DataStructure - Graphs

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Lecture outlines

Definition of Graphs and Related Concepts

Representation of Graphs Graph Applications

Graph Traversal


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Definition of Graphs

Graphs are non liner data structures.

A graph is a finite set of nodes with line between

nodes

Nodes are called vertices or points.

Lines are called edges or arcs.

Formally, a graph G is a structure (V,E) consisting of

a finite set V called the set of nodes, and a finite set E called the set of edges/links , of the form

(x,y) where x and y are nodes in V


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CS 103 3

Application of Graphs

Graph application area:

Computer science

Electrical engineering

Chemistry

Data base

Logic and design

computer networks

airline flights

road map

course prerequisite structure

tasks for completing a job

flow of control through a program

many more


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Graphs Applications

Electronic circuits

Networks

(roads, flights, communications)

CS16

LAX

JFK

LAX

DFW

STL

FTL


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Examples of Graphs

V={0,1,2,3,4}

E={(0,1), (1,2), (0,3), (3,0), (2,2), (4,3)}

01

4

2

3

When (x,y) is an edge,

we say that x is adjacent to

y, and y is adjacent fromx.

0 is adjacent to 1.

1 is not adjacent to 0.

2 is adjacent from 1.


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Intuition Behind Graphs

The nodes represent entities (such as people, cities,computers, words, etc.)

Edges (x,y) represent relationships between entities x and

y, such as:

x is a friend of y (note that this not necessarily reciprocal)

x is a child of y


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Definition and Terminology

What is a graph?

A graph G = (V,E) is composed of:

V: set of vertices

E: set ofedgesconnecting the verticesin VAn edgee = (u,v) is a pair of vertices

Example:

a b

c

d e

V= {a,b,c,d,e}

E= {(a,b),(a,c),(a,d),(b,e),(c,d),(c,e),

(d,e)}

a b

c

V={a, b, c}

E={,,

}


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Definition and Terminology Graph may be either directedor undirected

Directed graph (or Digraph)

Each line has a directionto its successor.

The lines in a directed graph are known as arcs

Note

Undirected graph Each line has nodirection.

Note =


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Definition and Terminology0

1 2

3

0

1

2

0

1 2

3 4 5 6G1

G2G3

V(G1)={0,1,2,3} E(G1)={(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)}

V(G2)={0,1,2,3,4,5,6} E(G2)={(0,1),(0,2),(1,3),(1,4),(2,5),(2,6)}V(G3)={0,1,2} E(G3)={,,}

A graph is said to b a complete graph if there is an edge

between every pair of vertices.

complete graph incomplete graph


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Loop edge and multiple edges

0

1

(a)

2

1

0

3

(b)self edge

2

multigraph:

multiple occurrences

of the same edge


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Definition and Terminology Path: sequence of vertices v1,v2,. . .vk such that consecutive

vertices viand vi+1are adjacent

{A, B, C, E} ?

{E,C,B,A}?

{A,B,C,B,E}?

simple path: no repeated vertices, except possibly the first and the

last

{A, B, C, E} ? {E,C,B,A}?{A,B,C,B,E}?

Cycle path: simple path, except that the last vertex is the same as

the first vertex

{B,C,D,E,B}?


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Weighted graph:When a no. or weight is associate with every

edge, is called weighted graph. Weight are

some times called cost. Source nodes:

The node that have a positive out degree but

zero in degree is called source node.

Sink nodes:

The node that has zero out degree but a positive

in degree.12


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Each edge ehas a weight, wt(e)

graph may be undirected or directedweight may represent length, cost, capacity, etc

adjacency matrix becomes weight matrix

adjacency lists include weight in node

4

6

6 7

4

5

75

5

8

v w

yx

z

a weighted graphG

Weighted Graphs


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Connectivity

Connected undirected graph: any two vertices are

connected by some path in undirected graph

Directed graph

Strongly connected: A directed graph is strongly connected if

there is a pathfrom each vertex to every other vertexin the

digraph.

Weakly connected: A directed graph is weakly connected if at

most two vertices are not connected.

Disjoint: A graph is disjoint if it is not connected.

connected not connected


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Degree in undirected graphThe degreeof a vertex is the number of edges

incident to that vertex

Example:

The degrees of the nodes A, C, D, F = 1

The degrees of the nodes B, E = 3


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Degree in directed graph

The degreeof a vertex is thesum of the

indegree and outdegreeof lines incident to it.

The outdegreeof a vertex in a digraph is thenumber of arcs leavingthe vertex.

Theindegreeis the number of arcs enteringthe

vertex.

Example: B&E


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Max and Min Degree of the

Graph

Max Degree:The highest degree of the node in the graph is called the

Max Degree on the Graph

Min Degree:

The minimum degree of the node in the graph is called the

Min Degree on the Graph


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Graph Traversal Techniques

There are two standard graph traversal

techniques:

Depth-First Search(DFS)

Breadth-First Search(BFS)


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Graph Traversal (Contd.)

In both DFS and BFS, the nodes of theundirected graph are visited in a systematic

manner so that every node is visited exactlyone.

Both BFS and DFS give rise to a tree:

When a node x is visited, it is labeled as visited,and it is added to the tree

If the traversal got to node x from node y, y isviewed as the parent of x, and x a child of y


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Graph searches A tree search starts at the root

and explores nodes from there,looking for a goal node (anode that satisfies certainconditions, depending on the

problem)

For some problems, any goalnode is acceptable (Nor J); forother problems, you want a

minimum-depth goal node,that is, a goal node nearest theroot (only J)L M N O P

G

Q

H JI K

FED

B C

A

Goal nodes


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Depth-first searching A depth-first search (DFS)

explores a path all the way to a

leaf before backtracking and

exploring another path

For example, after searching A,then B, then D, the search

backtracks and tries another

path from B

Node are explored in the orderA B D E H L M N I O P C F G J

K Q

Nwill be found before JL M N O P

G

Q

H JI K

FED

B C

A


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Depth-First Search

DFS follows the following rules:1. Select an unvisited node x, visit it, and treat as the

current node

2. Find an unvisited neighbor of the current node, visit it,

and make it the new current node;

3. If the current node has no unvisited neighbors,

backtrackto the its parent, and make that parent the

new current node;

4. Repeat steps 3 and 4 until no more nodes can be

visited.

5. If there are still unvisited nodes, repeat from step 1.


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Breadth-first searching A breadth-first search (BFS)

explores nodes nearest the root

before exploring nodes further

away

For example, after searching A,then B, then C, the search

proceeds with D,E,F,G

Node are explored in the order

A B C D E F G H I J K L M N OP Q

Jwill be found before NL M N O P

G

Q

H JI K

FED

B C

A


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Breadth-First Search

BFS follows the following rules:1. Select an unvisited node x, visit it, make it the root in a

BFS tree being formed. Its level is called the current

level.

2. From each node z in the current level, in the order inwhich the level nodes were visited, visit all the

unvisited neighbors of z. The newly visited nodes from

this level form a new level that becomes the next

current level.

3. Repeat step 2 until no more nodes can be visited.

4. If there are still unvisited nodes, repeat from Step 1.

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DataStructure - Graphs