01. Graphs

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(C) GOYANI MAHESH(C) GOYANI MAHESH 11

DATASTRUCTURES

MAHESH GOYANI

MAHATMA GANDHI INSTITUTE OF TECHNICAL EDUCATION & RESEARCH CENTER

[email protected]

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GRAPHS

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TERMINOLOGY

A Graph G consists of

1. Set of vertices V (Called node), (V = {v1, v2, v3, }) and

2. Set of Edged E (Called Arc) (i.e. E {e1, e2, e3, })

A graph can be described as G = (V, E), where V is a finite and non empty set ofvertices and E is a set of pairs of edges.

Tree must be a graph, but graph need not be a tree

E

D

C

A

B

Set of Vertices V = {A, B, C, D, E}

Set of Edges E = {(A,B), (A, C), (A, D), (C, D), (C, E) }

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a b

d c

a1

d1

c1 b1

ISOMORPHICGRAPH

a b

c d

e

a b

c d

SUBGRAPH

a b

cd

eSPANING SUB

GRAPH

TERMINOLOGY

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a

b

d

c

Two vertices are said to be adjacentif they are joined by an edge (i.e. (a, b)). The number of edges incident on a vertex is called degree

Indegree of a node is number of arcs that have head as incident on that vertex.

Outdegree of a vertex is number of arcs that have tail as incident on that vertex.

Degree of isolated vertex is ZERO. (e.g. Node X)

Node b has outdegree 1, indegree 2 and degree 3.

A path from mode to it self is called cycle (Node C & D)

If a graph contains cycle, than it is cyclic graph, other wise it is acyclic graph

A directed acyclic graph is called a DAG

TERMINOLOGY

X

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A Graph is said to be weighted graph (network) if every edge and / or vertices inthe graph is assigned with some weight or value.

A Weighted graph can be defined as G = (V, E, We, Wv) where,

V = Set of vertices

E = Set of Edges

We = Set of Weight of edges

Wv = Set of Weight of Vertices

N

C

KB

47

27

55

39

16

TERMINOLOGY

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Undirected graph is said to be Connected Graph if there exist a path from anyvertex to any other vertex. Other wise it is called Disconnected Graph.

Undirected graph is said to be Fully Connected Graph(strongly connected / complete graph) if there exist a

path from all vertex to all other vertex.

A complete graph with n vertices will have n(n-1)/2edges.

a

b

c

d

a

b

c

d

a

b

c

d

TERMINOLOGY

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v1 v2 v3

v5v6

v9 v4

v7

v8

e1 e2

e3

e9

e4

e12

e5e6

e11e7

e8

e10

In a directed graph, a path is a sequence of edges such that the edges areconnected with each other.

(e1, e3, e4, e5, e12, e9, e11, e6, e7, e8, e11)

A path is said to be elementary path if it does not meet the same vertex twice.

(e1, e3, e4, e5, e6, e7, e8)

A path is said to be simple path if it does not meet the same edges twice.

(e1, e3, e4, e5, e6, e7, e8, e11, e12)

A Circuitis a path in which terminal vertex en coincides with the initial vertex e1.

Simple circuit

(e1, e3, e4, e5, e12, e9, e10)

Elementary circuit

(e1, e3, e4, e5, e6, e7, e8, e10)

TERMINOLOGY

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GRAPH REPRESENTATION

1. Sequential representation (Using Array)2. Linked representation (Using Linked List)

1

3

2

4 5

100005

000014

010003

100002

001101

54321i

j

Directed Graph

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1

3

2

4 5100105

001014

010013

100012

011101

54321

Undirected Graph

i j

ARRAY REPRESENTATION

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0-1-1-1-15

-1-1-1-124

-14-1-1-137-1-1-1-12

-1-135-11

54321ij

1

3

2

4 5

3

4

2

5

7

0

Weighted Graph

ARRAY REPRESENTATION

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LINKED LIST REPRESENTATION

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4 5

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0

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4 5

1 2 3

2 5

3 4

4 1

5 5

1 2 5 3 3

2 5 7

3 4 4

4 1 2

5 5 0

Directed Graph

Weighted Graph

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OPERATION ON GRAPH

Create ( )

Search ( ) Delete ( )

Traverse ( )

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1. Input the Vertices & Edges of the Graph G =( V, E )

2. Input the source vertex and assign it to thevariable S

3. Push S to Queue

4. While Front

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A

B C

D E F

G H

I

Front = -1

Rear = -1

A

Front = 0

Rear = 0

A B C

Front = 1

Rear = 2

BREADTH FIRST SEARCH (BFS)

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A

B C

D E F

G H

I

A B C D E

Front = 2

Rear = 4

A B C D E F

Front = 3

Rear = 5

A B C D E F G

Front = 4

Rear = 6


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A B C D E F G

Front = 5

Rear = 6

A B C D E F G H

Front = 6

Rear = 7

A B C D E F G H

Front = 7

Rear = 7

A

B C

D E F

G H

I


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A B C D E F G H

Front = 8

Rear = 7A

B C

D E F

G H

I


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1. Input the Vertices & Edges of the Graph G =( V, E )

2. Input the source vertex and assign it to thevariable S

3. Push S to STACK

4. While TOP != -1, repeat step 5 & 6.

5. Pop front element and display it

6. Push the vertices, which are neighbor to justpopped element, if it is not in the STACK and

displayed

7. Exit.

DEPTH FIRST SEARCH (DFS)

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DEPTH FIRST SEARCH (DFS)

A

B C

D E F

G H

I

A

B

C

D

E

C

G

E

C

E

C

F

C

H

C C

A B D G E F H C

TOP

TOP

TOP TOP

TOP TOP TOP

TOP

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MINIMUM SPANNING TREE (MST)

A minimum spanning tree for a graph G = (V, E) is a sub graph G1 = (V1,E1) of G, contains all the vertices of G, and

The vertex set V1 is same as that at graph G

The Edge E1 is sub set of G

And there is no cycle.

If graph G is not connected graph than it can not have any spanning tree. Inthis case, it will have spanning forest.

Suppose if Graph G has n vertices than MST will have (n-1) edges.

A MST for a weighted graph is a spanning tree with minimum weight.

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SHORTEST PATH

A Path from source vertex a to b is said to be shortest path if there is no otherpath from a to b with lower weights.

Dijkstars Algorithm is used to find shortest path.

Set V = {V1, V2, V3, Vn} contains the vertices and the edges E = {e1, e2, en} ofthe graph G. We is the weight of an edge e, which contains the vertices V1 and V2. Q isa set of vertices, which are not visited. M is the vertex in Q for which weight Wm isminimum i.e. minimum cost edge. S is source vertex.

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DIJKSTRAS ALGPRITHM

1. Input the source vertex and assign it to S.

(a). Set W(s) = 0 and

(b). Set W(v) = ______ for all verttices V is not equal to S

2. Set Q = V which is set of vertices in the graph.

3. Suppose m be a vertices in Q for which W(m) is minimum.

4. Make the vertices m as visited and delete it from Q

5. Find the vertices I which are incident with m and member of Q

6. Update the weight of vertices I = {i1, i2, , ik} by

(a) W(i) = min [W(i1), W(m) + W(m,i1)]

7. If any changes is made in W(v), store the vertices to correspondingvertices I, using the array for tracing the sortest path

8. Repeat the process from strep 3 to 7 untill Q is empty

9. Exit

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A

B C

E D

F

6 5

5

32 3

1

2

4

3

Source vertex = A

V = (A, B, C, D, E, F) = Q

V A B C D E F

W(V) 0

Q A B C D E F

A

B

C

D

E

F

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ITERATION : 1

m = A

W (A, A) = 0Now Q = (B, C, D, E, F)

Two edges incident with m

i.e. I = (B, C)

W(B) = min( w(B), W(A) + W(A,B))

= (-, 0 + 6) = 6

W(C) = min (w(C) , W(A) + W(A,C))

= (-, 0 + 5) = 5

A

B C

E D

F

6 5

5

32 3

1

2

4

3

V A B C D E F

W(V) 0 6 5

Q B C D E F

A

B A

C A

D

E

F

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ITERATION : 2

m = C (Because W(C) is minimum & is member of Q)

Now Q = (B, D, E, F)Two edges incident with m

i.e. I = (D, F)

W(D) = min( w(D), W(C) + W(C,D))

= (-, 5 + 2) = 7

W(F) = min (w(F) , W(C) + W(C,F))

= (-, 5 + 3) = 8

A

B C

E D

F

6 5

5

32 3

1

2

4

3

V A B C D E F

W(V) 0 6 5 7 8

Q B D E F

A

B A

C A

D C

E

F C

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ITERATION : 3

m = B (Because W(B) is minimum and is member of Q)

Now Q = (D, E, F)Three edges incident with m (C, E, F) but C is not in Q

So, I = (E, F)

W(E) = min( w(E), W(B) + W(B,E))

= (-, 6 + 3) = 9

W(F) = min (w(F) , W(B) + W(B,F))

= (-, 6 + 2) = 8

A

B C

E D

F

6 5

5

32 3

1

2

4

3

V A B C D E F

W(V) 0 6 5 7 9 8

Q D E F

A

B A

C A

D C

E B

F C

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ITERATION : 4

m = D (Because W(D) is minimum and is member of Q)

Now Q = (E, F)One edges incident with m

So, I = (F)

W(F) = min (w(F) , W(D) + W(D,F))

= (-, 7 + 1) = 8

A

B C

E D

F

6 5

5

32 3

1

2

4

3

V A B C D E F

W(V) 0 6 5 7 9 8

Q E F

A

B A

C A

D C

E B

F C

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ITERATION : 5

m = F (Because W(F) is minimum and is member of Q)

s = FQ =(F)

One edges incident with m

So, I = (E)

W(E) = min (w(F) , W(F) + W(F, E))

= (9, 9 + 3) = 9

A

B C

E D

F

6 5

5

32 3

1

2

4

3

V A B C D E F

W(V) 0 6 5 7 9 8

Q E

A

B A

C A

D C

E B

F C

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Now E is the only chain, hence we stop the iteration andthe final table is as bellow

A

B C

E D

F

6 5

5

32 3

1

2

4

3

V A B C D E F

W(V) 0 6 5 7 9 8

A

B A

C A

D C

E B

F C

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APPLICATION

Shortest Path (City Map)

Flow Function (Water Flow)

Scheduling (Cook)

01. Graphs

Documents

Transcript of 01. Graphs