SLIDE 1
Exact Reasoning: AND/OR Search and Hybrids COMPSCI 276, Fall 2009 - - PowerPoint PPT Presentation
Exact Reasoning: AND/OR Search and Hybrids COMPSCI 276, Fall 2009 - - PowerPoint PPT Presentation
Exact Reasoning: AND/OR Search and Hybrids COMPSCI 276, Fall 2009 Set 8, Rina dechter Approximation Techniques bounded inference COMPSCI 276, Fall 2009 Set 6: Rina Dechter eading: Primary: Class Notes (7) econdary: , Darwiche chapters 14)
SLIDE 2
SLIDE 3
CS 276 3
Probabilistic Inference Tasks
X/A a * k * 1
e) , x P( max arg ) a ,..., (a evidence) | x P(X ) BEL(X
i i i
- Belief updating:
- Finding most probable explanation (MPE)
- Finding maximum a-posteriory hypothesis
- Finding maximum-expected-utility (MEU) decision
e) , x P( max arg * x
x
) x U( e) , x P( max arg ) d ,..., (d
X/D d * k * 1
variables hypothesis : X A function utility x variables decision : ) ( : U X D
SLIDE 4
CS 276 4
Belief Updating
lung Cancer Smoking X-ray Bronchitis Dyspnoea
P (lung cancer=yes | smoking=no, dyspnoea=yes ) = ?
SLIDE 5
CS 276 5
Conditioning generates the probability tree
) , | ( ) , | ( ) | ( ) | ( ) ( ) , (
e b c b
c b e P b a d P a c P a b P a P e a P Complexity of conditioning: exponential time, linear space
SLIDE 6
CS 276 6
Conditioning+Elimination
) , | ( ) , | ( ) | ( ) | ( ) ( ) , (
e d c b
c b e P b a d P a c P a b P a P e a P Idea: conditioning until of a (sub)problem gets small
* w
SLIDE 7
CS 276 7
Loop-cutset decomposition
You condition until you get a polytree
B C
B
F
A
B C
B
F
A=0 A=0
A=0
B C
B
F
A=1 A=1
A=1
P(B|F=0) = P(B, A=0|F=0)+P(B,A=1|F=0) A=0 A=1 Loop-cutset method is time exp in loop-cutset size And linear space
SLIDE 8
CS 276 8
OR search space
A D B C E F
1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
E C F D B A
1
Ordering: A B E C D F
SLIDE 9
CS 276 9
AND/OR search space
A
OR AND
1 B
OR
B
AND
1 1 E
OR
C E C E C E C
OR
D F D F D F D F D F D F D F D F
AND
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
AND
1 1 1 1 1 1 1 1
A D B C E F A D B C E F
Primal graph DFS tree
A D B C E F A D B C E F
SLIDE 10
CS 276 10
OR vs AND/OR
A
OR AND
1 B
OR
B
AND
1 1 E
OR
C E C E C E C
OR
D F D F D F D F D F D F D F D F
AND
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
AND
1 1 1 1 1 1 1 1
E
1 1 1 1
C
1 1 1 1 1 1 1 1
F
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
D
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
B
1 1
A
1
E
1 1 1 1
C
1 1 1 1 1 1 1 1
F
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
D
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
B
1 1
A
1
AND/OR OR
A D B C E F A D B C E F
1 1 1 1
SLIDE 11
CS 276 11
AND/OR vs. OR
E
1 1 1 1
C
1 1 1 1 1 1 1 1
F
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
D
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
B
1 1
A
1 A
OR AND
1 B
OR
B
AND
1 1 E
OR
C E C E C E C
OR
D F D F D F D F D F D F D F D F
AND
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
AND
1 1 1 1 1 1 1 1
E
1 1 1 1
C
1 1 1 1 1 1 1 1
F
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
D
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
B
1 1
A
1
E
1 1 1 1
C
1 1 1 1 1 1 1 1
F
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
D
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
B
1 1
A
1
AND/OR OR
A D B C E F A D B C E F
AND/OR size: exp(4), OR size exp(6)
SLIDE 12
CS 276 12
AND/OR vs. OR
E
1 1 1 1
C
1 1 1 1 1 1 1 1
F
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
D
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
B
1 1
A
1 A
OR AND
1 B
OR
B
AND
1 1 E
OR
C E C E C E C
OR
D F D F D F D F D F D F D F D F
AND
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
AND
1 1 1 1 1 1 1 1
E
1 1 1 1
C
1 1 1 1 1 1 1 1
F
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
D
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
B
1 1
A
1
E
1 1 1 1
C
1 1 1 1 1 1 1 1
F
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
D
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
B
1 1
A
1
AND/OR OR
A D B C E F A D B C E F No-goods (A=1,B=1) (B=0,C=0)
SLIDE 13
CS 276 13
AND/OR vs. OR
E
1 1 1
C
1 1 1 1 1 1
F
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
D
1 1 1 1 1 1 1 1
B
1
A
1 A
OR AND
1 B
OR
B
AND
1 E
OR
C E C E C
OR
D F D F D F D F
AND
1 1 1 1 1 1 1 1
AND
1 1 1 1 1 1
E
1 1 1
C
1 1 1 1 1 1
F
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
D
1 1 1 1 1 1 1 1
B
1
A
1
E
1 1 1
C
1 1 1 1 1 1
F
0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
D
1 1 1 1 1 1 1 1
B
1
A
1
AND/OR OR
A D B C E F A D B C E F (A=1,B=1) (B=0,C=0)
SLIDE 14
CS 276 14
OR space vs. AND/OR space
width height
OR space AND/OR space
time(sec.) nodes backtracks time(sec.) AND nodes OR nodes
5 10 3.154 2,097,150 1,048,575 0.03 10,494 5,247 4 9 3.135 2,097,150 1,048,575 0.01 5,102 2,551 5 10 3.124 2,097,150 1,048,575 0.03 8,926 4,463 4 10 3.125 2,097,150 1,048,575 0.02 7,806 3,903 5 13 3.104 2,097,150 1,048,575 0.1 36,510 18,255 5 10 3.125 2,097,150 1,048,575 0.02 8,254 4,127 6 9 3.124 2,097,150 1,048,575 0.02 6,318 3,159 5 10 3.125 2,097,150 1,048,575 0.02 7,134 3,567 5 13 3.114 2,097,150 1,048,575 0.121 37,374 18,687 5 10 3.114 2,097,150 1,048,575 0.02 7,326 3,663
SLIDE 15
CS 276 15
AND/OR search tree for graphical models
The AND/OR search tree of R relative to a spanning-tree, T, has:
Alternating levels of: OR nodes (variables) and AND nodes (values)
Successor function:
The successors of OR nodes X are all its consistent values along its path
The successors of AND <X,v> are all X child variables in T
A solution is a consistent subtree
Task: compute the value of the root node
A B E
OR AND OR AND OR AND
C
OR AND
D 1 F 1 1 D 1 F 1 1 1 E C D 1 F 1 1 D 1 F 1 1 1 B E C D 1 F 1 1 D 1 F 1 1 1 E C D 1 F 1 1 D 1 F 1 1
A D B C E F A D B C E F
SLIDE 16
CS 276 16
From DFS trees to pseudo-trees (Freuder 85, Bayardo 95)
(a) Graph
4 6 1 3 2 7 5
(b) DFS tree depth=3 (c) pseudo- tree depth=2 (d) Chain depth=6
4 6 1 3 2 7 5 2 7 1 4 3 5 6 4 6 1 3 2 7 5
SLIDE 17
CS 276 17
From DFS trees to Pseudo-trees
a 1 2 a 3 a 4
OR AND OR AND OR AND AND OR
b a b c c b 4 b a c c 4 b a c b 3 a 4 b a c b 4 b a c c 4 b a c c 3 a 4 b a c b 4 b a c c 4 b a c 7 a 5 a 6 b a c b 6 b a c c 6 b a c b 5 a 6 b a c b 6 b a c c 6 b a c c 5 a 6 b a c b 6 b a c c 6 b a c a 1 3 a 4 b c
OR AND OR AND OR AND
b c a 2 b c a b 4 b c a 2 b c a c 4 b c a 2 b c a 5 a 7 b c a 6 b c a b 7 b c a 6 b c a c 7 b c a 6 b c a
DFS tree depth = 3 pseudo- tree depth = 2
SLIDE 18
CS 276 18
Finding min-depth backbone trees
Finding min depth DFS, or pseudo tree is NP-
complete, but:
Given a tree-decomposition whose tree-width is
w*, there exists a pseudo tree T of G whose depth, satisfies (Bayardo and Mirankar, 1996, bodlaender and
Gilbert, 91):
m <= w* log n,
SLIDE 19
CS 276 19
Generating pseudo-trees from Bucket trees
F A C B D E E A C B D F (AF) (EF) (A) (AB) (AC) (BC) (AE) (BD) (DE)
Bucket-tree based on d d: A B C E D F
E A C B D F
Induced graph
E A C B D F
Bucket-tree used as pseudo-tree
AND AND AND AND
1
B
OR
B
1 1
C
OR
E C E C E C E
OR
D F D F D F D F D F D F D F D F
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
A
OR
AND/OR search tree Bucket-tree
ABE A ABC AB BDE AEF
bucket-A bucket-E bucket-B bucket-C bucket-D bucket-F
(AE) (BE)
SLIDE 20
CS 276 20
AND/OR Search-tree properties
(k = domain size, m = pseudo-tree depth. n = number of variables)
Theorem: Any AND/OR search tree based on a pseudo-
tree is sound and complete (expresses all and only solutions)
Theorem: Size of AND/OR search tree is O(n km)
Size of OR search tree is O(kn)
Theorem: Size of AND/OR search tree can be bounded
by O(exp(w* log n))
Related to:
(Freuder 85; Dechter 90, Bayardo et. al. 96, Darwiche 1999, Bacchus 2003)
When the pseudo-tree is a chain we get an OR space
SLIDE 21
CS 276 21
Tasks and value of nodes
V( n) is the value of the tree T(n) for the task:
Counting: v(n) is number of solutions in T(n)
Consistency: v(n) is 0 if T(n) inconsistent, 1 othewise.
Optimization: v(n) is the optimal solution in T(n)
Belief updating: v(n), probability of evidence in T(n).
Partition function: v(n) is the total probability in T(n).
Goal: compute the value of the root node recursively using
dfs search of the AND/OR tree.
Theorem: Complexity of AO dfs search is
Space:
O(n)
Time:
O(n km)
Time:
O(exp(w* log n))
SLIDE 22
SLIDE 23
CS 276 23
A C B D E A D B C E
A B E D 1 C 1 C 1 1 1 E D 1 C 1 C 1 1 1 B E D 1 C 1 C 1 1 1 E D 1 C 1 C 1 1
Belief-updating on example
SLIDE 24
CS 276 24
A B E D 1 C 1 C 1 1 1 E D 1 C 1 C 1 1 1 B E D 1 C 1 C 1 1 1 E D 1 C 1 C 1 1
PA(0) PA(1))
PB|A(0|0)
PB|A(1|0) PB|A(0|1)
PB|A(1|1)
PE|AB(0|0,0) PE|AB(1|0,0) PE|AB(0|0,1) PE|AB(1|0,1) PE|AB(0|1,0) PE|AB(1|1,0) PE|AB(0|1,1) PE|AB(1|1,1) PD|BC(0|0,0)× PC|A(0|0) A C B D E A D B C E
SLIDE 25
CS 276 25
A B E D 1 C 1 1 C 1 1 E D 1 C 1 1 C 1 P(A=0) P(B=0|A=0) P(B=1|A=0) P(E=0|A=0,B=0) P(D=0|B=0,C=0)× P(C=0|A=0) P(D=1|B=1,C=1)× P(C=1|A=0) P(D=0|B=0,C=1)× P(C=1|A=0) P(D=1|B=0,C=0)× P(C=0|A=0) P(D=1|B=0,C=1)× P(C=1|A=0) P(D=0|B=1,C=0)× P(C=0|A=0) P(D=0|B=1,C=1)× P(C=1|A=0) P(D=1|B=1,C=0)× P(C=0|A=0) P(E=1|A=0,B=0) P(E=0|A=0,B=1) P(E=1|A=0,B=1)
A C B D E A D B C E
SLIDE 26
CS 276 26
A B E C 1 D 1 1 D 1 1 E C 1 D 1 1 D 1
P(A=0) P(B=0|A=0) P(B=1|A=0) P(E=0|A=0,B=0) P(D=0|B=0,C=0) P(D=1|B=1,C=1) P(D=0|B=0,C=1) P(D=1|B=0,C=0) P(D=1|B=0,C=1) P(D=0|B=1,C=0) P(D=0|B=1,C=1) P(D=1|B=1,C=0) P(E=1|A=0,B=0) P(E=0|A=0,B=1) P(E=1|A=0,B=1) P(C=0|A=0) P(C=1|A=0) P(C=0|A=0) P(C=1|A=0)
(d)
SLIDE 27
CS 276 27
AND/OR Tree DFS Algorithm (Belief Updating)
AND node: Combination operator (product) OR node: Marginalization operator (summation) Value of node = updated belief for sub-problem below
A B E
OR AND OR AND OR AND
C
OR AND
D 1 1 D 1 1 1 E C D 1 1 D 1 1 1 B E C D 1 1 D 1 1 1 E C D 1 1 D 1 1
A D B C E A D B C E B C D=0 D=1 0 0 .2 .8 0 1 .1 .9 1 0 .3 .7 1 1 .5 .5
) , | ( C B D P
.7 .8 .9 .5 .7 .8 .9 .5
Evidence: D=1
A B E=0 E=1 0 0 .4 .6 0 1 .5 .5 1 0 .7 .3 1 1 .2 .8
) , | ( B A E P
Evidence: E=0
.4 .5 .7 .2 A B=0 B=1 .4 .6 1 .1 .9
) | ( A B P
A C=0 C=1 .2 .8 1 .7 .3
) | ( A C P
A P(A) .6 1 .4
) (A P
.2 .8 .2 .8 .1 .9 .1 .9 .4 .6 .1 .9 .6 .4 .8 .9 .8 .9 .7 .5 .7 .5 .8 .9 .8 .9 .7 .5 .7 .5 .4 .5 .7 .2 .88 .54 .89 .52 .352 .27 .623 .104 .3028 .1559 .24408 .3028 .1559
Result: P(D=1,E=0)
SLIDE 28
CS 276 28
Complexity of AND/OR Tree Search
AND/OR tree OR tree Space O(n) O(n) Time
O(n km) O(n kw* log n)
[Freuder & Quinn85], [Collin, Dechter & Katz91], [Bayardo & Miranker95], [Darwiche01]
O(kn)
k = domain size m = depth of pseudo-tree n = number of variables w*= treewidth
SLIDE 29
CS 276 29
From Search Trees to Search Graphs
Any two nodes that root identical subtrees
(subgraphs) can be merged
SLIDE 30
CS 276 30
From Search Trees to Search Graphs
Any two nodes that root identical subtrees
(subgraphs) can be merged
SLIDE 31
CS 276 31
AND/OR Tree
A D B C E F A D B C E F G H J K G H J K
A
OR AND
1 B
OR
B
AND
1 1 E
OR
C E C E C E C
OR
D F D F D F D F D F D F D F D F
AND AND
1 1 1 1 1 1 1 1
OR OR AND AND
G H H
0 1 0 1 1
1 G H H
0 1 0 1 1
J K K
0 1 0 1 1
1 J K K
0 1 0 1 1
G H H
0 1 0 1 1
1 G H H
0 1 0 1 1
J K K
0 1 0 1 1
1 J K K
0 1 0 1 1
G H H
0 1 0 1 1
1 G H H
0 1 0 1 1
J K K
0 1 0 1 1
1 J K K
0 1 0 1 1
G H H
0 1 0 1 1
1 G H H
0 1 0 1 1
J K K
0 1 0 1 1
1 J K K
0 1 0 1 1
G H H
0 1 0 1 1
1 G H H
0 1 0 1 1
J K K
0 1 0 1 1
1 J K K
0 1 0 1 1
G H H
0 1 0 1 1
1 G H H
0 1 0 1 1
J K K
0 1 0 1 1
1 J K K
0 1 0 1 1
G H H
0 1 0 1 1
1 G H H
0 1 0 1 1
J K K
0 1 0 1 1
1 J K K
0 1 0 1 1
G H H
0 1 0 1 1
1 G H H
0 1 0 1 1
J K K
0 1 0 1 1
1 J K K
0 1 0 1 1
SLIDE 32
CS 276 32
An AND/OR graph
A D B C E F A D B C E F G H J K G H J K
A
OR AND
1 B
OR
B
AND
1 1 E
OR
C E C E C E C
OR
D F D F D F D F D F D F D F D F
AND AND
1 1 1 1 1 1 1 1
OR OR AND AND
G H H
0 1 0 1 1
1 G H H
0 1 0 1 1
J K K
0 1 0 1 1
1 J K K
0 1 0 1 1
SLIDE 33
CS 276 33
Merging Based on Context
One way of recognizing nodes that can be merged:
context (X) = ancestors of X in pseudo tree that are connected to X, or to descendants of X
[ ] [A] [AB] [AE] [BC] [AB]
A D B E C F
pseudo tree
A E C B F D A E C B F D
SLIDE 34
CS 276 34
AND/OR Search Graph
A E C B F D
A D B E C F
A B C RABC 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 A B E RABE 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 A E F RAEF 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 B C D RBCD 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Constraint Satisfaction – Counting Solutions
[ ] [A] [AB] [AE] [BC] [AB] Context
pseudo tree context minimal graph
A
OR AND
B
OR AND OR
E
OR AND AND
1 C D D 1 1 1 1 E F F 1 1 1 C D D 1 1 1 1 B E F F 1 1 1 C 1 1 E 1 C 1
SLIDE 35
CS 276 35
.8 .9 .7 .7 .8
AND/OR Tree DFS Algorithm (Belief Updating)
AND node: Combination operator (product) OR node: Marginalization operator (summation) Value of node = updated belief for sub-problem below
A B E C D 1 1 D 1 1 1 E C D 1 1 D 1 1 1 B E C D 1 1 D 1 1 1 E C D 1 1 D 1 1
B C D=0 D=1 0 0 .2 .8 0 1 .1 .9 1 0 .3 .7 1 1 .5 .5
) , | ( C B D P
.7 .8 .9 .5 .7 .8 .9 .5
Evidence: D=1
A B E=0 E=1 0 0 .4 .6 0 1 .5 .5 1 0 .7 .3 1 1 .2 .8
) , | ( B A E P
Evidence: E=0
.4 .5 .7 .2 A B=0 B=1 .4 .6 1 .1 .9
) | ( A B P
A C=0 C=1 .2 .8 1 .7 .3
) | ( A C P
A P(A) .6 1 .4
) (A P
.2 .8 .2 .8 .1 .9 .1 .9 .4 .6 .1 .9 .6 .4 .9 .8 .9 .5 .7 .5 .8 .9 .5 .7 .5 .4 .5 .7 .2 .88 .54 .89 .52 .352 .27 .623 .104 .3028 .1559 .24408 .3028 .1559
Result: P(D=1,E=0)
A D B C E
[ ] [A] [AB] [BC] [AB] Context
A D B C E
SLIDE 36
CS 276 36
.7 .8
AND/OR Graph DFS Algorithm (Belief Updating)
A B E C D 1 1 D 1 1 1 E C D 1 1 D 1 1 1 B E C 1 1 1 E C 1 1
A D B C E B C D=0 D=1 0 0 .2 .8 0 1 .1 .9 1 0 .3 .7 1 1 .5 .5
) , | ( C B D P
.7 .8 .9 .5
Evidence: D=1
A B E=0 E=1 0 0 .4 .6 0 1 .5 .5 1 0 .7 .3 1 1 .2 .8
) , | ( B A E P
Evidence: E=0
.4 .5 .7 .2 A B=0 B=1 .4 .6 1 .1 .9
) | ( A B P
A C=0 C=1 .2 .8 1 .7 .3
) | ( A C P
A P(A) .6 1 .4
) (A P
.2 .8 .2 .8 .1 .9 .1 .9 .4 .6 .1 .9 .6 .4 .9 .8 .9 .5 .7 .5 .8 .9 .7 .5 .4 .5 .7 .2 .88 .54 .89 .52 .352 .27 .623 .104 .3028 .1559 .24408 .3028 .1559 A D B C E
[ ] [A] [AB] [BC] [AB] Context
B C Value 0 0 .8 0 1 .9 1 0 .7 1 1 .1
Cache table for D
Result: P(D=1,E=0)
SLIDE 37
CS 276 37
AND/OR context minimal graph
C K H L 1 N N 1 1 F F F 1 1 1 1 F G 0 1 1 A 1 B B 1 1 E E E E 1 1 J J J J 1 1 A 1 B B 1 1 E E E E 1 1 J J J J 1 1 G 0 1 G 0 1 G 0 1 M 0 1 M 0 1 M 0 1 M 0 1 P 0 1 P 0 1 O 0 1 O 0 1 O 0 1 O 0 1 L 1 N N 1 1 P 0 1 P 0 1 O 0 1 O 0 1 O 0 1 O 0 1 D 0 1 D 0 1 D 0 1 D 0 1 K H L 1 N N 1 1 1 1 A 1 B B 1 1 E E E E 1 1 J J J J 1 1 A 1 B B 1 1 E E E E 1 1 J J J J 1 1 P 0 1 P 0 1 O 0 1 O 0 1 O 0 1 O 0 1 L 1 N N 1 1 P 0 1 P 0 1 O 0 1 O 0 1 O 0 1 O 0 1 D 0 1 D 0 1 D 0 1 D 0 1
B A C E F G H J D K M L N O P C H K D M F G A B E J O L N P [AB] [AF] [CHAE] [CEJ] [CD] [CHAB] [CHA] [CH] [C] [ ] [CKO] [CKLN] [CKL] [CK] [C]
(C K H A B E J L N O D P M F G)
C H K D M F G A B E J O L N P [AB] [AF] [CHAE] [CEJ] [CD] [CHAB] [CHA] [CH] [C] [ ] [CKO] [CKLN] [CKL] [CK] [C]
(C K H A B E J L N O D P M F G)
SLIDE 38
CS 276 38
How Big Is the Context?
Theorem: The maximum context size for a pseudo tree is equal to the treewidth of the graph along the pseudo tree.
C H K D M F G A B E J O L N P [AB] [AF] [CHAE] [CEJ] [CD] [CHAB] [CHA] [CH] [C] [ ] [CKO] [CKLN] [CKL] [CK] [C]
(C K H A B E J L N O D P M F G)
B A C E F G H J D K M L N O P
max context size = treewidth
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G E K F L H C B A M J D
Treewidth vs. Pathwidth
E K L H C A M J ABC BDEF BDFG EFH FHK HJ KLM
treewidth = 3 = (max cluster size) - 1
ABC BDEFG EFH FHKJ KLM
pathwidth = 4 = (max cluster size) - 1
D G B F
TREE CHAIN
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Complexity of AND/OR Graph Search
AND/OR graph OR graph Space
O(n kw*) O(n kpw*)
Time
O(n kw*) O(n kpw*)
k = domain size n = number of variables w*= treewidth pw*= pathwidth
w* ≤ pw* ≤ w* log n
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AND/OR search algorithms
AO(i)
i = the max size of a cache table (i.e.
number of variables in a context)
i = 0 i = w* treewidth Space: Time: O(n exp w* ) O(n exp w* ) O( n ) O( exp (w* log n) )
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Searching AND/OR Graphs
AO(i): searches depth-first, cache i-context
i = the max size of a cache table (i.e.
number of variables in a context)
i=0 i=w* Space: O(n) Time: O(exp(w* log n)) Space: O(exp w*) Time: O(exp w*)
Space: O(exp(i) ) Time: O(exp(m_i+i )
i
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Caching
A D B C E F A D B C E F G H J K G H J K
A
OR AND
1 B
OR
B
AND
1 1 E
OR
C E C E C E C
OR
D F D F D F D F D F D F D F D F
AND AND
1 1 1 1 1 1 1 1
OR OR AND AND
G H H
0 1 0 1 1
1 G H H
0 1 0 1 1
J K K
0 1 0 1 1
1 J K K
0 1 0 1 1
G H H
0 1 0 1 1
1 G H H
0 1 0 1 1
J K K
0 1 0 1 1
1 J K K
0 1 0 1 1
G H H
0 1 0 1 1
1 G H H
0 1 0 1 1
J K K
0 1 0 1 1
1 J K K
0 1 0 1 1
G H H
0 1 0 1 1
1 G H H
0 1 0 1 1
J K K
0 1 0 1 1
1 J K K
0 1 0 1 1
G H H
0 1 0 1 1
1 G H H
0 1 0 1 1
J K K
0 1 0 1 1
1 J K K
0 1 0 1 1
G H H
0 1 0 1 1
1 G H H
0 1 0 1 1
J K K
0 1 0 1 1
1 J K K
0 1 0 1 1
G H H
0 1 0 1 1
1 G H H
0 1 0 1 1
J K K
0 1 0 1 1
1 J K K
0 1 0 1 1
G H H
0 1 0 1 1
1 G H H
0 1 0 1 1
J K K
0 1 0 1 1
1 J K K
0 1 0 1 1
context(D)={D} context(F)={F}
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All four search spaces
E C F D B A
1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
AND
1
B
OR
B
AND
1 1
E
OR
C E C E C E C
OR
D F D F D F D F D F D F D F D F
AND
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
AND
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
E C F D B A
1 1 1 1 1
A
OR AND
1 B
OR
B
AND
1 1 E
OR
C E C E C E C
OR
D F D F D F D F D F D F D F D F
AND
1 1
AND
1 1 1 1 1
Full OR search tree Full AND/OR search tree Context minimal OR search graph Context minimal AND/OR search graph
A D B C E F A D B C E F
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All four search spaces
E C F D B A
1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
AND
1
B
OR
B
AND
1 1
E
OR
C E C E C E C
OR
D F D F D F D F D F D F D F D F
AND
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
AND
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
E C F D B A
1 1 1 1 1
A
OR AND
1 B
OR
B
AND
1 1 E
OR
C E C E C E C
OR
D F D F D F D F D F D F D F D F
AND
1 1
AND
1 1 1 1 1
Full OR search tree Full AND/OR search tree Context minimal OR search graph Context minimal AND/OR search graph
A D B C E F A D B C E F
Time- space
SLIDE 46
CS 276 46
Available code
http://graphmod.ics.uci.edu/group/Software