JUNCTION TREE CONSTRUCTION

Practice exercise

2

Example 1: Junction Tree Construction

A B

D C

GHF

E

Step1: Insert link between parents of common child

3

Example 1: Junction Tree Construction

A B

D C

GHF

E

Step2: Convert into undirected graph

A B

D C

GHF

E

4

Example 1: Junction Tree Construction

A B

D C

GHF

E

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

Note : No need of triangulation in this example.

5

Example 1: Junction Tree Construction

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

A B

D C

GHF

E

1 2

345

6 78

6

Example 1: Junction Tree Construction

Step4: Ordered the node by max cardinality search.

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

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

ABCDE

ACD

CDF

CEH

EHG

4

5

6

7

8

A B

D C

GHF

E

1 2

345

6 78

7

Example 1: Junction Tree Construction

ABCDE

ACD CEH

CDF EHG

ABCDE

ACD

CDF

CEH

EHG

5

6

7

8

4Construct Tree

8

Example 2: Junction Tree Construction

A

B DC

G

H

FE

Step1: Insert link between parents of common child

I

9

Example 2: Junction Tree Construction

Step2: Convert into undirected graph

A

B DC

G

H

FE

I

A

B DC

G

H

FE

I

10

Example 2: Junction Tree Construction

A

B DC

G

H

FE

I

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

11

Example 2: Junction Tree Construction

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

B DC

G

H

FE

I

1

42 3

5

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.

12

Example 2: Junction Tree Construction

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

1

42 3

756

98

A

B DC

G

H

FE

I

13

Example 2: Junction Tree Construction

Step4: Ordered the node by max cardinality search.

Five cliques are found in Example 2:• Two cliques of four

nodes• Four cliques of three

nodes1

42 3

756

98

A

B DC

G

H

FE

I

Clique NodesMax Cardinality

BCDF

BEF

DFG

EFH

FGI

4

5

6

7

8

ABCD

9

14

Example 2: Junction Tree Construction

ABCD

BCDF

DFGBEF

FGI

BCDF

BEF

DFG

EFH

FGI

4

5

6

7

8

ABCD

9

EFH

Construct Tree

15

Example 2: Junction Tree Construction (with different triangulation )

A

B DC

G

H

FE

Step1: Insert link between parents of common child

I

16

Example 2: Junction Tree Construction

Step2: Convert into undirected graph

A

B DC

G

H

FE

I

A

B DC

G

H

FE

I

17

Example 2: Junction Tree Construction

A

B DC

G

H

FE

I

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

18

Example 2: Junction Tree Construction

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

1

42 3

6

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.

5 7

A

B DC

G

H

FE

I

19

Example 2: Junction Tree Construction

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

1

42 3

65

A

B DC

G

H

FE

I

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.

20

Example 2: Junction Tree Construction

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

1

42 3

5

A

B DC

G

H

FE

I

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.

21

Example 2: Junction Tree Construction

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

1

42 3

65

A

B DC

G

H

FE

I

7

8 9

22

Example 2: Junction Tree Construction

Step4: Ordered the node by max cardinality search.

Cliques are found in Example 2: • Four cliques of four

nodes.• Two cliques of three

nodes.Clique NodesMax

Cardinality

BCDE

CDEF

CDFG

EFH

FGI

4

5

6

7

8

ABCD

9

1

42 3

65

A

B DC

G

H

FE

I

7

8 9

23

Example 2: Junction Tree Construction

Construct Tree

BCDE

CDEF

CDFG

EFH

FGI

4

5

6

7

8

ABCD

9

ABCD

BCDE

CDEF

CDFG

FGI

EFH