Junction Tree Construction
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Transcript of Junction Tree Construction
JUNCTION TREE CONSTRUCTIONPractice exercise
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Example 1: Junction Tree Construction
A B
D C
GHF
E
Step1: Insert link between parents of common child
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Example 1: Junction Tree Construction
A B
D C
GHF
E
Step2: Convert into undirected graph
A B
D C
GHF
E
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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.
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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
3 45
6 78
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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
3 45
6 78
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Example 1: Junction Tree Construction
ABCDE
ACD CEH
CDF EHG
ABCDE
ACD
CDF
CEH
EHG
5
6
7
8
4Construct Tree
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Example 2: Junction Tree Construction
A
B DC
G
H
FE
Step1: Insert link between parents of common child
I
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Example 2: Junction Tree Construction
Step2: Convert into undirected graph
A
B DC
G
H
FE
I
A
B DC
G
H
FE
I
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Example 2: Junction Tree Construction
A
B DC
G
H
FE
I
Step3: Triangulate a graph if it has circle of length>3
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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.
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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
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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
nodes 1
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
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Example 2: Junction Tree Construction
ABCD
BCDF
DFGBEF
FGI
BCDF
BEF
DFG
EFH
FGI
4
5
6
7
8
ABCD
9
EFH
Construct Tree
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Example 2: Junction Tree Construction (with different triangulation )
A
B DC
G
H
FE
Step1: Insert link between parents of common child
I
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Example 2: Junction Tree Construction
Step2: Convert into undirected graph
A
B DC
G
H
FE
I
A
B DC
G
H
FE
I
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Example 2: Junction Tree Construction
A
B DC
G
H
FE
I
Step3: Triangulate a graph if it has circle of length>3
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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
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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.
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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.
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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
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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
45
6
7
8
ABCD
9
1
42 3
65
A
B DC
G
H
FE
I
7
8 9
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Example 2: Junction Tree Construction
Construct TreeBCD
ECDE
FCDF
GEFH
FGI
45
6
7
8
ABCD
9
ABCD
BCDE
CDEF
CDFG
FGI
EFH