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Junction Tree ConstructionPowerPoint Presentation

Junction Tree Construction

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### Junction Tree Construction

Practice exercise

Example 1: Junction Tree Construction

Step1: Insert link between parents of common child

A

B

D

E

C

G

F

H

Example 1: Junction Tree Construction

Step2: Convert into undirected graph

A

B

A

B

D

E

D

E

C

C

G

G

F

H

F

H

Example 1: Junction Tree Construction

Step3: Triangulate a graph if it has circle of length>3

A

B

D

E

C

G

F

H

Note : No need of triangulation in this example.

Example 1: Junction Tree Construction

Step3: Numbered the node if its numbered neighboured nodes are connected together.

1

2

A

B

4

5

3

D

E

C

8

6

7

G

F

H

Example 1: Junction Tree Construction

Definition:

A Clique in a graph is set of adjacent vertices which is a complete graph.

- Five cliques are found in Example 1:
- One clique of four nodes
- Four cliques of three nodes

Step4: Ordered the node by max cardinality search.

ABCDE

4

1

ACD

5

2

A

B

CDF

6

4

5

3

D

E

C

CEH

7

8

6

7

G

EHG

8

F

H

Example 1: Junction Tree Construction

Construct Tree

ABCDE

4

ACD

5

ABCDE

CDF

6

CEH

CEH

7

ACD

EHG

8

CDF

EHG

Example 2: Junction Tree Construction

Step1: Insert link between parents of common child

A

B

C

D

E

F

G

H

I

Example 2: Junction Tree Construction

Step2: Convert into undirected graph

A

A

B

C

D

B

C

D

E

F

G

E

F

G

H

I

H

I

Example 2: Junction Tree Construction

Step3: Triangulate a graph if it has circle of length>3

A

B

C

D

E

F

G

H

I

Example 2: Junction Tree Construction

Step3: Numbered the node if its numbered neighboured nodes are connected together.

1

A

2

4

3

F has three numbered neighbour B, C and D where B and D are not connected therefore add link between B and D and restart numbering again from beginning.

B

C

D

5

E

F

G

H

I

Example 2: Junction Tree Construction

Step3: Numbered the node if its numbered neighboured nodes are connected together.

1

A

2

4

3

B

C

D

5

6

7

E

F

G

H

I

8

9

Example 2: Junction Tree Construction

- Five cliques are found in Example 2:
- Two cliques of four nodes
- Four cliques of three nodes

Step4: Ordered the node by max cardinality search.

Clique Nodes

Max Cardinality

1

A

ABCD

4

BCDF

5

2

4

3

B

C

D

BEF

6

DFG

5

7

6

7

E

F

G

EFH

8

H

I

FGI

8

9

9

Example 2: Junction Tree Construction

Construct Tree

ABCD

4

ABCD

BCDF

5

BEF

6

BCDF

DFG

7

DFG

EFH

8

BEF

FGI

9

FGI

EFH

Example 2: Junction Tree Construction (with different triangulation )

Step1: Insert link between parents of common child

A

B

C

D

E

F

G

H

I

Example 2: Junction Tree Construction triangulation )

Step2: Convert into undirected graph

A

A

B

C

D

B

C

D

E

F

G

E

F

G

H

I

H

I

Example 2: Junction Tree Construction triangulation )

Step3: Triangulate a graph if it has circle of length>3

A

B

C

D

E

F

G

H

I

Example 2: Junction Tree Construction triangulation )

Step3: Numbered the node if its numbered neighboured nodes are connected together.

1

A

2

4

3

G has three numbered neighbour F, C and D where F and D are not connected therefore add link between F and D and restart numbering again from beginning.

B

C

D

6

5

7

E

F

G

H

I

Example 2: Junction Tree Construction triangulation )

Step3: Numbered the node if its numbered neighboured nodes are connected together.

1

A

2

4

3

F has three numbered neighbour E, C and D where E and D are not connected therefore add link between E and D and restart numbering again from beginning.

B

C

D

6

5

E

F

G

H

I

Example 2: Junction Tree Construction triangulation )

Step3: Numbered the node if its numbered neighboured nodes are connected together.

1

A

2

4

3

E has three numbered neighbour B, C and D where B and D are not connected therefore add link between B and D and restart numbering again from beginning.

B

C

D

5

E

F

G

H

I

Example 2: Junction Tree Construction triangulation )

Step3: Numbered the node if its numbered neighboured nodes are connected together.

1

A

2

4

3

B

C

D

6

5

7

E

F

G

H

8

I

9

Example 2: Junction Tree Construction triangulation )

- Cliques are found in Example 2:
- Four cliques of four nodes.
- Two cliques of three nodes.

Step4: Ordered the node by max cardinality search.

1

A

Clique Nodes

Max Cardinality

2

4

3

ABCD

4

B

C

D

BCDE

5

6

CDEF

5

6

7

E

F

G

CDFG

7

EFH

8

H

8

I

9

FGI

9

Example 2: Junction Tree Construction triangulation )

Construct Tree

ABCD

4

ABCD

BCDE

5

CDEF

6

BCDE

CDFG

7

CDEF

EFH

8

FGI

9

EFH

CDFG

FGI

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