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JUNCTION TREE CONSTRUCTION Practice exercise

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Example 1: Junction Tree Construction AB D C G HF E Step1: Insert link between parents of common child 2

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Example 1: Junction Tree Construction AB D C G HF E Step2: Convert into undirected graph AB D C G HF E 3

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Example 1: Junction Tree Construction AB D C G HF E Step3: Triangulate a graph if it has circle of length>3 Note : No need of triangulation in this example. 4

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Example 1: Junction Tree Construction Step3: Numbered the node if its numbered neighboured nodes are connected together. AB D C G HF E 1 2 3 4 5 6 7 8 5

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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 AB D C G HF E 1 2 3 4 5 6 7 8 6

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Example 1: Junction Tree Construction ABCDE ACD CEH CDF EHG ABCDE ACD CDF CEH EHG 5 6 7 8 4 Construct Tree 7

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Example 2: Junction Tree Construction A BDC G H FE Step1: Insert link between parents of common child I 8

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Example 2: Junction Tree Construction Step2: Convert into undirected graph A BDC G H FE I A BDC G H FE I 9

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Example 2: Junction Tree Construction A BDC G H FE I Step3: Triangulate a graph if it has circle of length>3 10

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Example 2: Junction Tree Construction Step3: Numbered the node if its numbered neighboured nodes are connected together. A BDC 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. 11

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Example 2: Junction Tree Construction Step3: Numbered the node if its numbered neighboured nodes are connected together. 1 42 3 7 5 6 98 A BDC G H FE I 12

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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 7 5 6 98 A BDC G H FE I Clique Nodes Max Cardinality BCDF BEF DFG EFH FGI 4 5 6 7 8 ABCD 9 13

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Example 2: Junction Tree Construction ABCD BCDF DFG BEF FGI BCDF BEF DFG EFH FGI 4 5 6 7 8 ABCD 9 EFH Construct Tree 14

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Example 2: Junction Tree Construction (with different triangulation ) A BDC G H FE Step1: Insert link between parents of common child I 15

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Example 2: Junction Tree Construction Step2: Convert into undirected graph A BDC G H FE I A BDC G H FE I 16

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Example 2: Junction Tree Construction A BDC G H FE I Step3: Triangulate a graph if it has circle of length>3 17

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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 BDC G H FE I 18

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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 5 A BDC 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. 19

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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 BDC 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. 20

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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 5 A BDC G H FE I 7 8 9 21

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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 Nodes Max Cardinality BCDE CDEF CDFG EFH FGI 4 5 6 7 8 ABCD 9 1 42 3 6 5 A BDC G H FE I 7 8 9 22

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Example 2: Junction Tree Construction Construct Tree BCDE CDEF CDFG EFH FGI 4 5 6 7 8 ABCD 9 23 ABCD BCDE CDEF CDFG FGI EFH

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