[PPT] - Graph Coloring: Comparing Cluster Graphs to Factor Graphs A,B,E PowerPoint Presentation

SLIDE 1

Simon Streicher and Johan du Preez Stellenbosch University

Graph Coloring: Comparing Cluster Graphs to Factor Graphs

A B C D E F G A B C D E F G

A,D E,D A,E,D A,E A,B,E B,E A,C,D,F B,E,G E,D,G G A,E,D A,B,E A,C,D,F B,E,G E,D,G A B F G C D E A B C D E F G A B D E G A D E

A B C D E F G

SLIDE 2

Graph Coloring

SLIDE 3

Practical example is a four coloring map problem: You only need four colors to color-in a map with no neighboring countries having the same color. Noted by Francis Guthrie in 1852 Theorem proven by Appel and Haken in 1976

A B C D E F G

Graph Coloring

SLIDE 4

Practical example is a four coloring map problem: You only need four colors to color-in a map with no neighboring countries having the same color. Noted by Francis Guthrie in 1852 Theorem proven by Appel and Haken in 1976

A B C D E F G A B C D E F G A B C D E F G

Graph Coloring

SLIDE 5

Practical example is a four coloring map problem: You only need four colors to color-in a map with no neighboring countries having the same color. Noted by Francis Guthrie in 1852 Theorem proven by Appel and Haken in 1976

A B C D E F G A B C D E F G A B C D E F G D F A C B G E A B C D E F G

Graph Coloring

SLIDE 6

D F A C B G E

Undirected graph

Graph Coloring

SLIDE 7

D F A C B G E D F A C

Maximal cliques

Graph Coloring

SLIDE 8

D F A C B G E D F A C

Maximal cliques

D A E D F A C B G E

Graph Coloring

SLIDE 9

D F A C B G E D F A C

Maximal cliques

D A E D F A C B G E A B E D F A C B G E

Graph Coloring

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D F A C B G E D F A C

Maximal cliques

D A E D F A C B G E A B E D F A C B G E D F A C B G E B G E

Graph Coloring

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D F A C B G E D F A C

Maximal cliques

D A E D F A C B G E A B E D F A C B G E D F A C B G E B G E D F A C B G E D G E

Graph Coloring

SLIDE 12

D F A C B G E D F A C

Maximal cliques

D A E D F A C B G E A B E D F A C B G E D F A C B G E B G E D F A C B G E D G E D F A C B G E

Graph Coloring

SLIDE 13

D F A C B G E D F A C

Maximal cliques

D A E D F A C B G E A B E D F A C B G E D F A C B G E B G E D F A C B G E D G E D F A C B G E D F A C B G E D F A C D A E A B E B G E D G E

Graph Coloring

SLIDE 14

D F A C B G E D F A C

Maximal cliques

D A E D F A C B G E A B E D F A C B G E D F A C B G E B G E D F A C B G E D G E D F A C B G E D F A C B G E D F A C D A E A B E B G E D G E D F A C B G E

Graph Coloring

SLIDE 15

D F A C B G E D F A C

Maximal cliques

D A E D F A C B G E A B E D F A C B G E D F A C B G E B G E D F A C B G E D G E D F A C B G E D F A C B G E D F A C D A E A B E B G E D G E D F A C B G E A B C D E F G

Graph Coloring

SLIDE 16

Sudoku is also a graph coloring problem

Graph Coloring

SLIDE 17

Sudoku is also a graph coloring problem 4x4 Sudoku example:

A B C D H G F E I J K L P O N M

Graph Coloring

SLIDE 18

Sudoku is also a graph coloring problem 4x4 Sudoku example:

A B C D H G F E I J K L P O N M

rows

Maximal cliques:

Graph Coloring

SLIDE 19

Sudoku is also a graph coloring problem 4x4 Sudoku example:

A B C D H G F E I J K L P O N M

rows

Maximal cliques:

A B F E C D H G I J N M K L P O

blocks

Graph Coloring

SLIDE 20

Sudoku is also a graph coloring problem 4x4 Sudoku example:

A B C D H G F E I J K L P O N M

rows

Maximal cliques:

A B F E C D H G I J N M K L P O

blocks

A E I M B F J N C G K O D H L P

columns

Graph Coloring

SLIDE 21

Sudoku is also a graph coloring problem 4x4 Sudoku example:

A B C D H G F E I J K L P O N M

rows

Maximal cliques:

A B F E C D H G I J N M K L P O

blocks

A E I M B F J N C G K O D H L P

columns

A B C D H G F E I J K L P O N M C D H G A B F E K L P O I J N M D H L P C G K O B F J N A E I M

Graph Coloring

SLIDE 22

Sudoku is also a graph coloring problem 4x4 Sudoku example:

A B C D H G F E I J K L P O N M

rows

Maximal cliques:

A B F E C D H G I J N M K L P O

blocks

A E I M B F J N C G K O D H L P

columns

A B C D H G F E I J K L P O N M C D H G A B F E K L P O I J N M D H L P C G K O B F J N A E I M 1 2 3 4 3 4 1 2 2 3 4 1 4 1 2 3 1 2 3 4 3 4 1 2 2 3 4 1 4 1 2 3 1 3 2 4 2 4 3 1 3 1 4 2 4 2 1 3

Graph Coloring

SLIDE 23

1 2 3 4 3 4 1 2 2 3 4 1 4 1 2 3 A B C D H G F E I J K L P O N M

Probabilistic Graphical Models Graph Coloring

SLIDE 24

1 2 3 4 3 4 1 2 2 3 4 1 4 1 2 3 A B C D H G F E I J K L P O N M 1 2 3 4 2 1 4 3 2 3 4 1 3 2 1 4

Probabilistic Graphical Models Graph Coloring

SLIDE 25

1 2 3 4 3 4 1 2 2 3 4 1 4 1 2 3 A B C D H G F E I J K L P O N M 1 2 3 4 2 1 4 3 2 3 4 1 3 2 1 4

But how do we take all constraints into account ?

Probabilistic Graphical Models Graph Coloring

SLIDE 26

Probabilistic Graphical Models

SLIDE 27

PROBABILISTIC GRAPHICAL MODELS

In a general sense, a PGM of a system

clusters information into local sections, and
let the sections communicate about their combined outcome

SLIDE 28

PROBABILISTIC GRAPHICAL MODELS

In a general sense, a PGM of a system

clusters information into local sections, and
let the sections communicate about their combined outcome

In a probabilistic sense, these "local sections" are

prior distributions,
marginal distributions, and/or
conditional distributions;

together, a compact representation of a larger space

SLIDE 29

PROBABILISTIC GRAPHICAL MODELS

In a general sense, a PGM of a system

clusters information into local sections, and
let the sections communicate about their combined outcome

SLIDE 30

PROBABILISTIC GRAPHICAL MODELS

In a general sense, a PGM of a system

clusters information into local sections, and
let the sections communicate about their combined outcome

A B C D E F G

SLIDE 31

PROBABILISTIC GRAPHICAL MODELS

In a general sense, a PGM of a system

clusters information into local sections, and
let the sections communicate about their combined outcome

A B C D E F G

D F A C D A E A B E D G E B G E

SLIDE 32

PROBABILISTIC GRAPHICAL MODELS

In a general sense, a PGM of a system

clusters information into local sections, and
let the sections communicate about their combined outcome

A B C D E F G

D F A C D A E A B E D G E B G E

P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)

1 2 3 4 5

SLIDE 33

PROBABILISTIC GRAPHICAL MODELS

In a general sense, a PGM of a system

clusters information into local sections, and
let the sections communicate about their combined outcome

A B C D E F G

D F A C D A E A B E D G E B G E

P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)

1 2 3 4 5

1 2 3 4 1 1 2 4 3 1 1 3 2 4 1 4 3 2 1 1

non normalized

...

P(A,B,C,D) elsewhere

A,C,D,F

P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)

1 2 3 4 5

SLIDE 34

PROBABILISTIC GRAPHICAL MODELS

In a general sense, a PGM of a system

clusters information into local sections, and
let the sections communicate about their combined outcome

A B C D E F G

D F A C D A E A B E D G E B G E

P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)

1 2 3 4 5

1 2 3 4 1 1 2 4 3 1 1 3 2 4 1 4 3 2 1 1

non normalized

...

P(A,B,C,D) elsewhere

A,C,D,F

P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)

1 2 3 4 5

1 2 3 1 1 2 4 1 1 3 2 1 4 3 2 1

non norm

...

P(B,E,G) elsewhere

1 2 3 4 1 1 2 4 3 1 1 3 2 4 1 4 3 2 1 1

non normalized

...

P(A,B,C,D) elsewhere

B,E,G

P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)

1 2 3 4 5

SLIDE 35

PROBABILISTIC GRAPHICAL MODELS

In a general sense, a PGM of a system

clusters information into local sections, and
let the sections communicate about their combined outcome

A B C D E F G

D F A C D A E A B E D G E B G E

P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)

1 2 3 4 5

1 2 3 4 1 1 2 4 3 1 1 3 2 4 1 4 3 2 1 1

non normalized

...

P(A,B,C,D) elsewhere

A,C,D,F

P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)

1 2 3 4 5

1 2 3 1 1 2 4 1 1 3 2 1 4 3 2 1

non norm

...

P(B,E,G) elsewhere

1 2 3 4 1 1 2 4 3 1 1 3 2 4 1 4 3 2 1 1

non normalized

...

P(A,B,C,D) elsewhere

B,E,G

P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)

1 2 3 4 5

A,B,E D,E,G A,D,E

P(A,B,C,D,E,F,G) = f (A,C,D,F) · f (A,D,E) · f (A,B,E) · f (B,E,G) · f (D,E,G)

1 2 3 4 5

SLIDE 36

In a general sense, a PGM of a system

clusters information into local sections, and
let the sections communicate about their combined outcome

A B C D E F G A,C,D,F B,E,G A,B,E D,E,G A,D,E

PROBABILISTIC GRAPHICAL MODELS

SLIDE 37

In a general sense, a PGM of a system

clusters information into local sections, and
let the sections communicate about their combined outcome

A B C D E F G A,C,D,F B,E,G A,B,E D,E,G A,D,E

Factor/Bethé graph

A,D,E A,B,E A,C,D,F B,E,G E,D,G A B F G C D E

A B C D E F G A B D E G A D E

PROBABILISTIC GRAPHICAL MODELS

SLIDE 38

In a general sense, a PGM of a system

clusters information into local sections, and
let the sections communicate about their combined outcome

A B C D E F G A,C,D,F B,E,G A,B,E D,E,G A,D,E

Factor/Bethé graph

A,D,E A,B,E A,C,D,F B,E,G E,D,G A B F G C D E

A B C D E F G A B D E G A D E

Cluster graph

A,D E,D A,D,E A,E A,B,E B,E A,C,D,F B,E,G E,D,G G

PROBABILISTIC GRAPHICAL MODELS

SLIDE 39

In a general sense, a PGM of a system

clusters information into local sections, and
let the sections communicate about their combined outcome

A B C D E F G A,C,D,F B,E,G A,B,E D,E,G A,D,E

Factor/Bethé graph

A,D,E A,B,E A,C,D,F B,E,G E,D,G A B F G C D E

A B C D E F G A B D E G A D E

Cluster graph

A,D E,D A,D,E A,E A,B,E B,E A,C,D,F B,E,G E,D,G G

Factor/Bethé graph

A,D,E A,B,E A,C,D,F B,E,G E,D,G A B F G C D E

A B C D E F G A B D E G A D E

Cluster graph

A,D E,D A,D,E A,E A,B,E B,E A,C,D,F B,E,G E,D,G G

PROBABILISTIC GRAPHICAL MODELS

SLIDE 40

In a general sense, a PGM of a system

clusters information into local sections, and
let the sections communicate about their combined outcome

A B C D E F G A,C,D,F B,E,G A,B,E D,E,G A,D,E

Factor/Bethé graph

A,D,E A,B,E A,C,D,F B,E,G E,D,G A B F G C D E

A B C D E F G A B D E G A D E

Cluster graph

A,D E,D A,D,E A,E A,B,E B,E A,C,D,F B,E,G E,D,G G

Factor/Bethé graph

A,D,E A,B,E A,C,D,F B,E,G E,D,G A B F G C D E

A B C D E F G A B D E G A D E

Cluster graph

A,D E,D A,D,E A,E A,B,E B,E A,C,D,F B,E,G E,D,G G

Cluster graph

A,D E,D A,D,E A,E A,B,E B,E A,C,D,F B,E,G E,D,G G

PROBABILISTIC GRAPHICAL MODELS

SLIDE 41

In a general sense, a PGM of a system

clusters information into local sections, and
let the sections communicate about their combined outcome

A B C D E F G A,C,D,F B,E,G A,B,E D,E,G A,D,E

Factor/Bethé graph

A,D,E A,B,E A,C,D,F B,E,G E,D,G A B F G C D E

A B C D E F G A B D E G A D E

Cluster graph

A,D E,D A,D,E A,E A,B,E B,E A,C,D,F B,E,G E,D,G G

Factor/Bethé graph

A,D,E A,B,E A,C,D,F B,E,G E,D,G A B F G C D E

A B C D E F G A B D E G A D E

Cluster graph

A,D E,D A,D,E A,E A,B,E B,E A,C,D,F B,E,G E,D,G G

Cluster graph

A,D E,D A,D,E A,E A,B,E B,E A,C,D,F B,E,G E,D,G G

PROBABILISTIC GRAPHICAL MODELS

SLIDE 42

In a general sense, a PGM of a system

clusters information into local sections, and
let the sections communicate about their combined outcome

A B C D E F G A,C,D,F B,E,G A,B,E D,E,G A,D,E

Factor/Bethé graph

A,D,E A,B,E A,C,D,F B,E,G E,D,G A B F G C D E

A B C D E F G A B D E G A D E

Cluster graph

A,D E,D A,D,E A,E A,B,E B,E A,C,D,F B,E,G E,D,G G

Factor/Bethé graph

A,D,E A,B,E A,C,D,F B,E,G E,D,G A B F G C D E

A B C D E F G A B D E G A D E

Cluster graph

A,D E,D A,D,E A,E A,B,E B,E A,C,D,F B,E,G E,D,G G

Cluster graph

A,D E,D A,D,E A,E A,B,E B,E A,C,D,F B,E,G E,D,G G

PROBABILISTIC GRAPHICAL MODELS

SLIDE 43

In a general sense, a PGM of a system

clusters information into local sections, and
let the sections communicate about their combined outcome

A B C D E F G A,C,D,F B,E,G A,B,E D,E,G A,D,E

Factor/Bethé graph

A,D,E A,B,E A,C,D,F B,E,G E,D,G A B F G C D E

A B C D E F G A B D E G A D E

Cluster graph

A,D E,D A,D,E A,E A,B,E B,E A,C,D,F B,E,G E,D,G G

Factor/Bethé graph

A,D,E A,B,E A,C,D,F B,E,G E,D,G A B F G C D E

A B C D E F G A B D E G A D E

Cluster graph

A,D E,D A,D,E A,E A,B,E B,E A,C,D,F B,E,G E,D,G G

Cluster graph

A,D E,D A,D,E A,E A,B,E B,E A,C,D,F B,E,G E,D,G G

PROBABILISTIC GRAPHICAL MODELS

SLIDE 44

In a general sense, a PGM of a system

clusters information into local sections, and
let the sections communicate about their combined outcome

A B C D E F G A,C,D,F B,E,G A,B,E D,E,G A,D,E

Factor/Bethé graph

A,D,E A,B,E A,C,D,F B,E,G E,D,G A B F G C D E

A B C D E F G A B D E G A D E

Cluster graph

A,D E,D A,D,E A,E A,B,E B,E A,C,D,F B,E,G E,D,G G

Factor/Bethé graph

A,D,E A,B,E A,C,D,F B,E,G E,D,G A B F G C D E

A B C D E F G A B D E G A D E

Cluster graph

A,D E,D A,D,E A,E A,B,E B,E A,C,D,F B,E,G E,D,G G

Cluster graph

A,D E,D A,D,E A,E A,B,E B,E A,C,D,F B,E,G E,D,G G

PROBABILISTIC GRAPHICAL MODELS

SLIDE 45

A B C D E F G

PROBABILISTIC GRAPHICAL MODELS

SLIDE 46

A B C D E F G

PROBABILISTIC GRAPHICAL MODELS

SLIDE 47

A B C D E F G

Factor/Bethé graph

A,D,E A,B,E A,C,D,F B,E,G E,D,G A B F G C D E

A B C D E F G A B D E G A D E

A B C D E F G A B C D E F G

PROBABILISTIC GRAPHICAL MODELS

SLIDE 48

A B C D E F G

Factor/Bethé graph

A,D,E A,B,E A,C,D,F B,E,G E,D,G A B F G C D E

A B C D E F G A B D E G A D E

A B C D E F G A B C D E F G

Cluster graph

A,D E,D A,D,E A,E A,B,E B,E A,C,D,F B,E,G E,D,G G

A B C D E F G A B C D E F G

PROBABILISTIC GRAPHICAL MODELS

SLIDE 49

CLUSTER GRAPHS

SLIDE 50

CLUSTER GRAPHS

We found that

Graph structure influence convergence speed and accuracy
Factor graphs are predominant in PGM literature
Cluster graphs outperform factor graphs

SLIDE 51

CLUSTER GRAPHS

We found that

Graph structure influence convergence speed and accuracy
Factor graphs are predominant in PGM literature
Cluster graphs outperform factor graphs

Why is cluster graphs the underdog?

Multiple solutions for the same clusters
Absence of a generic construction procedure

SLIDE 52

CLUSTER GRAPHS

We found that

Graph structure influence convergence speed and accuracy
Factor graphs are predominant in PGM literature
Cluster graphs outperform factor graphs

Why is cluster graphs the underdog?

Multiple solutions for the same clusters
Absence of a generic construction procedure

We propose the LTRIP procedure as a solution

SLIDE 53

B,C,D,E,F A,B,C,D A,B,G B,C,G B,E,F

Factor/Bethé graph:

A B C D E F G variables clusters

CLUSTER GRAPHS

SLIDE 54

B,C,D,E,F A,B,C,D A,B,G B,C,G B,E,F

Factor/Bethé graph:

A B C D E F G variables clusters connection layer

G A B C D E F

CLUSTER GRAPHS

SLIDE 55

B,C,D,E,F A,B,C,D A,B,G B,C,G B,E,F

Factor/Bethé graph:

A B C D E F G variables clusters connection layer

G A B C D E F

*running intersection property sepsets A

A A

CLUSTER GRAPHS

SLIDE 56

B,C,D,E,F A,B,C,D A,B,G B,C,G B,E,F

Factor/Bethé graph:

A B C D E F G variables clusters connection layer

G A B C D E F

*running intersection property sepsets B

B B B B B

CLUSTER GRAPHS

SLIDE 57

B,C,D,E,F A,B,C,D A,B,G B,C,G B,E,F

Factor/Bethé graph:

A B C D E F G variables clusters connection layer

G A B C D E F

*running intersection property sepsets

C C C

C

CLUSTER GRAPHS

SLIDE 58

B,C,D,E,F A,B,C,D A,B,G B,C,G B,E,F

Factor/Bethé graph:

A B C D E F G variables clusters connection layer

G A B C D E F

*running intersection property sepsets A B C D E F G

G A B C E F A B B B B C C D D E F G

CLUSTER GRAPHS

SLIDE 59

B,C,D,E,F A,B,C,D A,B,G B,C,G B,E,F

Cluster graph:

A B C D E F G variables clusters connection layer

G A B C D E F

*running intersection property

CLUSTER GRAPHS

SLIDE 60

B,C,D,E,F A,B,C,D A,B,G B,C,G B,E,F

Cluster graph:

A B C D E F G variables clusters connection layer

G A B C D E F

*running intersection property

CLUSTER GRAPHS

SLIDE 61

*running intersection property

multivar. sepsets

LTRIP procedure:

A B C D E F G variables clusters

B,E,F B,C,D,E,F A,B,C,D A,B,G B,C,G

CLUSTER GRAPHS

SLIDE 62

*running intersection property

multivar. sepsets

LTRIP procedure:

A B C D E F G variables clusters A

B,E,F B,C,D,E,F A,B,C,D A,B,G B,C,G A

CLUSTER GRAPHS

SLIDE 63

*running intersection property

multivar. sepsets

LTRIP procedure:

A B C D E F G variables clusters

B,E,F B,C,D,E,F A,B,C,D A,B,G B,C,G B B B A B

B

CLUSTER GRAPHS

SLIDE 64

*running intersection property

multivar. sepsets

LTRIP procedure:

A B C D E F G variables clusters C

B,E,F B,C,D,E,F A,B,C,D A,B,G B,C,G B,C B B,C A B

CLUSTER GRAPHS

SLIDE 65

*running intersection property

multivar. sepsets

LTRIP procedure:

A B C D E F G variables clusters D

B,E,F B,C,D,E,F A,B,C,D A,B,G B,C,G B,C,D B B,C A B

CLUSTER GRAPHS

SLIDE 66

*running intersection property

multivar. sepsets

LTRIP procedure:

A B C D E F G variables clusters E

B,E,F B,C,D,E,F A,B,C,D A,B,G B,C,G B,C,D B B,C A B,E

CLUSTER GRAPHS

SLIDE 67

*running intersection property

multivar. sepsets

LTRIP procedure:

A B C D E F G variables clusters F

B,E,F B,C,D,E,F A,B,C,D A,B,G B,C,G B,C,D B B,C A B,E,F

CLUSTER GRAPHS

SLIDE 68

*running intersection property

multivar. sepsets

LTRIP procedure:

A B C D E F G variables clusters G

B,E,F B,C,D,E,F A,B,C,D A,B,G B,C,G B,C,D B B,C G A B,E,F

CLUSTER GRAPHS

SLIDE 69

*running intersection property

multivar. sepsets

LTRIP procedure:

A B C D E F G variables clusters

B,E,F B,C,D,E,F A,B,C,D A,B,G B,C,G B,C,D B B,C G A B,E,F

CLUSTER GRAPHS

SLIDE 70

A B C D H G F E I J K L P O N M

Sudoku cluster graph:

CLUSTER GRAPHS

SLIDE 71

A B C D H G F E I J K L P O N M

Sudoku cluster graph:

A B C D H G F E I J K L P O N M A B F E C D H G I J N M K L P O A E I M B F J N C G K O D H L P

CLUSTER GRAPHS

SLIDE 72

A B C D H G F E I J K L P O N M

Sudoku cluster graph:

A,B,C,D E,F,G,H A,E,I,M B,F,J,N A,B E,F A,E B,F C,D G,H I,M J,N A,B,E,F C,D,G,H I,J,M,N C,G D,H I,J M,N K,L,O,P K,O L,P K,L O,P I,J,K,L C,G,K,O D,H,L,P M,N,O,P

CLUSTER GRAPHS

SLIDE 73

A

A,B,C,D E,F,G,H A,E,I,M B,F,J,N A,B,E,F C,D,G,H K,L,O,P I,J,K,L C,G,K,O D,H,L,P M,N,O,P I,J,M,N A E I M B F J N C G K O D H L P

E I M F J N C G K O D H L P A B E I M F J N C G K O D H L P A B E I M F J N C G K O D H L P B

Sudoku factor/Bethé graph:

CLUSTER GRAPHS

SLIDE 74

CLUSTER GRAPHS RESULTS

SLIDE 75

A comparison of cluster graphs vs. factor graphs on PGMs build from Sudoku puzzles Datasets used:

Project Euler @ projecteuler.net/problem=96
Sterten's 95 hardest Sudokus @ magictour.free.fr/top95

Tested with different cluster sizes by splitting-up clusters

CLUSTER GRAPHS RESULTS

SLIDE 76

A B C D E F G H I

9x9 Sudoku puzzle

... J K L M N O P Q R

CLUSTER GRAPHS RESULTS

SLIDE 77

A B C D E F G H I

9x9 Sudoku puzzle

... J K L M N O P Q R A B C D E F G H I ... J K L M N O P Q R

CLUSTER GRAPHS RESULTS

SLIDE 78

A B C D E F G H I

Split each 9 variable clique

C B A D E A F G A H I A H I B F B G D E B C B D F E C H C G D I C D G H D E F F E G H E I F I H F G H H I G

into cliques of 3 variables

CLUSTER GRAPHS RESULTS

SLIDE 79

A B C D E F G H I

Split each 9 variable clique into cliques of 5 variables

E D B A C F I G A H F B I G H E D F B C D I G H C E D F G H E F I G H

CLUSTER GRAPHS RESULTS

SLIDE 80

A B C D E F G H I

Split each 9 variable clique

E D F B G A C E F B I G A H E D F B G H C E D F I G H C

into cliques of 7 variables

CLUSTER GRAPHS RESULTS

SLIDE 81

A B C D E F G H I

Split each 9 variable clique

A B C D E F G H I

into cliques of 9 variables

CLUSTER GRAPHS RESULTS

SLIDE 82

A B C D E F G H I

Split each 9 variable clique

A B C D E F G H I

into cliques of 9 variables

A,B,C,D E,F,G,H A,E,I,M B,F,J,N A,B E,F A,E B,F C,D G,H I,M J,N A,B,E,F C,D,G,H I,J,M,N C,G D,H I,J M,N K,L,O,P K,O L,P K,L O,P I,J,K,L C,G,K,O D,H,L,P M,N,O,P A A,B,C,D E,F,G,H A,E,I,M B,F,J,N A,B,E,F C,D,G,H K,L,O,P I,J,K,L C,G,K,O D,H,L,P M,N,O,P I,J,M,N A E I M B F J N C G K O D H L P E I M F J N C G K O D H L P A B E I M F J N C G K O D H L P A B E I M F J N C G K O D H L P B

Build these clusters into both a factor graph and a cluster graph

CLUSTER GRAPHS RESULTS

SLIDE 83

A B C D E F G H I

Split each 9 variable clique

A B C D E F G H I

into cliques of 9 variables

A,B,C,D E,F,G,H A,E,I,M B,F,J,N A,B E,F A,E B,F C,D G,H I,M J,N A,B,E,F C,D,G,H I,J,M,N C,G D,H I,J M,N K,L,O,P K,O L,P K,L O,P I,J,K,L C,G,K,O D,H,L,P M,N,O,P A A,B,C,D E,F,G,H A,E,I,M B,F,J,N A,B,E,F C,D,G,H K,L,O,P I,J,K,L C,G,K,O D,H,L,P M,N,O,P I,J,M,N A E I M B F J N C G K O D H L P E I M F J N C G K O D H L P A B E I M F J N C G K O D H L P A B E I M F J N C G K O D H L P B

Build these clusters into both a factor graph and a cluster graph Run message passing on both graphs

(note, for a solved Suduko all clusters is reduced to a single entry)

P(A,B,C,D,E,F,G,H,I)

A 1 B 3 C 2 D 6 E 4 F 5 G 7 H 9 I 8 elsewhere 1

CLUSTER GRAPHS RESULTS

SLIDE 84

Project Euler: solution count

36 13 1 36 14 13 37 factor graph cluster graph

Size 3

CLUSTER GRAPHS RESULTS

SLIDE 85

Project Euler: solution count

36 13 1 36 14 13 37 factor graph cluster graph

Size 3

CLUSTER GRAPHS RESULTS

SLIDE 86

Project Euler: solution count

36 13 1 36 14 13 37 factor graph cluster graph

Size 3

cluster graph slower

CLUSTER GRAPHS RESULTS

SLIDE 87

Project Euler: solution count

36 13 1 36 14 13 37 factor graph cluster graph

Size 3

cluster graph slower 25 23 2 25 25 23 27 factor graph cluster graph

Size 5

CLUSTER GRAPHS RESULTS

SLIDE 88

Project Euler: solution count

36 13 1 36 14 13 37 factor graph cluster graph

Size 3

cluster graph slower 25 23 2 25 25 23 27 factor graph cluster graph

Size 5

CLUSTER GRAPHS RESULTS

SLIDE 89

Project Euler: solution count

36 13 1 36 14 13 37 factor graph cluster graph

Size 3

cluster graph slower 25 23 2 25 25 23 27 factor graph cluster graph

Size 5

cluster graph slower

CLUSTER GRAPHS RESULTS

SLIDE 90

Project Euler: solution count

36 13 1 36 14 13 37 factor graph cluster graph

Size 3

cluster graph slower 25 23 2 25 25 23 27 factor graph cluster graph

Size 5

cluster graph slower 5 41 4 5 45 41 9 46 4 50 46 4 cluster graph faster cluster graph faster factor graph factor graph cluster graph cluster graph

Size 7 Size 9

CLUSTER GRAPHS RESULTS

SLIDE 91

Project Euler: solution count

36 13 1 36 14 13 37 factor graph cluster graph

Size 3

cluster graph slower 25 23 2 25 25 23 27 factor graph cluster graph

Size 5

cluster graph slower 5 41 4 5 45 41 9 46 4 50 46 4 cluster graph faster cluster graph faster factor graph factor graph cluster graph cluster graph

Size 7 Size 9

Cluster graphs more successful than factor graphs Naive solver to test graph structures - can improve!

CLUSTER GRAPHS RESULTS

SLIDE 92

CONCLUSION

Main contribution is LTRIP for constructing cluster graphs
These cluster graphs show great promise over factor graphs
We hope LTRIP will enhance the popularity of cluster graphs

SLIDE 93

CONCLUSION

Main contribution is LTRIP for constructing cluster graphs
These cluster graphs show great promise over factor graphs
We hope LTRIP will enhance the popularity of cluster graphs

FUTURE WORK

Investigate more advance techniques for graph coloring PGMs
Mutual information approach for LTRIP's max spanning trees
Investigate cluster graphs on wider set of problems