11 Lecture – Graph

8/19/2019 11 Lecture – Graph

1/53

Data Structures

Dr Kazi A Kalpoma

Associate Professor Faculty of Science and Information Technology

American International University Bangladesh (AIUB)

Contact: [email protected]

11 Lecture – Graph

mailto:[email protected]:[email protected]


2/53

Graphs - Background

Graphs = a set of nodes (vertices) with edges(links) between them.

Notations:

• G = (V, E) - graph

• V = set of vertices V = n

• E = set of edges E = m

1 2

3 4

1 2

3 4

Directed

graph

Undirected

graph

1 2

3 4

Acyclic

graph


3/53

Graph Representation

• Adjacency Matrix

• Adjacency Lists

Linked Adjacency Lists

Array Adjacency Lists


4/53

Adjacency Matrix

• 0/1 n x n matrix, where n = # of vertices• A(i,j) = 1 iff (i,j) is an edge

23

1

45

1 2 3 4 5

1

23

4

5

0 1 0 1 0

1 0 0 0 10 0 0 0 1

1 0 0 0 1

0 1 1 1 0


5/53

Adjacency Matrix Properties

23

1

45

1 2 3 4 5

1

2

34

5

0 1 0 1 0

1 0 0 0 1

0 0 0 0 11 0 0 0 1

0 1 1 1 0

•

Diagonal entries are zero.•Adjacency matrix of an undirected graph is

symmetric.

A(i,j) = A(j,i) for all i and j.


6/53

Adjacency Lists

• Adjacency list for vertex i is a linear list of vertices adjacent from vertex i.

• An array of n adjacency lists.

23

1

45

aList[1] = (2,4)

aList[2] = (1,5)

aList[3] = (5)

aList[4] = (5,1)

aList[5] = (2,4,3)


7/53

Array Adjacency Lists

• Each adjacency list is an array list.

23

1

45

aList[1]

aList[5]

[2]

[3][4]

2 4

1 5

55 1

2 4 3

Array Length = n

# of list elements = 2e (undirected graph)

# of list elements = e (digraph)


8/53

Path Finding

Path between node 1 and node 8.

23

8

101

45

911

67

4

8

6

6

7

5

2

4

4 53

Path length is 20.


9/53

Another Path Between 1 and 8

23

8

101

45

911

67

4

8

6

6

7

5

2

4

4 53

Path length is 28.


10/53

Example Of No Path

No path between 2 and 9.

23

8

101

45

911

67


11/53

Spanning Tree

• Subgraph that includes all vertices of the

original graph.

• Subgraph is a tree.

If original graph has n vertices, the spanning

tree has n vertices and n-1 edges.


12/53

Minimum Cost Spanning Tree

• Tree cost is sum of edge weights/costs.

23

8

101

45

911

67

4

8

6

6

7

5

24

4 53

8

2


13/53

A Spanning Tree

Spanning tree cost = 51.

23

8

101

45

911

67

4

8

6

6

7

5

24

4 53

8

2


14/53

Minimum Cost Spanning Tree

Spanning tree cost = 41.

23

8

101

45

911

67

4

8

6

6

7

5

24

4 53

8

2


15/53

Graph Search Methods

• Many graph problems solved using a searchmethod.

Path from one vertex to another.

Is the graph connected?

Find a spanning tree.

etc.

•Commonly used search methods: Breadth-first search.

Depth-first search.


16/53

Breadth-First Search

• Visit start vertex and put into a FIFO queue.

• Repeatedly remove a vertex from the queue, visit

its unvisited adjacent vertices, put newly visitedvertices into the queue.


17/53

Breadth-First Search Example

Start search at vertex 1.

23

8

10

1

45

9

116

7


18/53


Visit/mark/label start vertex and put in a FIFO queue.

OUT PUT: 1

23

8

10

1

45

9

116

7

1

FIFO Queue

1


19/53


Remove 1 from Q; visit adjacent unvisited vertices; put in Q.

OUT PUT: 1

23

8

10

1

45

9

116

7

1

FIFO Queue

1


20/53



OUT PUT: 1 2 4

23

8

10

1

45

9

116

7

1

FIFO Queue2

2

4

4


21/53



OUT PUT: 1 2 4

23

8

10

1

45

9

116

7

1

FIFO Queue2

2

4

4


22/53



OUT PUT: 1 2 4 5 3 6

23

8

10

1

45

9

116

7

1

FIFO Queue2

4

4

5

53

3

6

6


23/53



OUT PUT: 1 2 4 5 3 6

23

8

10

1

45

9

116

7

1

FIFO Queue2

4

4

5

53

3

6

6


24/53



OUT PUT: 1 2 4 5 3 6

23

8

10

1

45

9

116

7

1

FIFO Queue2

45

53

3

6

6


25/53



OUT PUT: 1 2 4 5 3 6

23

8

10

1

45

9

116

7

1

FIFO Queue2

45

53

3

6

6


26/53



OUT PUT: 1 2 4 5 3 6 9 7

23

8

10

1

45

9

116

7

1

FIFO Queue2

45

3

3

6

6

9

9

7

7


27/53



OUT PUT: 1 2 4 5 3 6 9 7

23

8

10

1

45

9

116

7

1

FIFO Queue2

45

3

3

6

6

9

9

7

7


28/53



OUT PUT: 1 2 4 5 3 6 9 7

23

8

10

1

45

9

116

7

1

FIFO Queue2

45

3

6

6

9

9

7

7


29/53



OUT PUT: 1 2 4 5 3 6 9 7

23

8

10

1

45

9

116

7

1

FIFO Queue2

45

3

6

6

9

9

7

7


30/53



OUT PUT: 1 2 4 5 3 6 9 7

23

8

10

1

45

9

116

7

1

FIFO Queue2

45

3

6

9

9

7

7


31/53



OUT PUT: 1 2 4 5 3 6 9 7

23

8

10

1

45

9

116

7

1

FIFO Queue2

45

3

6

9

9

7

7


32/53



OUT PUT: 1 2 4 5 3 6 9 7 8

23

8

10

1

45

9

116

7

1

FIFO Queue2

45

3

6

9

7

78 8


33/53



OUT PUT: 1 2 4 5 3 6 9 7 8

23

8

10

1

45

9

116

7

1

FIFO Queue2

45

3

6

9

7

78 8


34/53



OUT PUT: 1 2 4 5 3 6 9 7 8

23

8

10

1

45

9

116

7

1

FIFO Queue2

45

3

6

9

7

8 8


35/53



OUT PUT: 1 2 4 5 3 6 9 7 8

23

8

10

1

45

9

116

7

1

FIFO Queue2

45

3

6

9

7

8 8


36/53


Queue is empty. Search terminates.

OUT PUT: 1 2 4 5 3 6 9 7 8

23

8

10

1

45

9

116

7

1

FIFO Queue2

45

3

6

9

7

8


37/53

Breadth-First Search Property

• All vertices reachable from the start vertex

(including the start vertex) are visited.

OUT PUT: 1 2 4 5 3 6 9 7 8


38/53

Depth-First Search

depthFirstSearch(v)

{

Label vertex v as reached.

for (each unreached vertex u

adjacenct from v)

depthFirstSearch(u);

}

D th Fi t S h E l


39/53

Depth-First Search Example

Start search at vertex 1.

2

38

10

1

45

9

116

7

Label vertex 1 and do a depth first search

from either 2 or 4.

1

2

Suppose that vertex 2 is selected.

1stack


40/53

Depth-First Search Example2

38

10

1

45

9

116

7


from either 3, 5, or 6.

1

22

5


12stack


41/53


38

10

1

45

9

116

7


from either 3, 7, or 9.

1

22

55

9


125stack


42/53


38

10

1

45

9

116

7


from either 6 or 8.

1

22

55

99

8


1259stack


43/53


38

10

1

45

9

116

7

Label vertex 8 and return to vertex 9.

1

22

55

99

88

From vertex 9 do a dfs(6).

6

12598stack


44/53


38

10

1

45

9

116

7

1

22

55

99

88

Label vertex 6 and do a depth first search fromeither 4 or 7.

66

4


12596stack


45/53


38

10

1

45

9

116

7

1

22

55

99

88

Label vertex 4 and return to 6.

66

44

From vertex 6 do a dfs(7).

7

125964stack


46/53


38

10

1

45

9

116

7

1

22

55

99

88

Label vertex 7 and return to 6.

66

44

77

Return to 9.

125967

stack


47/53


38

10

1

45

9

116

7

1

22

55

99

88

66

44

77

Return to 5.

125stack


48/53


38

10

1

45

9

116

7

1

22

55

99

88

66

44

77

Do a dfs(3).

3


49/53


38

10

1

45

9

116

7

1

22

55

99

88

66

44

77

Label 3 and return to 5.

33

Return to 2.

1253stack


50/53


38

10

1

45

9

116

7

1

22

55

99

88

66

44

77

33

Return to 1.

12stack


51/53


38

10

1

45

9

116

7

1

22

55

99

88

66

44

77

33

Return to invoking method.

stack


52/53

Depth-First Search Properties

•

Same complexity as BFS.• Same properties with respect to path

finding, connected components, and

spanning trees.• Edges used to reach unlabeled vertices

define a depth-first spanning tree when the

graph is connected.• There are problems for which bfs is better

than dfs and vice versa.


53/53

http://www.youtube.com/watch?v=zLZhSSXAwxI
http://www.youtube.com/watch?v=zLZhSSXAwxIhttp://www.youtube.com/watch?v=zLZhSSXAwxI

Documents

11 Lecture – Graph