lecture graphs

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Outline

Introduction tograph theory and algorithms

Jean-Yves LExcellent and Bora Ucar

GRAAL, LIP, ENS Lyon, France

CR-07: Sparse Matrix Computations, September 2010http://graal.ens-lyon.fr/~bucar/CR07/

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http://graal.ens-lyon.fr/~bucar/CR07/http://graal.ens-lyon.fr/~bucar/CR07/http://graal.ens-lyon.fr/~bucar/CR07/http://graal.ens-lyon.fr/~bucar/CR07/http://find/http://goback/


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Outline

Outline

1 Definitions and some problems

2 Basic algorithmsBreadth-first searchDepth-first searchTopological sortStrongly connected components

3 Questions

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http://find/http://goback/


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Definitions and some problemsBasic algorithms

Questions

Graph notations and definitions

A graph G = (V,E) consists of a finite set V, called the vertex set and afinite, binary relation E on V, called the edge set.

Three standard graph models

Undirected graph: The edges are unordered pair of vertices, i.e.,{u, v} E for some u, v V.

Directed graph: The edges are ordered pair of vertices, that is, (u, v) and(v, u) are two different edges.

Bipartite graph: G = (U V,E) consists of two disjoint vertex sets U

and V such that for each edge (u, v) E, u U and v V.

An ordering or labelling ofG = (V,E) having n vertices, i.e., |V| = n, isa mapping ofV onto 1, 2, . . . , n.

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Questions

Matrices and graphs: Rectangular matrices

The rows/columns and nonzeros of a given sparse matrix correspond (withnatural labelling) to the vertices and edges, respectively, of a graph.

Rectangular matrices

A =

0@

1 2 3 41 2 3

1A

Bipartite graph

1

r2

r3

1c

2c

3c

4c

r

The set of rows corresponds to one of the vertex set R, the set of columns

corresponds to the other vertex set C such that for each aij = 0, (ri, cj) is an

edge.

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Questions

Matrices and graphs: Square unsymmetric pattern

The rows/columns and nonzeros of a given sparse matrix correspond (with

natural labelling) to the vertices and edges, respectively, of a graph.

Square unsymmetricpattern matrices

A =

0@

1 2 31 2 3

1A

Graph models

Bipartite graph as before.

Directed graph

v2 3v

1v

The set of rows/cols corresponds the vertex set V such that for each aij = 0,

(vi, vj) is an edge. Transposed view possible too, i.e., the edge (vi, vj) directed

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Questions

Matrices and graphs: Square unsymmetric pattern

A special subclass

Directed acyclic graphs (DAG):A directed graphs with no loops(maybe except for self-loops).

DAGs

We can sort the vertices such thatif (u, v) is an edge, then u appearsbefore v in the ordering.

Question: What kind of matrices have a DAG?

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Questions

Matrices and graphs: Symmetric pattern

The rows/columns and nonzeros of a given sparse matrix correspond (with

natural labelling) to the vertices and edges, respectively, of a graph.

Square symmetricpattern matrices

A =

0@

1 2 31 2 3

1A

Graph models

Bipartite and directed graphs as before.

Undirected graph

v2 3v

1v

The set of rows/cols corresponds the vertex set V such that for each

aij, aji = 0, {vi, vj} is an edge. No self-loops; usually the main diagonal is

assumed to be zero-free.7/124 CR09


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Questions

Definitions: Edges, degrees, and paths

Many definitions for directed and undirected graphs are the same. Wewill use (u, v) to refer to an edge of an undirected or directed graph toavoid repeated definitions.

An edge (u, v) is said to incident on the vertices u and v.

For any vertex u, the set of vertices in adj(u) = {v : (u, v) E} arecalled the neighbors ofu. The vertices in adj(u) are said to beadjacent to u.

The degree of a vertex is the number of edges incident on it.

A path p of length k is a sequence of vertices v0, v1, . . . , vk where(vi1, vi) E for i = 1, . . . , k. The two end points v0 and vk aresaid to be connected by the path p, and the vertex vk is said to bereachable from v0.

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D fi iti d bl


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Questions

Definitions: Components

An undirected graph is said to be connected if every pair of verticesis connected by a path.

The connected components of an undirected graph are the

equivalence classes of vertices under the is reachable fromrelation.

A directed graph is said to be strongly connected if every pair ofvertices are reachable from each other.

The strongly connected components of a directed graph are theequivalence classes of vertices under the are mutually reachablerelation.

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Definitions and some problems


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Questions

Definitions: Trees and spanning trees

A tree is a connected, acyclic, undirected graph. If an undirected graph isacyclic but disconnected, then it is a forest.

Properties of trees

Any two vertices are connected by a unique path.

|E| = |V| 1

A rooted tree is a tree with a distinguished vertex r, called the root.

There is a unique path from the root r to every other vertex v. Anyvertex y in that path is called an ancestor ofv. Ify is an ancestor ofv,

then v is a descendant ofy.The subtree rooted at v is the tree induced by the descendants ofv,rooted at v.

A spanning tree of a connected graph G = (V,E) is a tree T = (V,F),such that F E.

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Questions

Ordering of the vertices of a rooted tree

A topological ordering of a rooted tree is an ordering that numberschildren vertices before their parent.

A postorder is a topological ordering which numbers the vertices inany subtree consecutively.

u w

x

y

z

v

with topological ordering

1 3

2

4

5

6

Rooted spanning tree

w

yz

xu

v

Connected graph G

u w

x

y

z

v

1

6

54

3

2

Rooted spanning trewith postordering

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Questions

Postordering the vertices of a rooted tree I

The following recursive algorithm will do the job:

[porder]=PostOrder(T, r)

for each child c of r doporder [porder, PostOrder(T, c)]

porder [porder, r]

We need to run the algorithm for each root r when T is a forest.

Usually recursive algorithms are avoided, as for a tree with large numberof vertices can cause stack overflow.

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Questions

Postordering the vertices of a rooted tree II

[porder]=PostOrder(T, r)

porder []seen(v) False for all v Tseen(r) TruePush(S, r)

while NotEmpty(S) dov Pop(S)if a child c of v with seen(c) = False thenseen(c) TruePush(S, c)

elseporder [porder, v]

Again, have to run for each root, ifT is a forest.

Both algorithms run in O(n) time for a tree with n nodes.

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pBasic algorithms

Questions

Permutation matrices

A permutation matrix is a square (0, 1)-matrix where each row and

column has a single 1.

IfP is a permutation matrix, PPT = I, i.e., it is an orthogonal matrix.Let,

A =

1 2 3

1 2 3

and suppose we want to permute columns as [2, 1, 3]. Define p2,1 = 1,p1,2 = 1, p3,3 = 1, and B= AP (if column j to be at position i, set

pji = 1)

B=

2 1 3

1 2 3

=

1 2 3

1 2 3

1 2 3

1 12 13 1

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Basic algorithmsQuestions

Matching in bipartite graphs and permutations

A matching in a graph is a set of edges no two of which share a commonvertex. We will be mostly dealing with matchings in bipartite graphs.

In matrix terms, a matching in the bipartite graph of a matrixcorresponds to a set of nonzero entries no two of which are in the samerow or column.

A vertex is said to be matched if there is an edge in the matchingincident on the vertex, and to be unmatched otherwise. In a perfectmatching, all vertices are matched.

The cardinality of a matching is the number of edges in it. A maximumcardinality matching or a maximum matching is a matching of maximumcardinality. Solvable in polynomial time.

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Definitions and some problemsB i l i h


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Matching in bipartite graphs and permutations

Given a square matrix whose bipartite graph has a perfect matching, such

a matching can be used to permute the matrix such that the matchingentries are along the main diagonal.

1

r2

r3 3

c

2c

1cr

1 2 3

1 2 3

2 1 3

1 2 3

=

1 2 3

1 2 3

1 2 3

1 12 13 1

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Definitions and some problemsB i l ith


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Definitions: Reducibility

Reducible matrix: An n n square matrix is reducible if there exists ann n permutation matrix P such that

PAPT = A11 A12

O A22 ,

where A11 is an r r submatrix, A22 is an (n r) (n r) submatrix,where 1 r< n.

Irreducible matrix: There is no such a permutation matrix.

Theorem: An n n square matrix is irreducible iff its directed graph isstrongly connected.

Proof: Follows by definition.

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Definitions: Fully indecomposability

Fully indecomposable matrix: There is no permutation matrices P and Qsuch that

PAQ=

A11 A12O A22

,

with the same condition on the blocks and their sizes as above.

Theorem: An n n square matrix A is fully indecomposable iff for somepermutation matrix P, the matrix PA is irreducible and has a zero-freemain diagonal.

Proof: We will come later in the semester to the if part.

Only if part (by contradiction): Let B= PA be an irreducible matrix withzero-free main diagonal. B is fully indecomposable iffA is (why?).Therefore we may assume that A is irreducible and has a zero-freediagonal. Suppose, for the sake of contradiction, A is not fullyindecomposable.

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Fully indecomposable matrices

Fully indecomposable matrix

There is no permutation matrices P and Q such that

PAQ= A11 A12

O A22 ,

with the same condition on the blocks and their sizes as above.

Proof cont.: Let P1AQ1 be of the form above with A11 of size r r. Wemay write P

1AQ

1= AQ, where A = P

1AP

1

T with zero-free diagonal(why?), and Q = P1Q1 is a permutation matrix which has to permute(why?) the first r columns among themselves, and similarly the last n rcolumns among themselves. Hence, A is in the above form, and A isreducible: contradiction. 2

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Definitions: Cliques and independent sets

Clique

In an undirected graph G = (V,E), a set of vertices S V is a clique if for alls, t S, we have (s, t) E.

Maximum clique: A clique of maximum cardinality (finding a maximum cliquein an undirected graph is NP-complete).

Maximal clique: A clique is a maximal clique, if it is not contained in anotherclique.

In a symmetric matrix A, a clique corresponds to a subset of rows R and thecorresponding columns such that the matrix A(R,R) is full.

Independent set

A set of vertices is an independent set if none of the vertices are adjacent toeach other. Can we find the largest one in polynomial time?

In a symmetric matrix A, an independent set corresponds to a subset of rows Rand the corresponding columns such that the matrix A(R,R) is either zero, ordiagonal.

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s c go t sQuestions

Definitions: More on cliques

Clique: In an undirected graph G = (V,E), a set of vertices S V is aclique if for all s, t S, we have (s, t) E.

In a symmetric matrix A corresponds to a subset of rows R and thecorresponding columns such that the matrix A(R,R) is full.

Cliques in bipartite graphs: Bi-cliquesIn a bipartite graph G = (U V,E), a pair of sets R,C where R Uand C V is a bi-clique if for all a R and b C, we have (a, b) E.

In a matrix A, corresponds to a subset of rows R and a subset of columnsC such that the matrix A(R,C) is full.

The maximum node bi-clique problem asks for a bi-clique of maximumsize (e.g., |R| + |C|), and it is polynomial time solvable, whereasmaximum edge bi-clique problem (e.g., asks for a maximum |R| |C|) isNP-complete.

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gQuestions

Definitions: Hypergraphs

Hypergraph: A hypergraph H= (V,N) consists of a finite set V called the

vertex set and a set of non-empty subsets of vertices N called the hyperedgeset or the net set. A generalization of graphs.

For a matrix A, define a hypergraph whose vertices correspond to the rows andwhose nets correspond to the columns such that vertex vi is in net nj iffaij = 0(the column-net model).

A sample matrix

0@

1 2 3 4

1 2 3

1A

The column-net hypergraph model

4

3n

1n

1v

2v

3v

n

n2

v

2n

4n

3n

1n

1

2

3

v

v

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Breadth-first searchDepth-first searchTopological sort


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QuestionsTopological sortStrongly connected components

Basic graph algorithms

Searching a graph: Systematically following the edges of the graph so asto visit all the vertices.

Breadth-first search,

Depth-first search.

Topological sort (of a directed acyclic graph): It is a linear ordering of allthe vertices such that if (u, v) directed is an edge, then u appears beforev in the ordering.

Strongly connected components (of a directed graph; why?): Recall thata strongly connected component is a maximal set of vertices for whichevery pair its vertices are reachable. We want to find them all.

We will use some of the course notes by Cevdet Aykanat(http://www.cs.bilkent.edu.tr/~aykanat/teaching.html)

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Q i

http://www.cs.bilkent.edu.tr/~aykanat/teaching.htmlhttp://www.cs.bilkent.edu.tr/~aykanat/teaching.htmlhttp://www.cs.bilkent.edu.tr/~aykanat/teaching.htmlhttp://www.cs.bilkent.edu.tr/~aykanat/teaching.htmlhttp://find/http://goback/


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QuestionsTopological sortStrongly connected components

Breadth-first search: Idea

Graph G =(V,E),directed or undirected with adjacency list repres.

GOAL: Systematically explores edges ofG to

discover every vertex reachable from the source vertex s

compute theshortest path distance of every vertex

from the source vertexs

produce a breadth-first tree (BFT) G!

with root s

"BFT contains all vertices reachable from s

" the unique path from any vertex v to s in G!

constitutes a shortest path from s to v in G

IDEA: Expanding frontier across the breadth -greedy- propagate a wave 1 edge-distance at a time

using a FIFO queue: O(1) time to update pointers to both ends

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Q ti



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Questionsp g

Strongly connected components

Breadth-first search: Key components

Maintains the following fields for each u ! V

color[u]:color ofu

"WHITE : not discovered yet

"GRAY : discovered and to be or being processed

"BLACK: discovered and processed

#[u]: parent ofu (NIL ofu = s or u is not discovered yet)

d[u]: distance ofu from s

Processing a vertex = scanning its adjacency list

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Questions



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Questionsp g


Breadth-first search: Algorithm

BFS(G, s)for each u ! V"{s} do

color[u]#WHITE$[u]# NIL; d[u]#%

color[s]# GRAY$[s]# NIL; d[s]# 0

Q# {s}while Q & ' do

u# head[Q]for each v inAdj[u] do

ifcolor[v] = WHITE thencolor[v]# GRAY

$[v]# ud[v]# d[u] + 1ENQUEUE(Q, v)

DEQUEUE(Q)color[u]# BLACK

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Questions



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QuestionsStrongly connected components

Breadth-first search: Example

Sample Graph:

a

b

g

c

f

d

e

h

i

0

sFIFO just after

queue Q processing vertex

!a" -

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Questions

Breadth-first searchDepth-first searchTopological sortS l d


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QuestionsStrongly connected components


a

b

g

c

f

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e

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0

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1

sFIFO just after


!a" -

!a,b,c" a

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Questions

Breadth-first searchDepth-first searchTopological sortSt l t d t


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QStrongly connected components


a

b

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c

f

d

e

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0

1

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2

sFIFO just after


!a" -

!a,b,c" a

!a,b,c,f" b

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Questions

Breadth-first searchDepth-first searchTopological sortStrongly connected components


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a

b

g

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f

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e

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0

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sFIFO just after


!a" -

!a,b,c" a

!a,b,c,f" b

!a,b,c,f,e" c

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Questions



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a

b

g

c

f

d

e

h

i

0

1

1

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s

3 3

FIFO just after


!a" -

!a,b,c" a

!a,b,c,f" b

!a,b,c,f,e" c

!a,b,c,f,e,g,h" f

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Questions



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a

b

g

c

f

d

e

h

i

0

1

1

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2

s

3 3

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FIFO just after


!a" -

!a,b,c" a

!a,b,c,f" b

!a,b,c,f,e" c

!a,b,c,f,e,g,h" f

!a,b,c,f,e,g,h,d,i" e

all distances are filled in after processing e

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Questions



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g y p


a

b

g

c

f

d

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0

1

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s

3 3

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3

FIFO just after


!a" -

!a,b,c" a

!a,b,c,f" b

!a,b,c,f,e" c

!a,b,c,f,e,g,h" f

!a,b,c,f,e,g,h,d,i" g

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Questions



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g y p


a

b

g

c

f

d

e

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0

1

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FIFO just after


!a" -

!a,b,c" a

!a,b,c,f" b

!a,b,c,f,e" c

!a,b,c,f,e,g,h" f

!a,b,c,f,e,g,h,d,i" h

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Questions



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a

b

g

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f

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e

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i

0

1

1

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s

3 3

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FIFO just after


!a" -!a,b,c" a

!a,b,c,f" b

!a,b,c,f,e" c

!a,b,c,f,e,g,h" f

!a,b,c,f,e,g,h,d,i" d

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Questions



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a

b

g

c

f

d

e

h

i

0

1

1

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2

s

3 3

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FIFO just after


!a" -

!a,b,c" a

!a,b,c,f" b

!a,b,c,f,e" c

!a,b,c,f,e,g,h" f

!a,b,c,f,e,g,h,d,i" i

algorithm terminates: all vertices are processed

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Questions



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Breadth-first search: Analysis

Running time: O(V+E) = considered linear time in graphs

initialization: !(V)

queue operations: O(V)

" each vertex enqueued and dequeued at most once" both enqueue and dequeue operations take O(1) time

processinggray vertices: O(E)

" each vertex is processed at most once and

!#

!=

Vu

EuAdj )(|][|

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Questions



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Breadth-first search: The paths to the root

BFS(G, s) , where V!={v " V: ![v] # NIL}${s} andE

! ={(![v],v) "E: v " V

!%{s}}

is a breadth-firsttree such that

% V!

consistsof all vertices in Vthat are reachable from s

% &v " V!

,unique path p(v, s) in G!

constitutes a sp(s, v) in G

PRINT-PATH(G, s, v)

ifv = s then print selse if![v] = NIL then

print no s!v path

else

PRINT-PATH(G, s, ![v] )print v

Prints out vertices on a

s!v shortest path

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Questions



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Breadth-first search: The BFS tree

Breadth-First Tree Generated by BFS

a

b

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s

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a

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s

3 3

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3

BFS(G,a) terminated BFT generated by BFS(G,a)

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Questions



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Depth-first search: Idea

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Questions



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Depth-first search: Key components

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Questions



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Depth-first search: Algorithm

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);" -.*/0E=!>" %$'&" %$'& %-

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Definitions and some problemsBasic algorithmsQuestions


D h fi h A l i


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Depth-first search: Analysis

# #

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D h fi h E l


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Depth-first search: Example

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D th fi t h E l


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Breadth-first search

Depth-first searchTopological sortStrongly connected components

Depth first search: Example


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Questions





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Questions





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p p

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Depth-first search: Example# #


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Depth-first search: DFT and DFF


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Depth-first search: Parenthesis theorem


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Thm: In any DFS ofG=(V,E), let int[v] = [d[v], f[v]]

then exactly one of the following holds

for any u and v !V

int[u] and int[v] are entirely disjoint int[v] is entirely contained in int[u] and

v is a descendant ofu in a DFT

int[u] is entirely contained in int[v] and

u is a descendant ofv in a DFT

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Depth-first search: Parenthesis theorem


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ParenthesisTheorem

x y z

s t

w v u

2

3 4 5 6

7 9 1 0

8 1 11 1 2

1 4 1 5

1 3 1 6

1 2 3 4 5 6 7 8 9

x z

s y u

w v t

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6

1 0 1 1 1 2 1 3 1 4 1 5 1 6

( x ( s ( w w ) ( v v ) s ) ( y ( t t) y ) x ) ( z ( u u ) z )

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Depth-first search: Edge classification


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Tree Edge: discover a new (WHITE) vertex!GRAY to WHITE"

Back Edge: from a descendent to an ancestor in DFT!GRAY to GRAY"

Forward Edge:from ancestor to descendent in DFT!GRAY to BLACK"

Cross Edge: remaining edges (btwn trees and subtrees)!GRAY to BLACK"

Note: ancestor/descendent is wrt Tree Edges

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Depth-first search: Edge classification


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How to decide which GRAY to BLACK edgesare forward, which are cross

Let BLACK vertex v !Adj[u] is encounteredwhile processing GRAY vertex u

(u,v) isaforward edge ifd[u] > d[v]

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Depth-first search: Edge classification example


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Breadth-first searchDepth-first search

Topological sortStrongly connected components



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Depth-first search: Undirected graphs

Edge classification


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g

Any DFS on an undirected graph produces only Tree and Back edges.

16/17

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Depth-first search: Non-recursive algorithm


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[, d, f]=DFS(G, v)

top 1stack(top) vd(v) ctime 1while top> 0 dou stack(top)

if there is a vertex w Adj(u) where (w) is not set thentop top+ 1stack(top) w(w) ud(w) ctime ctime+ 1

elsef(u) ctime ctime+ 1top top 1

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Topological sort


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Topological sort (of a directed acyclic graph): It is a linear ordering of allthe vertices such that if (u, v) is a directed edge, then u appears before v

in the ordering.

Ordering is not necessarily unique.

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Questions



Topological sort: Example


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3/4

shortsocks watch

pants shoes shirt

tiebelt jacket

pantsundershort

socks shoes watch shirt belt tie jacket

6/7 3/4

11/16 17/18 9/10

13/14

2/5

1/812/15

17/18 11/16 12/15 13/14 9/10 1/8 6/7 2/5

under

100/124 CR09


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Topological sort: Algorithm

The algorithm
http://find/http://find/http://goback/


101/124

run DFS(G)when a vertex is finished, output it

vertices are output in the reverse topologically sorted order

Runs in O(V + E) time a linear time algorithm.

The algorithm: Correctness

if (u, v) E, then f[u] > f[v]

Proof: Consider the color ofv during exploring the edge (u, v), where uis Gray. 2

v cannot be Gray (otherwise a Back edge in an acyclic graph !!!).Ifv is White, then u is an ancestor ofv, hence f[u] > f[v].

Ifv is Black, f[v] is computed already, f[u] is going to be computed,hence f[u] > f[v].

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Questions



Strongly connected components (SCC)


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The strongly connected components of a directed graph are theequivalence classes of vertices under the are mutually reachablerelation.

For a graph G = (V,E), the transpose is defined as GT

= (V,ET

),where ET = {(u, v) : (v, u) E}.

Constructing GT from G takes O(V + E) time with adjacency list (likethe CSR or CSC storage format for sparse matrices) representation.

Notice that G and GT have the same SCCs.

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Questions



Strongly connected components: Algorithm


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(1) Run DFS(G) to compute finishing times for all u!V

(2) Compute GT

(3) Call DFS(GT) processing vertices in main loop indecreasing f[u] computed in Step (1)

(4) Output vertices of each DFT in DFF of Step (3) as a

separate SCC

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Strongly connected components: Analysis

Lemma 1: no path between a pair of vertices in


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Lemma 1: no path between a pair of vertices in

the same SCC, ever leaves the SCC

Proof: let u and v be in the same SCC! u! v

let w be on some path u!w! v! u!w

but v! u! ! a path w!v! u! w!u

therefore u and w are in the same SCC

QED

u

v

wSCC

104/124 CR09


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Strongly connected components: Example


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a b c

e

d

f g h

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a b c

e

d

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1

(1)Run DFS(G) to compute finishing times for all u!V

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a b c

e

d

f g h

1

1

10

2 73 4 5 6

8 9

(1)Run DFS(G) to compute finishing times for all u!V

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!


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a b c

e

d

f g h

1

2

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2 73 4 5 6

8 911

(1)Run DFS(G) to compute finishing times for allu!

V

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2 1


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a b c

e

d

f g h

1 10

2 73 4 5 6

8 916111413

1512

Vertices sorted according to the finishing times:

!!!!b,e,a,c,d,g,h,f""""

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T


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a b c

e

d

f g h

(2)Compute GT

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T


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a b c

e

d

f g h

(3) Call DFS(GT

) processing vertices in main loop indecreasingf[u] order: !!!!b,e,a,c,d,g,h,f""""

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T


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a b c

e

d

f g h

r1=

(3) Call DFS(G ) processing vertices in main loop indecreasingf[u] order: !!!!b,e,a,c,d,g,h,f""""

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DFS T


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a b c

e

d

f g h

r1=

(3) CallDFS

(G ) processing vertices in main loop indecreasingf[u] order: !!!!b,e,a,c,d,g,h,f""""

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T


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a b c

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r1

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=


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T


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=


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DFS T


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a b c

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d

f g h

r1

= r2

=

r3

=

(3) CallDFS

(G ) processing vertices in main loop indecreasingf[u] order: !!!!b,e,a,c,d,g,h,f""""

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Questions




(3) Call DFS(GT) processing vertices in main loop in


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a b c

e

d

f g h

r1

= r2

=

r3

=

(3) Call DFS(G ) processing vertices in main loop in

decreasingf[u] order: !!!!b,e,a,c,d,g,h,f""""

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Questions




(3) Call DFS(GT) processing vertices in main loop in


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a b c

e

d

f g h

r1

= r2

=

r3

= r4

=

( ) ( ) p g p

decreasingf[u] order: !!!!b,e,a,c,d,g,h,f""""

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Questions




(4) Output vertices of each DFT in DFF as a separate SCC


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a b c

e

d

fg

h

r1

= r2

=

r3

= r4

=

Cb={b,a,e}Cg={g,f} Ch={h}

Cc={c,d}

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Questions




a b c d


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e f g h

hf,g

c,d

a,b,e

Acyclic component

graph

Cb Cg

Cc

Ch

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Questions



Strongly connected components: Observations


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In any DFS(G), all vertices in the same SCC are placed in the sameDFT.

In the DFS(G) step of the algorithm, the last vertex finished in an

SCC is the first vertex discovered in the SCC.

Consider the vertex r with the largest finishing time. It is a root of aDFT. Any vertex that is reachable from r in GT should be in theSCC of r (why?)

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Questions



SCC and reducibility


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To detect if there exists a permutation matrix P such that

PAPT =

A11 A12O A22

,

where A11 is an r r submatrix, A22 is an (n r) (n r) submatrix,where 1 r< n:

run SCC on the directed graph ofA to identify each strongly connectedcomponent as an irreducible block (more than one SCC?). Hence A11,

too, can be in that form (how many SCCs?).

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Questions

Could not get enough of it: Questions


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How would you describe the following in the language of graphs

the structure ofPAPT for a given square sparse matrix A and apermutation matrix P,

the structure ofPAQ for a given square sparse matrix A and two

permutation matrices P and Q,

the structure ofAk, for k> 1,

the structure ofAAT,

the structure of the vector b, where b= Ax for a given sparsematrix A, and a sparse vector x.

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Questions

Could not get enough of it: Questions


124/124

Can you define:

the row-net hypergraph model of a matrix.

a matching in a hypergraph (is it a hard problem?).

Can you relate:

the DFS or BFS on a tree to a topological ordering? postordering?

Find an algorithm

how do you transpose a matrix in CSR or CSC format?

124/124 CR09

lecture graphs

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