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3'&;51+)"11(+2Z'(#+(11+JL+Q1P&?&)&#?+DM7M87M9

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Cmput 651 - Clique Trees {20,24}/10/2008

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Cmput 651 - Clique Trees {20,24}/10/2008

3'5()1"+I"%-.+P#"+Y%"&%$'1+1'&4&?%)&#?+2O11+JL+NM8M89

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8A

Ci scope() Ci

Ci C j

Cmput 651 - Clique Trees {20,24}/10/2008

H5??&?I+&?)1"(1*)&#?+-"#-1")d

Q1P&?&)&#?: + ,&Y1?+*'5()1"+)"11+<+#Y1"+P%*)#"(+LC + <+.%(+).1+)%,,#,1'#,(&)*&.(#0,'2)02&)(9+41%?(: + TP+Y%"&%$'1+V+&(+&?+)=#+*'5()1"(+3&+%?0+3cC + ).1?+V+&(+&?+1Y1"d+*'5()1"+&?+25?&;519+-%).+P"#4+3&+)#+ 3cM

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H5??&?I+&?)1"(1*)&#?+-"#-1")d

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3'5()1"+,"%-. 3'5()1"+)"11+(%)&(P&1(+"5??&?I+&?)1"(1*)&#?+-"#-1")dM 1I:+,+&(+&?+*'5()1"(+A+%?0+DC+%(+=1''+%(+).1+&?)1"Y1?&?I+*'5()1"(+_+%?0+ 7M+

Cmput 651 - Clique Trees {20,24}/10/2008

H5??&?I+&?)1"(1*)&#?+-"#-1")d+%?0 Y%"&%$'1+1'&4&?%)&#?

8/&0)&5: + TP+<+&(+%+*'5()1"+)"11+&?05*10+$d+Y%"&%$'1+1'&4&?%)&#?C+ ).1?+<+(%)&(P&1(+).1+"5??&?I+&?)1"(1*)&#?+-"#-1")dM +

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86

Cmput 651 - Clique Trees {20,24}/10/2008

3'&;51+)"11(:+<=#+1;5&Y%'1?)+01P&?&)&#?(

8e

3'&;51+<"11 3'&;51+)"11+(%)&(P&1(: 01Pf?+89+$%(10+#?+*.#"0%' 01Pf?+A9+$%(10+#?+Y%"&%$'1+1'&4&?%)&#? l#)1+1X)"%+10I1(+)#+ 4%j1+).&(+*.#"0%'M

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8G

Cmput 651 - Clique Trees {20,24}/10/2008

\%"&%$'1+1'&4&?%)&#?+$%(10+#?+*'&;51+)"11( 2O11+JL+NMAM89

R(1+*'&;51+)"11(+)#+-1"P#"4+Y%"&%$'1+1'&4&?%)&#?+Y&%+ (54+-"#05*)+41((%I1+-%((&?I (5$()%?)&%'+-1"P#"4%?*1+$1?1P&)(

8N

3'&;51+)"11+(54K-"#05*)+5-=%"0+-%((+%'I#"&).4: !%((+41((%I1(+5-+)#+"##)+*'&;51+3"M L&?%'+*'&;51+-#)1?)&%'+m"n3"^+"1-"1(1?)(++ + + + + + + + 2&M1M+4%"I&?%'+-"#$%$&'&)d+#Y1"+"##)+*'&;519 5-()"1%4+Y(M+0#=?()"1%4

3'&;51+)"11+41((%I1+-%((&?I

2(11+JL+NMAM8MA9

AB

Cmput 651 - Clique Trees {20,24}/10/2008

3'&;51+)"11+41((%I1+-%((&?I+2Z'(#+(11+JL+NMAM8M8C+NMAM8MA9

A8

3'&;51+<"11 UX%4-'1+)%(j:+,&Y1?+).1+*'&;51+)"11+$1'#=+ 2$%(10+#?+?1)=#"j+%)+'1P)9C+*#4-5)1+!2i9M

3'&;51+)"11+41((%I1+-%((&?I

2(11+JL+NMAM8M8C+NMAM8MA9

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AA

Cmput 651 - Clique Trees {20,24}/10/2008

3'&;51+)"11+41((%I1+-%((&?I+2Z'(#+(11+JL+NMAM8M8C+NMAM8MA9

A_

3'&;51+<"11 &?&)&%' P%*)#"( )00('."#$%&

3'&;51+)"11+41((%I1+-%((&?I

2(11+JL+NMAM8MA9

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AM+(54+#5)+%''+Y%"&%$'1(+&?+3&+1X*1-)+(1-(1)+2&?)1"(1*)&#?9+ $1)=11?+3&+%?0+3-"

AD

23-"o5-()"1%4+?1&I.$#5"9

Cmput 651 - Clique Trees {20,24}/10/2008

3'&;51+)"11+41((%I1+-%((&?I+2Z'(#+(11+JL+NMAM8M8C+NMAM8MA9

A7

3'&;51+<"11 8M+T?+*'&;51+38C+1'&4&?%)1+3:++++++++++++++++++++++++p+(1?0+"1(5')&?I+P%*)#" + +++++++++++%(+%+41((%I1+)#+3A AM+T?+3AC+01P&?1+ + + + + + + + + +M+U'&4&?%)1+Q+)#+

• "#05*1+P%*)#"+

+ + ++C+=.&*.+&(+(1?)+)#+3_M _M+T?+3_C+01P&?1++ + + + + + + + + ++++M+U'&4&?%)1+T+"#+

• "#05*1++ +

+ +++C+=.&*.+&(+(1?)+)#+7+P#"+5(1+&?+O)1-+7M &?&)&%' P%*)#"(

3'&;51+)"11+41((%I1+-%((&?I+2O11+JL+NMAM8M8C+NMAM8MA9

A6

3'&;51+<"11 DM+T?+3D+C+1'&4&?%)1+b:+++ + + + C+(1?0+"1(5')&?I++ + + ++ )#+37M 7M+T?+37C+$%(10+#?+41((%I1(+P"#4+O)1-(+_+%?0+DC+=1+01P&?1 &?&)&%' P%*)#"(

Cmput 651 - Clique Trees {20,24}/10/2008

3'&;51+)"11+41((%I1+-%((&?I

2(11+JL+NMAM8M8C+NMAM8MA9

3'&;51+)"11+(54K-"#05*)+5-=%"0+-%((+2*#?)f09: 6M+a&).+P&?%'+*'&;51+-#)1?)&%'+m"n3"^ + &P+?1*1((%"d+(54+#5)+3"KV + &P+?1*1((%"d+?#"4%'&h1 + + +

• %")&)&#?+P5?*)&#?++

+ T?+1X%4-'1C+m7+1?*#01(+).1+c#&?)+!2,CiCSCO9M+<#+I1)+! 2i9C+(54+#5)+,C+SC+OM

Ae

Z = r[Cr]

Cr

• ]+8+1'&4&?%)&#?+#"01"+2O11+JL+NMAM8M89

AG

Z+*'&;51+&(+)&+-9'#?*1+&)f(+"1*1&Y10+%''+#P+&)(+&?*#4&?I+41((%I1+ 2q0#=?()"1%4r+41((%I1(9M+<.1?+&)+*%?+*#4-5)1+&)(+#=?+41((%I1+ %?0+(1?0+&)+q5-()"1%4r+2)#=%"0+).1+"##)+?#019M U'&4&?%)&#?+*%?+-"#*110+&?+%?d+#"01"+).%)+"1(-1*)(+"1%0&?1((+ "1;5&"141?)(M

Cmput 651 - Clique Trees {20,24}/10/2008

[5)'&?1

3'&;51+)"11(+%?0+*.#"0%'+I"%-.( \%"&%$'1+1'&4&?%)&#?+K]+*'&;51+)"11 3'&;51+)"11+K]+Y%"&%$'1+1'&4&?%)&#? !+"#4)+(#0, >1'&1P+5-0%)1+41((%I1+-%((&?I ,1?1"%'+;51"&1( >5&'0&?I+*'&;51+)"11(

AN

3'&;51+)"11+*%'&$"%)&#?+2Z'(#+(11+JL+NMAMA9

a%?)+%''+4%"I&?%'(+!2Vc9: ?%&Y1+s8:+*'&;51+)"11+&?P1"1?*1+P#"+1%*.+Vc + *#():+*#?()t?54uVc ?%&Y1+sA:+*'&;51+)"11+&?P1"1?*1+P#"+1%*.+*'&;51+3& + *#():+*#?()t?54u3& $1))1":+5-=%"0+(=11-C+).1?+$%*j=%"0+(=11- + At2?54u?#01(K89+41((%I1+*#4-5)%)&#?( + *#():+At*#?()

_B

Cmput 651 - Clique Trees {20,24}/10/2008

[?'d+A+0&PP1"1?)+41((%I1+-1"+10I1

_8

> 3

Q

U L Z

U0I1+3KQ: + H##)+&(+#?+'1P)+#"+"&I.) + [?'d+(=&)*.+2'1P)vK]"&I.)9+*.%?I1(+&?*#4&?I+41((%I1( +

• ]+A+5?&;51+41((%I1(+-#((&$'1+P#"+1%*.+10I1

+

• ]+At2?54u?#01(+w+89+41((%I1(+)#)%'+

R-=%"0+).1?+0#=?=%"0+2Z'(#+(11+JL+NMAMA9

_A

"##)

Cmput 651 - Clique Trees {20,24}/10/2008

H1%0&?1((

__

Z+*'&;51+&+&(+)&+-9')#+*#4-5)1+41((%I1+)#+?1&I.$#5"+c+%P)1"+ "1*1&Y&?I+).1+41((%I1(+P"#4+%''+&)f(+?#?Kc+?1&I.$#5"(M

/1((%I1(+%?0+41((%I1+-%((&?I

_D

8 A _ D 7 6 e N G

H##) 3'&;51+?#01+D R-=%"0+-%((: "1*1&Y1(+A+2"100&(.9+41((%I1( (1?0(+8+2-5"-'19+41((%I1+)#=%"0+"##) Q#=?=%"0+-%((: "1*1&Y1(+8+2#"%?I19+41((%I1+P"#4+"##)+$"%?*. (1?0(+A+0&PP1"1?)+2$'519+41((%I1(+0#=?=%"0

Cmput 651 - Clique Trees {20,24}/10/2008

R-=%"0+).1?+0#=?=%"0+2Z'(#+(11+JL+NMAMA9

ZP)1"+(&?I'1+5-=%"0+).1?+(&?I'1+0#=?=%"0+-%((: + U%*.+*'&;51+3&+.%(+%''+41((%I1( + o]+P&?%'+P%*)#"+m&n3&^+P#"+%''+& + o]+*%?+*#4-5)1+%?d+4%"I&?%'+!2Vc9

_7

3%'&$"%)&#?

Q1P&?&)&#?:+Z+*'&;51+)"11+=&).+*'&;51+-#)1?)&%'(+m&n3&^+ &(+.+"#4)+(&-+&P+P#"+%''+?1&I.$#5"&?I+*'&;51(+3&C+3c

• Ci−Si,j

βi[Ci] =

• Cj−Si,j

βj[Cj]

_6

Cmput 651 - Clique Trees {20,24}/10/2008

8/&0)&5:+O54K-"#05*)+41((%I1+-%((&?I+-"#05*1(+%+ *%'&$"%)10+*'&;51+)"11M !"##P:

ZP)1"+(54K-"#05*)+2Y%"&%$'1+1'&4&?%)&#?x9+$1'&1P+5-0%)&?I 2?#)1:+++++++++++++++*%?+$1+5?K?#"4%'&h109 L#"+?1&I.$#5"(+3&+%?0+3cC+=&).+

3%'&$"%)&#?

Si,j = Ci ∩ Cj PF(Ci)

• Ci−Si,j

βi[Ci] = PF(Si,j) =

• Cj−Si,j

βj[Cj] βi[Ci] =

• X−Ci

PF(X) = PF(Ci)

_e

3%'&$"%)10+*'&;51+)"11+o+%')1"?%)&Y1+

• %"%41)1"&h%)&#?+2%'(#+(11+JL+NMAM_9

_G

Cmput 651 - Clique Trees {20,24}/10/2008

[5)'&?1

3'&;51+)"11(+%?0+*.#"0%'+I"%-.( \%"&%$'1+1'&4&?%)&#?+K]+*'&;51+)"11 3'&;51+)"11+K]+Y%"&%$'1+1'&4&?%)&#? 3%'&$"%)&#? ;&"#&<'%2-+(&'5&**+1&'2+**#,1 ,1?1"%'+;51"&1( >5&'0&?I+*'&;51+)"11(

_N

>1'&1P+5-0%)1+41((%I1+-%((&?I+2%'(#+(11+JL+NM_M89

O54K-"#05*)+41((%I1+-%((&?I

&?(-&"10+$d+Y%"&%$'1+1'&4&?%)&#?

>1'&1P+5-0%)1+41((%I1+-%((&?I:

1;5&Y%'1?)+)#+(54K-"#05*)+41((%I1+-%((&?I #PP1"(+0&PP1"1?)+-1"(-1*)&Y1

DB

Cmput 651 - Clique Trees {20,24}/10/2008

L&?%'+*'&;51+-#)1?)&%'

D8

8 A _ D 7 6 e N G

H##)

L&?%'+*'&;51+-#)1?)&%': y&+o+&?&)&%'+-#)1?)&%'C+zj{&+o+41((%I1C+l&+o+?1&I.$#5"(+#P+&

βi = φi ·

• k∈Ni

δk→i

<=#+=%d(+)#+*#4-5)1+41((%I1(+2%'(#+(11+JL+NM_M89

O54K-"#05*)

δi→j =

• Ci−Si,j

φi ·

• k∈(Ni−{j})

δk→i

>1'&1PK5-0%)1 δi→j =

• Ci−Si,j βi

δj→i Z''+&?K*#4&?I+41((%I1(+1X*1-)+zc{&

@

L&?%'+*'&;51+-#)1?)&%'+0&Y&010+$d+zc{&

l#)1: βi = φi ·

• k∈Ni

δk→i

DA

Cmput 651 - Clique Trees {20,24}/10/2008

L%*)#"+0&Y&(&#?+2%'(#+(11+JL+Q1P&?&)&#?+NM_M89

Q1P&?&)&#?:+,&Y1?+0&(c#&?)+Y%"&%$'1+(1)(+V+%?0+W+%?0+ P%*)#"(+82VCW9+%?0+A2W9C+<+.(0)'-#:#*#0,+-"#05*1(+%+ P%*)#" =.1"1 BFB+o+B 2X|B9FB+&(+5?01P&?10

D_

>1'&1P+5-0%)1+41((%I1+-%((&?I+2%'(#+(11+JL+NM_M89

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>1'&1P+5-0%)1+41((%I1+-%((&?I

2%j%+S%5"&)h1?KO-&1I1'.%')1"+%'I#"&).49:

T)1"%)&Y1+-"#*1((: l#01+4%&?)%&?(+$1'&1P+-#)1?)&%'

• "#05*)+#P+&?&)&%'+-#)1?)&%'+%?0+ZSS+41((%I1(+(#+P%"

U0I1+()#"1(+-"1Y&#5(+41((%I1+2"1I%"0'1((+#P+0&"1*)&#?9

?1X)+41((%I1+&(+0&Y&010+$d+()#"10+41((%I1+2%?0+).1?+()#"109

J11-+-%((&?I+41((%I1(+5?)&'+%''+?#01(+.%Y1+%''+41((%I1( %?0+*%?+*#4-5)1+βi = φi ·

• k∈Ni

δk→i

DD

Cmput 651 - Clique Trees {20,24}/10/2008

>1'&1P+5-0%)1+41((%I1+-%((&?I+1X%4-'1

2%'(#+(11j+JL+1X%4-'1+NM_M_9 38 3A 3_

8 A _

8 8 }8 }A }_ 8 ~A{_ }8 }A }_+•}A €A{_ m&:

38o@ZC>EC+3Ao@>C3EC+3_o@>CQE

µ2→3 = σ2→3

D7

}A+•}_

>1'&1P+5-0%)1+41((%I1+-%((&?I+1X%4-'1

2%'(#+(11j+JL+1X%4-'1+NM_M_9 38 3A 3_

8 A _

8 8 }8 }A }_ 8 ~A{_ }8 }A }_+•}A €A{_ m&:

38o@ZC>EC+3Ao@>C3EC+3_o@>CQE

8 A _

8 ~_{A }8 €_{A

σ3→2(C) =

• D

β3[C, D]

}_+•}A

R-0%)1+)#+3A:

µ3→2 =

D6

Cmput 651 - Clique Trees {20,24}/10/2008

3'&;51+)"11+41%(5"1+2%'(#+(11+JL+01P&?&)&#?+NMAM889

Q1P&?&)&#?: <.1+."#$%&'()&&'5&+*%)&+#P+%+*%'&$"%)10+)"11+<+&(: =0(&

˜ PF(X) = βT

De

3'&;51+)"11+&?Y%"&%?)+2%'(#+(11+JL+NM_MA9

Q1P&?&)&#?: !"#$%&'()&&'#,:+)#+,.&+41%?(+).%) %)+1%*.+-#&?)+&?+$1'&1P+5-0%)1+*%'&$"%)&#?M

DG

Cmput 651 - Clique Trees {20,24}/10/2008

!"##P+#P+*'&;51+)"11+&?Y%"&%?*1+2%'(#+(11+JL+<.1#"14+NM_MD9

++++++++++++++++++++++++++++++++++++++++++++++++++++++.#'0(+05"&?I+ ++++++++++++++++++++++++++++++++++++++++++++++++++++++$1'&1P+5-0%)1+41((%I1 ++++++++++++++++++++++++++++++++++++++++++++++++++++++-%((&?I !"##P:

/1((%I1+€&{c+*.%?I1(+#?'d+mc+%?0+~&Cc+)#+mfc+%?0+~f&CcM a%?): >5)+$1'&1P+5-0%)1+41((%I1+-%((&?I+=%(+01P&?10+(#+).%)

DN

/1((%I1+-%((&?I+1;5&Y%'1?*1+2%'(#+(11+JL+NM_M_9

8/&0)&5> (54K-"#05*)+41((%I1+-%((&?I+2Y%"&%$'1+1'&4&?%)&#?9 &(+1;5&Y%'1?)+)#+ $1'&1PK5-0%)1+41((%I1+-%((&?I 2O11+JL+NM_M_+P#"+-"##PM9

7B

Cmput 651 - Clique Trees {20,24}/10/2008

[5)'&?1

3'&;51+)"11(+%?0+*.#"0%'+I"%-.( \%"&%$'1+1'&4&?%)&#?+K]+*'&;51+)"11 3'&;51+)"11+K]+Y%"&%$'1+1'&4&?%)&#? 3%'&$"%)&#? >1'&1P+5-0%)1+41((%I1+-%((&?I ?&,&)+"'$%&)#&* >5&'0&?I+*'&;51+)"11(

78

,1?1"%'+;51"&1(+#?+*'&;51+)"11(+2%'(#+(11+JL+NM_MDM89

3#?0&)&#?&?I+Y&%+P%*)#"+"105*)&#?+=&).&?+%+*'&;51:

38 3A 3_

3#?0&)&#?&?I+#?+Q:+P%*)#"+"105*)&#?+#?++m_ =.1"1+++++++++++++++++&(+%?+&?0&*%)#"+P5?*)&#?

˜ PF(C, D = d) = β3[C, D = d] = I{D = d} · β3[C, D] I{D = d} 38+o+@ZC>E+C+3A+o+@>C3E+C+3_+o+@3CQE 2l#)1:+++++++++&(+5?K?#"4%'&h10M9 ˜ PF

7A

Cmput 651 - Clique Trees {20,24}/10/2008

,1?1"%'+;51"&1(+#?+*'&;51+)"11(+2%'(#+(11+JL+NM_MDM89

!"#-%I%)&?I+*#?0&)&#?&?I+)#+1?)&"1+)"11:

38 3A 3_

38+o+@ZC>E+C+3A+o+@>C3E+C+3_+o+@3CQE ˜ PF(A, B, C, D = d) = I{D = d} ·

• Ci∈T βi[Ci]
• (Ci—Cj)∈T µi,j(Si,j)

a+=%?) 8()+()1-:+-%((+41((%I1+P"#4+3_+)#+3A:

2"1414$1"+-"#-1"+$1'&1P+5-0%)&?I+K+I#+)#+?1X)+('&019 σ3→2(C) =

• D

I{D = d} · β3[C, D]

7_

,1?1"%'+;51"&1(+#?+*'&;51+)"11(+2%'(#+(11+JL+NM_MDM89

!"#-%I%)&?I+*#?0&)&#?&?I+)#+1?)&"1+)"11:

38 3A 3_

38+o+@ZC>E+C+3A+o+@>C3E+C+3_+o+@3CQE

>1'&1P+5-0%)&?I+%)+3A+20&Y&01+41((%I1+$d+(1-(1)+

• #)1?)&%'9:

β2[B, C] · σ3→2(C) µ2,3(C) = β2[B, C] ·

• D I{D = d} · β3[C, D]
• D β3[C, D]

7D

Cmput 651 - Clique Trees {20,24}/10/2008

,1?1"%'+;51"&1(+#?+*'&;51+)"11(+2%'(#+(11+JL+NM_MDM89

!"#-%I%)&?I+*#?0&)&#?&?I+)#+1?)&"1+)"11:

• *%?+-"#-%I%)1+*#?0&)&#?&?I+)#+=.#'1+)"11
• #"+%+(5$(1)+#P+).1+)"11+2&P+).%)f(+%''+d#5+?1109
• *%??#)+1%(&'d+"1)"%*)+1Y&01?*1
• .%Y1+)#+()#"1+#"&I&?%'+)"11+2$1P#"1+*#?0&)&#?&?I9
• %--"#%*.+?#)+'&4&)10+c5()+)#++&?0&*%)#"+P%*)#"(

77

,1?1"%'+;51"&1(+#?+*'&;51+)"11(+2%'(#+(11+JL+NM_MDMA9

3#?0&)&#?%'+"1%(#?&?I+%*"#((+0&PP1"1?)+*'&;51(: !2W•19C+=.1"1+W+⊄+3&+∀& l%&Y1:+"1$5&'0+*'&;51+)"11+(#+).%)+W+⊆+3&+P#"+(#41+&

2>1))1"+%--"#%*.+#?+?1X)+('&01M9

76

Cmput 651 - Clique Trees {20,24}/10/2008

,1?1"%'+;51"&1(+#?+*'&;51+)"11(:+1X%4-'1

2%'(#+(11+JL+NM_MDMA9

3#?0&)&#?%'+"1%(#?&?I+%*"#((+0&PP1"1?)+*'&;51(:

38 3A 3_

38+o+@ZC>E+C+3A+o+@>C3E+C+3_+o+@3CQE

a%?) ˜

PF(B, D) T ′ = C2—C3

Q#+Y%"&%$'1+ 1'&4&?%)&#?+#?+ ?1=+P%*)#"(

7e

,1?1"%'+;51"&1(+2%'(#+(11+JL+NM_MDM_9

/5')&-'1+;51"&1(+%*"#((+0&PP1"1?)+*'&;51(: a%?)+!2W•19C+=.1"1+W+⊄+3&+∀&+P#"+?+0&PP1"1?)+Wf( l%&Y1+8:+P#"+1%*.+WC+"1$5&'0+*'&;51+)"11+(#+).%)+W+⊆+3&+ P#"+(#41+&+K]+OTSSWx l%&Y1+A:+\%"&%$'1+1'&4&?%)&#?+%--"#%*.+2P"#4+-"1Y&#5(+

('&019+++++++)&41(+K]+1X-1?(&Y1x

>1))1": Qd?%4&*+-"#I"%44&?I:+()%")+=&).+?1&I.$#5"+*'&;51(C+ $5&'0+#5)=%"0C+*%*.&?I+%(+=1+I#

n 2

• 7G

Cmput 651 - Clique Trees {20,24}/10/2008

[5)'&?1

3'&;51+)"11(+%?0+*.#"0%'+I"%-.( \%"&%$'1+1'&4&?%)&#?+K]+*'&;51+)"11 3'&;51+)"11+K]+Y%"&%$'1+1'&4&?%)&#? 3%'&$"%)&#? >1'&1P+5-0%)1+41((%I1+-%((&?I ,1?1"%'+;51"&1( ;%#"-#,1'."#$%&'()&&*

7N

3'&;51+)"11(+P"#4+Y%"&%$'1+1'&4&?%)&#?

2%'(#+(11+JL+NMDM8C+GMDMAM_9

Q1P&?&)&#?: <.1+#,-%.&-'1)+2/+P#"+%+Y%"&%$'1+1'&4&?%)&#?

• 5?0&"1*)10+I"%-.
• 10I1+$1)=11?+%''+-%&"(+#P+Y%"&%$'1(+).%)+%--1%"+&?+(#41+

&?)1"410&%)1+P%*)#"+y

2%'(#+(11+JL+01P&?&)&#?+GMDM_9

T?05*10+I"%-.

6B

Cmput 651 - Clique Trees {20,24}/10/2008

3'&;51+)"11(+P"#4+Y%"&%$'1+1'&4&?%)&#?

2%'(#+(11+JL+NMDM8C+GMDMAM_9

8/&0)&5:+T?05*10+I"%-.+&(+*.#"0%'M

2L#"+-"##P+(11+JL+<.1#"14+GMDMe9M

T?05*10+I"%-.

68

3'&;51+)"11(+P"#4+Y%"&%$'1+1'&4&?%)&#?

2%'(#+(11+JL+NMDM89

>5&'0+*'&;51+)"11+P"#4+&?05*10+I"%-. T?05*10+I"%-.

3'&;51+<"11

l#)1:+=1+(%=+).&(+1%"'&1"

<d-&*%''d+5(1+#?'d 4%X&4%'+*'&;51( &?+*'&;51+)"11M

6A

Cmput 651 - Clique Trees {20,24}/10/2008

3'&;51+)"11(+P"#4+Y%"&%$'1+1'&4&?%)&#?

2%'(#+(11+JL+NMDM89

>5&'0+*'&;51+)"11+P"#4+&?05*10+I"%-.+K+1X%4-'1+A 2l#)+&?05*10+I"%-.9

3'&;51+<"11

l#)1:+=1+(%=+).&(+1%"'&1"

l#+c#$+K]+.%--d+10I1

6_

3'&;51+)"11(+P"#4+Y%"&%$'1+1'&4&?%)&#?

2%'(#+(11+JL+NMDM89

>5&'0&?I+*'&;51+)"11(+P"#4+&?05*10+I"%-.(+&(+P#"4%''d+ =1''KI"#5?010+2(11+JL+NMDM8C+GMDMAM_9M

6D

Cmput 651 - Clique Trees {20,24}/10/2008

3'&;51+)"11(+P"#4+*.#"0%'+I"%-.(+2%'(#+(11+JL+NMDMA9

+ 8/&0)&5:+UY1"d+*.#"0%'+I"%-.+.%(+%+*'&;51+)"11M+2O11+

JL+<.1#"14+DM7M869

67

> Z 3 Q L U

a

V W `

3'&;51+)"11(+P"#4+*.#"0%'+I"%-.(+2%'(#+(11+JL+NMDMA9

,&Y1?+P%*)#"(++++++=.&*.+P%*)#"&h1+$%(10+#?+

8M $5&'0+*.#"0%'+I"%-.++++++++++P"#4+++++++++++2)"&%?I5'%)&#?9 AM P&?0+4%X&4%'+*'&;51(+&?+ _M $5&'0+*'&;51+)"11++++++P"#4+++++++‚(+4%X&4%'+*'&;51( HF H∗ F HF H∗ T H∗

66

Cmput 651 - Clique Trees {20,24}/10/2008

<"&%?I5'%)&#?+2%'(#+(11+JL+NMDMA9

• P&?0&?I+4&?&454+)"&%?I5'%)&#?+&(+l!K.%"0
• 4&?&454+)"&%?I5'%)&#?+o+#?1+&?+=.&*.+'%"I1()+*'&;51+.%(+4&?+(&h1
• 1X&()+1X%*)+%'I#"&).4(
• 1X-#?1?)&%'+&?+(&h1+#P+'%"I1()+*'&;51
• .15"&()&*+%'I#"&).4(+)d-&*%''d+5(10

6e

Z > 3 Q U L Z > 3 Q U L 8 A _

b15"&()&*+)"&%?I5'%)&#?

2%'(#+(11+JL+NMDMAC+GMDM_MA9

H1*%'':+&?05*10+I"%-.+P"#4+Y%"&%$'1+1'&4&?%)&#?+&(+*.#"0%' O#C+0#+Y%"&%$'1+1'&4&?%)&#?+)#+)"&%?I5'%)1

• 0#+?#)+0#+%*)5%'+(54+%?0+-"#05*)+*#4-5)%)&#?(
• c5()+j11-+)"%*j+#P+&?)1"410&%)1+P%*)#"(f+(*#-1(
• *#??1*)+%''+(1)(+#P+?#01(+=&).&?+%+I&Y1?+(*#-1
• (11+1X%4-'1+)=#+('&01(+$1'#=

\%"&%$'1+1'&4&?%)&#?+"1;5&"1(+%?+1'&4&?%)&#?+#"01"

• (11+?1X)+('&01

6G

Cmput 651 - Clique Trees {20,24}/10/2008

b15"&()&*+1'&4&?%)&#?+#"01"&?I+%'I#"&).4

2%'(#+(11+JL+NMDMAC+GMDM_MA9

a%?)+%?+#"01"&?I+).%)+-"#05*1(+%+(4%''+*.#"0%'+I"%-.M Q1P&?1+*#()+P5?*)&#?+2%'(#+(11+JL+GMDM_MA9

• 1I:+*#()2?#01uV9+o+s?1&I.$#5"(

R(1+I"110d+#"01"&?I+41).#0+2%'(#+(11+JL+L&I5"1+GM8e9 %''+?#01(+()%")+5?4%"j10 P#"+jo8MMMs?#01( (1'1*)+5?4%"j10+?#01+V+=&).+4&?+*#() ƒ2V9+o+j 4%"j+V "1)5"?+#"01"&?I+ƒ

6N

b15"&()&*+)"&%?I5'%)&#?+Y&%+Y%"&%$'1+1'&4&?%)&#? 1X%4-'1+2%'(#+(11+JL+NMDMAC+GMDM_MA9

@)-&)#,1:+Z>3QUL L%*)#"(: 82ZC>9C+A2ZC39C+_2>CQ9C D23CU9C+72QCL9C+62UCL9

P(A, B, C, D, E, F) = 1 Z

• i=1...6

φi

UX%4-'1+*#?)&?51(+#Y1"+?1X)+(&X+('&01(M

eB

Z > 3 Q U L

Cmput 651 - Clique Trees {20,24}/10/2008

b15"&()&*+)"&%?I5'%)&#?+Y&%+Y%"&%$'1+1'&4&?%)&#? 1X%4-'1+2*#?)f09+2%'(#+(11+JL+NMDMAC+GMDM_MA9

@)-&)#,1:+Z>3QUL L%*)#"(: 82ZC>9C+A2ZC39C+_2>CQ9C D23CU9C+72QCL9C+62UCL9

P(A, B, C, D, E, F) = 1 Z

• i=1...6

φi

U'&4&?%)1+Z:+ZC>C3+&?++}8f(+(*#-1+K]+%00+10I1+>K3

• A

P(A, B, C, D, E, F) = φ3(B, D)φ4(C, E)φ5(D, F)φ6(E, F)

• A

φ1(A, B)φ2(A, C) = φ3(B, D)φ4(C, E)φ5(D, F)φ6(E, F)

• A

ψ1(A, B, C) = φ3(B, D)φ4(C, E)φ5(D, F)φ6(E, F)τ1(B, C)

e8

Z > 3 Q U L Z > 3 Q U L 8

b15"&()&*+)"&%?I5'%)&#?+Y&%+Y%"&%$'1+1'&4&?%)&#? 1X%4-'1+2*#?)f09+2%'(#+(11+JL+NMDMAC+GMDM_MA9

@)-&)#,1:+Z>3QUL L%*)#"(: 82ZC>9C+A2ZC39C+_2>CQ9C D23CU9C+72QCL9C+62UCL9

P(A, B, C, D, E, F) = 1 Z

• i=1...6

φi

U'&4&?%)1+>:+>C3CQ+&?++}Af(+(*#-1+K]+%00+10I1+3KQ

• B

P(B, C, D, E, F) = φ4(C, E)φ5(D, F)φ6(E, F)

• B

φ3(B, D)τ1(B, C) = φ4(C, E)φ5(D, F)φ6(E, F)

• B

ψ2(B, C, D) = φ4(C, E)φ5(D, F)φ6(E, F)τ2(C, D)

eA

Z > 3 Q U L 8 Z > 3 Q U L 8 A

Cmput 651 - Clique Trees {20,24}/10/2008

b15"&()&*+)"&%?I5'%)&#?+Y&%+Y%"&%$'1+1'&4&?%)&#? 1X%4-'1+2*#?)f09+2%'(#+(11+JL+NMDMAC+GMDM_MA9

@)-&)#,1:+Z>3QUL L%*)#"(: 82ZC>9C+A2ZC39C+_2>CQ9C D23CU9C+72QCL9C+62UCL9

P(A, B, C, D, E, F) = 1 Z

• i=1...6

φi

U'&4&?%)1+3:+3CQCU+&?++}_f(+(*#-1+%'"1%0d+*#??1*)10

• C

P(C, D, E, F) = φ5(D, F)φ6(E, F)

• C

φ4(C, E)τ2(C, D) = φ5(D, F)φ6(E, F)

• C

ψ3(C, D, E) = φ5(D, F)φ6(E, F)τ3(D, E)

e_

Z > 3 Q U L 8 A

b15"&()&*+)"&%?I5'%)&#?+Y&%+Y%"&%$'1+1'&4&?%)&#? 1X%4-'1+2*#?)f09+2%'(#+(11+JL+NMDMAC+GMDM_MA9

@)-&)#,1:+Z>3QUL L%*)#"(: 82ZC>9C+A2ZC39C+_2>CQ9C D23CU9C+72QCL9C+62UCL9

P(A, B, C, D, E, F) = 1 Z

• i=1...6

φi

U'&4&?%)1+Q:+QCUCL+&?++}Df(+(*#-1+K]+%00+10I1+QKU

• D

P(D, E, F) = φ6(E, F)

• D

φ5(D, F)τ3(D, E) = φ6(E, F)

• D

ψ4(D, E, F) = φ6(E, F)τ4(E, F)

eD

Z > 3 Q U L 8 A Z > 3 Q U L 8 A _

Cmput 651 - Clique Trees {20,24}/10/2008

b15"&()&*+)"&%?I5'%)&#?+Y&%+Y%"&%$'1+1'&4&?%)&#? 1X%4-'1+2*#?)f09+2%'(#+(11+JL+NMDMAC+GMDM_MA9

@)-&)#,1:+Z>3QUL L%*)#"(: 82ZC>9C+A2ZC39C+_2>CQ9C D23CU9C+72QCL9C+62UCL9

P(A, B, C, D, E, F) = 1 Z

• i=1...6

φi

U'&4&?%)1+U:+UCL+&?++}7f(+(*#-1+%'"1%0d+*#??1*)10

• E

P(E, F) =

• E

φ6(E, F)τ4(E, F) =

• E

ψ5(E, F) = τ5(F)

e7

Z > 3 Q U L 8 A _

b15"&()&*+)"&%?I5'%)&#?+Y&%+Y%"&%$'1+1'&4&?%)&#? 1X%4-'1+2*#?)f09+2%'(#+(11+JL+NMDMAC+GMDM_MA9

@)-&)#,1:+Z>3QUL L%*)#"(: 82ZC>9C+A2ZC39C+_2>CQ9C D23CU9C+72QCL9C+62UCL9

P(A, B, C, D, E, F) = 1 Z

• i=1...6

φi

U'&4&?%)1+L:+L+o+'%()+Y%"&%$'1C+(#+=1+()#-

• F

P(F) =

• F

τ5(F)

e6

Z > 3 Q U L 8 A _

Cmput 651 - Clique Trees {20,24}/10/2008

b15"&()&*+)"&%?I5'%)&#?+Y&%+Y%"&%$'1+1'&4&?%)&#?

2%'(#+(11+JL+NMDMAC+GMDM_MA9

["01"&?I:+Z>3QUL ["01"&?I:+ZQU>3L

ee

Z > 3 Q U L Z > 3 Q U L 8 A _ Z > 3 Q U L Z > 3 Q U L 8 A _

L&?0+4%X&4%'+*'&;51(

2%'(#+(11+JL+NMDMAC+GMDM_MA9

<%(j:+I&Y1?+*.#"0%'+I"%-.C+P&?0+4%X&4%'+*'&;51(

„+l!K.%"0+P#"+I1?1"%'+I"%-.( „+>R<+1%(d+=&).+*.#"0%'+I"%-.(

eG

Cmput 651 - Clique Trees {20,24}/10/2008

L&?0+4%X&4%'+*'&;51(

2%'(#+(11+JL+NMDMAC+GMDM_M89

/%X&454+*%"0&?%'&)d+(1%"*.+2%'(#+(11+JL+L&I+GM869:+

+++*'&;51u'&()+o+@EC+*5""1?)u*'&;51+o+@E +++=.&'1+()&''+5?4%"j10+?#01(: ++++++(1'1*)+25?4%"j109+V+=&).+4%X+s+4%"j10+?1&I.$#5"( ++++++&P+V+P5''d+*#??1*)10+)#+*5""1?)u*'&;51: +++++++++%00+V+)#+*5""1?)u*'&;51 ++++++1'(1: +++++++++%00+*5""1?)u*'&;51+)#+*'&;51u'&() +++++++++*5""1?)u*'&;51+o+@V+%?0+Vf(+4%"j10+?1&I.$#5"(E ++++++4%"j+V +++"1)5"?+*'&;51u'&()++++++

eN

L&?0+4%X&4%'+*'&;51(

2%'(#+(11+JL+NMDMAC+GMDM_MA9

l#01(:++++++++++Q++++>+++++3++++++Z+++++++U+++++++L

*5""1?)u*'&;51:+@QE+@>QE+@>3QE+@Z>3E+@3QUE+@QULE %00&)&#?+)#+*'&;51u'&():

l#01(:++++++++++Q++++>+++++L++++++3+++++++Z++++++U

*5""1?)u*'&;51:+@QE+@>QE+@>QLE+@>3LE+@Z>3E+@3ULE %00&)&#?+)#+*'&;51u'&():

GB

Z > 3 Q U L Z > 3 Q U L

Cmput 651 - Clique Trees {20,24}/10/2008

L&?0+*'&;51+)"11+10I1(

2%'(#+(11+JL+NMDMA9

<%(j:+I&Y1?+*'&;51(C+*#??1*)+).14+)#+$5&'0+*'&;51+)"11 R(1+4%X&454+(-%??&?I+)"11+%'I#"&).4:

„+()%")+=&).+*#4-'1)1+*'&;51+I"%-. „+%((&I?+10I1+=1&I.)(:+ „+"14#Y1+10I1(+)#+-"#05*1+)"11+=&).+4%X+(54+#P+

10I1+=1&I.)( ⇒+10I1(+"1-"1(1?)+(1-(1)(+

|Ci ∩ Cj|

G8

>5&'0+*'&;51+)"11

2%'(#+(11+JL+NMDMAC+GMDM_MA9

@Z>3E @>3QE @3QUE @QULE @Z>3E @>QLE @>3LE @3ULE

GA

Z > 3 Q U L Z > 3 Q U L

a

V W `

a

V W `

Cmput 651 - Clique Trees {20,24}/10/2008

3'&;51+)"11(+P"#4+*.#"0%'+I"%-.(+2%'(#+(11+JL+NMDMA9

O544%"d: ,&Y1?+P%*)#"(++++++=.&*.+P%*)#"&h1+$%(10+#?+

8M $5&'0+*.#"0%'+I"%-.++++++++++P"#4+++++++++++2)"&%?I5'%)&#?9 AM P&?0+4%X&4%'+*'&;51(+&?+ _M $5&'0+*'&;51+)"11++++++P"#4+++++++‚(+4%X&4%'+*'&;51(

Q#+&?P1"1?*1+#?+*'&;51+)"11M

HF H∗ F HF H∗ T H∗

G_

Cmput 651 - Clique Trees {20,24}/10/2008