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

Graph Coloring: Comparing Cluster Graphs to Factor Graphs A,B,E B,E,G B,E A,E G A,C,D,F A,D A,E,D E,D E,D,G A A A B B B E E C C D D C D E F F F A,C,D,F A,B,E A,E,D B,E,G E,D,G G G G E B E C D E G A A G B D D F A A B C D E F G Simon Streicher and Johan du Preez Stellenbosch University

Graph Coloring

Graph Coloring 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. A B C D E F G Noted by Francis Guthrie in 1852 Theorem proven by Appel and Haken in 1976

Graph Coloring 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. A A A B B B C D E C C D D E E F F F G G G Noted by Francis Guthrie in 1852 Theorem proven by Appel and Haken in 1976

Graph Coloring 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. A A A A A B B B B B C C C D D D E E E C C D D E E F F F F F G G G G G Noted by Francis Guthrie in 1852 Theorem proven by Appel and Haken in 1976

Graph Coloring A B D E F C G Undirected graph

Graph Coloring Maximal cliques A F C D A B D E F C G

Graph Coloring A Maximal cliques A D E F C D A A B B D D E E F F C C G G

Graph Coloring A Maximal cliques A D E F C D A B E A A A B B B D D D E E E F F F C C C G G G

Graph Coloring A Maximal cliques A D E F C D A B E A A A A B B B B B D D D D E E E E E F F F F C C C C G G G G G

Graph Coloring A Maximal cliques A D E F C D A B E A A A A A B B B B B B D D D D D E E E E E E F F F F F C C C C C G G G G G G D E G

Graph Coloring A Maximal cliques A D E F C D A B E A A A A A A B B B B B B B D D D D D D E E E E E E E F F F F F F C C C C C C G G G G G G G D E G

Graph Coloring A A Maximal cliques A A D D E E F F C C D D A A B B E E A A A A A A A B B B B B B B B B D D D D D D D E E E E E E E E E F F F F F F F C C C C C C C G G G G G G G G G D D E E G G

Graph Coloring A A Maximal cliques A A D D E E F F C C D D A A B B E E A A A A A A A A B B B B B B B B B B D D D D D D D D E E E E E E E E E E F F F F F F F F C C C C C C C C G G G G G G G G G G D D E E G G

Graph Coloring A A Maximal cliques A A D D E E F F C C D D A A B B E E A A A A A A A A B A B B B B B B B B B B C D E D D D D D D D D E E E E E E E E E E F F F F F F F F C C C C C C C C F G G G G G G G G G G G D D E E G G

Graph Coloring Sudoku is also a graph coloring problem

Graph Coloring Sudoku is also a graph coloring problem 4x4 Sudoku example: A B C D E F G H I K L J M N O P

Graph Coloring Sudoku is also a graph coloring problem Maximal cliques: 4x4 Sudoku example: rows A B C D A B C D E F G H E F G H I K L J I J K L M N O P M N O P

Graph Coloring Sudoku is also a graph coloring problem Maximal cliques: 4x4 Sudoku example: rows blocks A B C D A B C D A B C D E F G H E F G H E F G H I K L J I J K L I K L J M N O P M N O P M N O P

Graph Coloring Sudoku is also a graph coloring problem Maximal cliques: 4x4 Sudoku example: rows blocks columns A B C D A B C D A B C D A B C D E F G H E F G H E F G H E F G H I J K L I K L J I J K L I K L J M N O P M N O P M N O P M N O P

Graph Coloring Sudoku is also a graph coloring problem Maximal cliques: 4x4 Sudoku example: rows blocks columns A B C D A B C D A A B B C C D D A B C D A B C D A B C D E F G H E F G H H E F G H E F G H E F G E F G H E F G H I J K L I K L J I J K L I L I J K L I K L J K I J J K L M N O P M N O P M N O P M N O P M N O P M N O P M N O P

Graph Coloring Sudoku is also a graph coloring problem Maximal cliques: 4x4 Sudoku example: rows blocks columns A B C D A 1 B 2 C 3 4 D 4 A A 1 B B 2 C C 3 D 4 D A 1 B 2 C 3 D A B C D A B C D E F G H E F G H H 2 E 3 F 4 G 1 H 2 E 3 4 F G 1 H 2 3 E F 4 G 1 E F G H E F G H I J K L I K L J I J K L 2 I 3 4 L 1 2 I 3 J K 4 L 1 I K L J K 2 I J 3 J K 4 1 L M N O P M N O P M N O P M 4 N 1 O 2 P 3 M 4 N 1 O 2 3 P M N O P M 4 N 1 O 2 P 3

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

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

Probabilistic Graphical Models Graph Coloring 1 2 3 4 But how do A 1 B 2 C 3 D 4 we take all 3 4 1 2 E 3 F 4 G 1 H 2 constraints I K L 2 J 3 4 1 into account 2 3 4 1 M 4 N 1 O 2 P 3 ? 4 1 2 3

Probabilistic Graphical Models

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

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

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

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

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

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

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 A A A B D B E C D E A,C,D,F F C D D E E E G F G 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 ) 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 1 2 2 3 3 4 4 5 5 1 2 3 4 1 1 2 4 3 1 1 3 2 4 1 P ( A , B , C , D ) non normalized ... 4 3 2 1 1 elsewhere 0

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 A A A B D B E C D E A,C,D,F B,E,G F C D D E E E G F G 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 ) 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 ) 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 1 1 2 2 2 3 3 3 4 4 4 5 5 5 1 1 2 2 3 3 4 4 1 1 1 2 3 1 1 1 2 2 4 4 3 3 1 1 1 2 4 1 1 1 3 3 2 2 4 4 1 1 1 3 2 1 P ( A , B , C , D ) P ( A , B , C , D ) P ( B , E , G ) non normalized non normalized non norm ... ... ... 4 4 3 3 2 2 1 1 1 1 4 3 2 1 elsewhere elsewhere elsewhere 0 0 0