFS of G iffσ satisfies property (LexB).



Why LexBFS behaves so nicely on well-structured graphs

A nice recursive property :On every tie-break set S , LexBFS operates on G (S) as a LexBFS.Analogous properties are false for other classical searches.



Applications of LexBFS

1. Most famous one : Chordal graph recognition

2. For many classes of graphs using LexBFS ordering”backward” provides structural information on the graph.

3. Last visited vertex (or clique) has some property (examplesimplicial for chordal graph)



Theorem [Tarjan et Yannakakis, 1984]

G is chordal iff every LexBFS ordering yields a simplicialelimination scheme.

Proof :

Let c be the leftmost non simplicial vertex.There exist a < b ∈ N(c) avec ab /∈ E .Using characterization of LexBFS orderings, it exists d < a withdb ∈ E and dc /∈ E . Since G is chordal, necessarily ad /∈ E .

d cba



d cba

But then from the triple d , a, b, it exists d ′ < d with d ′a ∈ E andd ′b /∈ E . Furthermore d ′d /∈ E . . .And using the triple d’, d, a , we start an infinite chain .....

Remark

Most of the proofs based on some characteristic ordering of thevertices are like that, with no extra reference to the algorithm itself.



LexDFS Lexicographic Depth First Search

BFS vs LexBFS

BFS

d cba

LexBFS

d cba

DFS vs LexDFS

DFS

d cba

LexDFS

d cba



Property (LDFS)

For an ordering σ on V , if a < b < c and ac ∈ E and ab /∈ E , thenit must exist a vertex d such that a < d < b and db ∈ E anddc /∈ E .

d cba

Theorem

For a graph G = (V ,E ), an ordering σ sur V is a LexDFS of G iffσ satisfies property (LDFS).



Lexicographic Depth First Search (LexDFS)

Data: a graph G = (V ,E ) and a start vertex s

Result: an ordering σ of V

Assign the label ∅ to all verticeslabel(s)← {0}for i ← 1 a n do

Pick an unumbered vertex v with lexicographically largest labelσ(i)← vforeach unnumbered vertex w adjacent to v do

label(w)← {i}.label(w)end

end



LexDFS

Complexity

Is it possible to compute a LexDFS in O(n + m) ?

Spinrad 2008

Best implementation so far needs O(n + mloglogn) using Van derBoas trees.

First application : D. Corneil, B. Dalton, M. H. 2010

Hamiltonicity on co-comparability graphs via LexDFS.



Applications

I BFS to compute distances, diameter, centersHeuristics for diameter

I DFS planarity, strongly connected components, 2-SAT, . . .

I LexBFS, recognition of chordal graphs, interval graphs . . .Heuristics for one consecutiveness property

I LexDFS, long paths, minimum path coverFor co-comparability graphs LexDFS computes layeredordering of the complement partial order.Heuristics for graph clustering



Conclusions

Using the 4-points configurations we have the following inclusionordering between searches

Inclusions

Gen↗ ↑ ↖↗ ↑ ↖↗ ↑ ↖

BFS MNS DFS↑ ↗ ↑ ↖ ↑↑ ↗ ↑ ↖ ↑↑ ↗ ↑ ↖↑

LexBFS MCS LexDFS



Search classification

Search Tie-break management

Generic search none (random)

BFS queue

DFS stack

LexBFS Lexicographic maximal

LexDFS Lexicographic maximal

MNS Maximal under inclusion

MCS Maximal for the cardinality



Importance of 4 points conditions for graph recognition

Many classes of graphs or partial orders can be characterized bythe existence of a particular ordering of the vertices with someforbidden configuration on three points.Examples with forbidden configuration on three points :

1. Interval graphs : ordering of the left ends of the intervals.

2. Chordal : simplicial elimination ordering.

3. Co-comparability : transitivity violation of the complementgraph

4. Permutation : transitivity violation of the graph and itscomplement.



Forbidden 3 points suborderings



Consequences

LexBFS is involved in many recognition algorithms for these classesof graphs.

I Apply a LexBFS on G giving an ordering σ

I If G is a comparability graph the last vertex of σ, can betaken as a source in a transitive orientation of G .

I The starting point for comparability and permutation graphrecognition algorithms.



Introduction

4 points conditions






Since we focus on the ordering of the vertices as the result of agraph search, now we can compose graph searches in a natural way.Therefore we can denote by M(G , x0) the order of the verticesobtained by applying M on G starting from the vertex x0.

Definition of the + Rule

Let M be a graph search and σ an ordering of the vertices of G ,M+(G , σ) be the ordering of the vertices obtained by applying Mon G starting from the vertex σ(1) and tie-breaking using σ indecreasing order.



Why this Rule ?

The + Rule forces to keep the ordering of the previous sweep incase of tie-break



I Graph searches operate on total orderings :Step 0 : σ = M(G , x0)Step 1 : M(G , σ)Step 2 : M2(G , σ) = M(G ,M(G , σ)). . .Step i : M i (G , σ) = M(G ,M i−1(G , σ)). . .

I For which search M and graph G does there exist fixedpoints ?

I Unfortunately a formal study of this composition remains tobe done !

I Also called multisweep algorithms.



1. Such an idea was already used for planarity testing in somealgorithm (de Fraysseix and Rosentiehl 1980) with 2consecutive DFS.

2. Algorithms for strongly connected components by Kosaraju1978, Sharir 1981In our framework,1) DFS(G )2) DFS(G−, postd)

3. To compute efficiently the diameter of a graph usingsuccessive BFS



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 of 5+1special consecutive LexBFS, published in 2010.

I M.H, McConnell, Paul and Viennot 2000, using LexBFS andpartition refinement on maximal cliques (a kind of LexBFS oncliques and minimal separators).

I New ! Peng Li, Yaokun Wu, using 3 +1 special LexBFS, 2011



Repeated LexBFS

Require: G = (V ,E )Ensure: an ordering σσ ← LexBFS(G)for i = 2 to |V | doσ ← LexBFS+(G,σ)

end for

Algorithm 1: LexBFS+ multi-sweep



Property

If G = (V ,E ) is an interval graph, that the previous algorithmfinds a fixed point in less than |V | iterations



C1

C2

C3C4

C5

C6

C7

ab

cde f

gh

i

j

k

LexBFS(G, c) : c, d, e, a, f, g, h, i, j, k, b

LexBFS(G, c) C2 C3 C4 C5 C6 C7 C1



C1

C2

C3C4

C5

C6

C7

ab

cde f

gh

i

j

k

LexBFS(G , c) : c, d, e, a, f, g, h, i, j, k, bLexBFS+(LexBFS(G , c)) : b, a, i, h, d, c, g, f, e, j, k

LexBFS(G , c) C2 C3 C4 C5 C6 C7 C1

LexBFS+(LexBFS(G , c)) C1 C5 C4 C3 C2 C6 C7



C1

C2

C3C4

C5

C6

C7

ab

cde f

gh

i

j

k

LexBFS+(LexBFS(G , c)) : b, a, i, h, d, c, g, f, e, j, kLexBFS3+(G , c) : k, c, j, h, g, d, a, i, f, e, b


LexBFS2+(G , c) C1 C5 C4 C3 C2 C6 C7




C1

C2

C3C4

C5

C6

C7

ab

cde f

gh

i

j

k

LexBFS3+(G , c) : k, c, j, h, g, d, a, i, f, e, bLexBFS4+(G , c) : b, a, e, d, c, f, g, h, i, j, k







Generalization to comparability graphs

Question

If G = (V ,E ) is co-comparability graph, what can be said ?Is the series (LBFS+(G ))i periodic (with a period less than |V | ) ?i.e. ∃i < j ≤ n such that :(LBFS+(G ))i = (LBFS+(G ))j .Is (LBFS+(G ))i a cocomp ordering ?



Lemma 1

For every cocomp graph G , it exists a cocomp order which is aLexBFS order.

Lemma 2

If σ is a cocomp order, then (LBFS+(G , σ) is still a cocomp order.



LBFS OrderingsLBFS OrderingsCocomp ordersCocomp orders

Another way to produce a transitive orientation in linear time



Some applications

Three applications :

1. Hamiltonicity on co-comparability graphs, D. Corneil, B.Dalton, M.H. 2010

2. New linear extensions for partial orders D. Corneil, M.H., E.Kohler, 2012

3. Exact diameter computations using a series of BFS, P.Crescenzi, R. Grossi, M.H., L. Lanzi, A. Marino, 2011.



Introduction

4 points conditions






Diameter computations

” While theoretical work on models of computation and methodsfor analyzing algorithms has had enormous payoffs, we are notdone. In many situations, simple algorithms do well. We don’tunderstand why ! Developing means for predicting the performanceof algorithms and heuristics on real data and on real computers isa grand challenge in algorithms.” Challenges for Theory ofComputing by :CONDON, EDELSBRUNNER, EMERSON, FORTNOW, HABER,KARP, LEIVANT, LIPTON, LYNCH, PARBERRY,PAPADIMITRIOU, RABIN, ROSENBERG, ROYER, SAVAGE,SELMAN, SMITH, TARDOS, AND VITTER, Report for anNSF-sponsored workshop on research in theoretical computerscience.



Basics Definitions

Definitions :

Let G be an undirected graph :

I exc(x) = maxy∈G{distance(x , y)} excentricity

I diam(G ) = maxx∈G{exc(x)} diameter

I radius(G ) = minx∈G{exc(x)}I x ∈ V is a center of G , if exc(x) = radius(G )

First remarks of the definitions

distance computed in # edgesIf x and y belong to different connected components d(x , y) =∞.diameter : Max Max Minradius : Min Max Min



Trivial bounds

For any graph G :radius(G ) ≤ diam(G ) ≤ 2radius(G ) and ∀e ∈ G ,diam(G ) ≤ diam(G − e)

These bounds are tight

I If G is a path of length 2K, then diam(G ) = 2k = 2radius(G ),and G admits a unique center, i.e. the middle of the path.

I If radius(G ) = diam(G ), then Center(G ) = V . All vertices arecenters (as for example in a cycle).



If 2.radius(G ) = diam(G ), then *roughly* G has a tree shape (atleast it works for trees).But there is no nice characterization of this class of graphs.



Diameter

Applications

1. A graph parameter which measures the quality of services of anetwork, in terms of worst cases, when all have a unitary cost.Find critical edges e s.t. diam(G − e) > diam(G )

2. Many distributed algorithms can be analyzed with thisparameter (when a flooding technique is used to spreadinformation over the network or to construct routing tables).

3. Verify the small world hypothesis in some large socialnetworks, using J. Kleinberg’s definition of small world graphs.

4. Compute the diameter of the Internet graph, or some Webgraphs, i.e. massive data.



FAQ

Usual questions on diameter, centers and radius :

I What is the best Program (resp. algorithm) available ?

I What is the complexity of diameter, center and radiuscomputations ?

I How to compute or approximate the diameter of huge graphs ?

I Find a center (or all centers) in a network, (in order to installserveurs).



Some notes

1. I was asked first this problem in 1980 by France Telecom forthe phone network (FT granted a PhD).

2. Marc Lesk obtained his PhD in 1984 with the title :Couplages maximaux et diametres de graphes.Maximum matchings and diameter computations

3. But, with very little practical results.



I Our aim is to design an algorithm or heuristic to compute thediameter of very large graphs

I Any algorithm that computes all distances between all pairs ofvertices, complexity O(n3) or O(nm). As for example with |V |successive Breadth First Searches in O(n(n + m)).

I Best known complexity for an exact algorithm is O( n3

log2n)

I At most O(Diam(G )) matrix multiplications.



1. Let us consider the procedure called : 2 consecutive BFS

Data: A graph G = (V ,E )

Result: u, v two vertices

Choose a vertex w ∈ Vu ← BFS(w)v ← BFS(u)

Where BFS stands for Breadth First Search.Therefore it is a linear procedure



Intuition behind the procedure



I FolkloreIf G is a tree, diam(G ) = d(u, v)Easy using Jordan’s theorem.



First theorem

Camille Jordan 1869 :

A tree admits one or two centers depending on the parity of itsdiameter and furthermore all chains of maximum length starting atany vertex contain this (resp. these) centers.

And radius(G ) = ddiam(G)2 e



Unfortunately it is not an algorithm !



Certificates for the diameter

To give a certificate diam(G ) = k , it is enough to provide :

I two vertices x , y s.t. d(x , y) = k (diam(G ) ≥ k).

I a subgraph H ⊂ G with diam(H) = k (diam(G ) ≤ k).H may belong to a class of graphs on which diametercomputations can be done in linear time.



Experimental results : M.H., M.Latapy, C. Magnien 2007

Randomized BFS procedure

Data: A graph G = (V ,E )

Result: u, v two vertices

Repeat α times :Randomly Choose a vertex w ∈ Vu ← BFS(w)v ← BFS(u)Select the vertices u0, v0 s.t. distance(u0, v0) is maximal.



1. This procedure gives a vertex u0 such that :exc(u0) ≤ diam(G ) i.e. a lower bound of the diameter.

2. Use a spanning tree as a partial subgraph to obtain an upperbound by computing its exact diameter in linear time.

3. Spanning trees given by the BFS.



I The Program and some Data on Web graphs or P-2-Pnetworks can be found

I http://www-rp.lip6.fr/~magnien/Diameter

I 2 millions of vertices, diameter 32 within 1

I Further experimentations by Crescenzi, Grossi, Marino (inESA 2010)which confirm the excellence of the lower bound usingBFS ! ! ! !

http://www-rp.lip6.fr/~magnien/Diameter



I Since α is a constant (≤ 1000), this method is still in lineartime and works extremely well on huge graphs (Web graphs,Internet . . .)

I How can we explain the success of such a method ?

I Due to the many counterexamples for the 2 consecutive BFSprocedure. An explanation is necessary !



2 kind of explanations

The method is good or the data used was good.

Partial answer

The method also works on several models of random graphs.So let us try to prove the first fact

Restriction

First we are going to focus our study on the 2 consecutive BFS.



Chordal graphs

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

2. If G is a chordal graph, Corneil, Dragan, H., Paul 2001, usinga variant called 2 consecutive LexBFSd(u, v) ≤ diam(G ) ≤ d(u, v) + 1

3. Generalized by Corneil, Dragan, Kohler 2003 using 2consecutive BFS :d(u, v) ≤ diam(G ) ≤ d(u, v) + 1



The 4-sweep method :Crescenzi, Grossi, MH, Lanzi,Marino 2011

L = max{ecc(a1), ecc(a2)} and R = min{ecc(r), ecc(m1)}



Ce n’est pas un algorithme !



An exact algorithm !

Compute the excentricity of the leaves of the treewith a stop condition.Complexity is O(nm) in the worst case, but often linear in practice.



iFub an exact O(mn) algorithm



Bad example



Results,

Diametre Facebook = 41 ! Backstrom, Boldi, Rosa, Uganden,Vigna 2011



Up to now

At least 2 new searches with no applications yet2 consecutive LBFS enough to recognize cographs4-5 consecutive LBFS are enough to capture the structure intervalgraphs(similar result for co-comparability graphs) ?



Easy extensions

1. To weighted graphs by replacing BFS with Dijkstra’s algorithm

2. To directed graphs



Some Perspectives1. Extend diameter results to efficient computation of centers

(one or all)2. Understand why the 4-sweep method works so well (analogy

with Quicksort)3. Uses a series of graph search for preprocessing when faced to

hard combinatorial problems (already used in biologicalapplications of interval graphs and for quadratic integerprogramming)Consecutive Ones property (C1P) (find a good ordering of thecolumns of the matrix such that in each row the ones areconsecutive)A kind of filtering process !

4. A theory of graph searches : searchoids special cases ofgreedoids ? Develop theoretical tools to analyze multisweepalgorithms and prove fixed points properties using latticetheory.



Hints for future work

1. Develop approximation algorithms even for polynomialproblems but applied on huge data on which only linear timealgorithms can be processed.

2. Apply to max flow problems in competition with spectralgraph theory a la PageRank

3. Reconsider consecutive ones property and planarity

4. Community detection in networks (using LexDFS ?)



Thank you for your attention !

Merci de votre ecoute !

du nouveau sur les parcours de graphes...3.using bfs to search an augmenting path provides a...

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