Reasoning with Deterministic and Probabilistic graphical models

Class 3: Constraints; Search

Rina Dechter

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Road Map

n  Graphical models n  Constraint networks Model n  Inference

n  Variable elimination: n  Tree-clustering n  Constraint propagation

n  Search n  Probabilistic Networks

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Road Map: Search in Constraint Networks

n  Improving search by bounded-inference in

branching ahead n  Improving search by looking-back n  The alternative AND/OR search space

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Road Map: Search in Constraint Networks

n  Improving search by bounded-inference in

branching ahead n  Improving search by looking-back n  The alternative AND/OR search space

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Backtracking Search for a Solution

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Backtracking Search for a Solution

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Backtracking Search for All Solutions

The Search Space Before Arc-Consistency

X!

..... ..... .....!

Y!

T!

X Y

T Z

<

=

<

∧

X=1 2 3

Y=2 3 3

T=2 3

Z

D={1,2,3}

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The Search Space After Arc-Consistency

X!

..... ..... .....!

Y!

T!

X=1

Y=2

T=2

Z

X Y <

=

<

∧ 1 3

2 3 T Z

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The Effect of Variable Ordering

z divides x, y and t

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The Effect of Consistency Level

n  After arc-consistency z=5 and l=5 are removed

n  After path-consistency n  R’_zx n  R’_zy n  R’_zl n  R’_xy n  R’_xl n  R’_yl

Tighter networks yield smaller search spaces

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Improving Backtracking O(exp(n))

n  Before search: (reducing the search space) n  Arc-consistency, path-consistency, i-consistency n  Variable ordering (fixed)

n  During search: n  Look-ahead schemes:

n  value ordering/pruning (choose a least restricting value), n  variable ordering (Choose the most constraining

variable) n  Look-back schemes:

n  Backjumping n  Constraint recording n  Dependency-directed backtracking

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Looking-Ahead: Constraint Propagation in Search

n  Apply some level of constraint propagation at each node, n  Forward-checking (FC) n  Arc-consistency (MAC)

n  Then: n  Value pruning or ordering : prune values that lead

to deadend n  Variable ordering: choose a variable that leaves

least options open

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Forward-Checking for Value Ordering

)( 2ekO

)()(3

2

ekOekOFC overhead:

MAC overhead:

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Forward-Checking for Value Ordering

)( 2ekO

)()(3

2

ekOekOFW overhead:

MAC overhead:

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Forward-Checking, Variable Ordering

)( 2ekO

)()(3

2

ekOekOFW overhead:

MAC overhead:

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Forward-Checking, Variable Ordering

)( 2ekO

)()(3

2

ekOekOFW overhead:

MAC overhead:

After X1 = red choose X3 and not X2

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Forward-Checking, Variable Ordering

)( 2ekO

)()(3

2

ekOekOFW overhead:

MAC overhead:

After X1 = red choose X3 and not X2

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Forward-Checking, Variable Ordering

)( 2ekO

)()(3

2

ekOekOFW overhead:

MAC overhead:

After X1 = red choose X3 and not X2

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Arc-consistency for Value Ordering

)( 2ekO

)()(3

2

ekOekOFW overhead:

MAC overhead:

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Arc-Consistency for Value Ordering

)( 2ekO

)()(3

2

ekOekOFW overhead:

MAC overhead:

Arc-consistency prunes x1=red Prunes the whole tree Not searched

By MAC

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Constraint Programming

n  Constraint solving embedded in programming languages

n  Allows flexible modeling with algorithms n  Logic programs + forward checking n  Eclipse, ILog, OPL n  Using only look-ahead schemes

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Branching-Ahead for SAT: DLL example: (~AVB)(~CVA)(AVBVD)(C)

Only enclosed area will be explored with unit-propagation

Backtracking look-ahead with Unit propagation= Generalized arc-consistency

(Davis, Logeman and Laveland, 1962)

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Road Map: Search in Constraint Networks

n  Improving search by bounded-inference in

branching ahead n  Improving search by looking-back n  The alternative AND/OR search space

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Look-back: Backjumping / Learning

n  Backjumping: n  In deadends, go back to

the most recent culprit.

n  Learning: n  constraint-recording, no-

good recording. n  good-recording

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Look-Back: Backjumping

n  (X1=r,x2=b,x3=b,x4=b,x5=g,x6=r,x7={r,b}) n  (r,b,b,b,g,r) conflict set of x7 n  (r,-,b,b,g,-) c.s. of x7 n  (r,-,b,-,-,-,-) minimal conflict-set n  Leaf deadend: (r,b,b,b,g,r) n  Every conflict-set is a no-good

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Jumps at dead-ends (Gascnnig-style 1977)

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Jumps at Dead-Ends (Gascnnig 1977)

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Complexity of Backjumping

Simple: always jump back to parent in pseudo tree Complexity for csp: exp(w*log n) From exp(n) to exp(w*logn) while linear space

Graph-based and conflict-based backjumpint

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Look-back: No-good Learning

n  (x1=2,x2=2,x3=1,x4=2) is a dead-end

n  Conflicts to record: n  (x1=2,x2=2,x3=1,x4=2) 4-ary n  (x3=1,x4=2) binary n  (x4=2) unary

Learning means recording conflict sets used as constraints to prune future search space.

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No-good Learning Example

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Complexity of Nogood-Learning for consistency

n  The complexity of learning along d is time and space exponential in w*(d):

n  The number of dead-ends is bounded by n  Number of constraint tests per dead-end are

Space complexity is Time complexity is No-good Learning reduces time to O(exp(w*)) but requires O(exp(w*)) space.

)()(

)*(2

)*(

dw

dw

kenOnkO

⋅

)( )*(dwnkO)(eO

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Moving to new queries

n  Consistency and one solution. n  Counting n  enumerating

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Bucket-elimination for counting

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:

:

:

:

:

E

RC

RRA

RRD

RRB

CE

AEAC

ADDE

BDAB

A

C B

D E

≠

}3,2,1{

}2,1{

}2,1{}2,1{

}2,1{

≠ ≠

≠ ≠

≠

Bucket-elimination for counting

∑ ∗= B DBRBARDAR ),(),(),("↓$,%  "↓',$ 

’"′↓('  R_E

#CSP - Tree DFS Traversal

0 1

0

0 1

0 1

0

C

D

F

E

B

A 0

A

E

C

B

F

D

A B C RABC 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0

A B E RABE 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0

A E F RAEF 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0

B C D RBCD 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 1

0 1

1 0

1

0 1

0 1

0 1

1 0

1 1

2

0

0 1

0 1

0 1

1 0

1 0 1

1

3

Value of node = number of solutions below it

5

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 0 1

1

0 1 0 1 1 1 1 1 1 0 1 0 1 0 1 1

1 2 1 2

0

0 1

0 1

0 2

0

0

0

2

2

2

1

0 3 3

6

6

9

1 1 1

1 2

3

14

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#CSP - OR Search Tree

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 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

C

D

F

E

B

A 0 1

A

E

C

B

F

D

A B C RABC 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0

B C D RBCD 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 1

A B E RABE 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0

A E F RAEF 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0

14 solutions

1 2 3 4 6 7 9 10 11 12 13 14 5 8

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Summary: Search Principles

n  Constraint propagation prunes search space n  Constraint propagation yields good advise for

how to branch and where to go n  Backjumping and no-good learning helps

prune search space and revise problem. n  Good learning cache results but helps only

counting, enumeration… as we see shortly.

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Thank You

n  “Constraint Processing”, Morgan Kaufmann, 2003

n  Probabilistic networks: Transferring these ideas to Probabilistic network, helping unifying the principles.

n 

Ijcai 2011 39

Road Map: Search in Graphical Models

n  Improving search by bounded-inference in

branching ahead n  Improving search by looking-back n  The alternative AND/OR search space

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OR Search Space A

D

B C

E

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 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

E

C

F

D

B

A 0 1

Ordering: A B E C D F

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AND/OR Search Space

A OR

0 AND 1

B OR B

0 AND 1 0 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 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

AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 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

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AND/OR vs. OR

E 0 1 0 1 0 1 0 1

0 C 1 0 1 0 1 0 1 0 1 0 1 0 1 0 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 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 B 1 0 1

A 0 1

A OR

0 AND 1

B OR B

0 AND 1 0 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 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

AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

E 0 1 0 1 0 1 0 1

0 C 1 0 1 0 1 0 1 0 1 0 1 0 1 0 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 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 B 1 0 1

A 0 1

E 0 1 0 1 0 1 0 1

0 C 1 0 1 0 1 0 1 0 1 0 1 0 1 0 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 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 B 1 0 1

A 0 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)

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AND/OR vs. OR

E 0 1 0 1 0 1 0 1

0 C 1 0 1 0 1 0 1 0 1 0 1 0 1 0 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 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 B 1 0 1

A 0 1

A OR

0 AND 1

B OR B

0 AND 1 0 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 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

AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

E 0 1 0 1 0 1 0 1

0 C 1 0 1 0 1 0 1 0 1 0 1 0 1 0 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 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 B 1 0 1

A 0 1

E 0 1 0 1 0 1 0 1

0 C 1 0 1 0 1 0 1 0 1 0 1 0 1 0 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 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 B 1 0 1

A 0 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)

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AND/OR vs. OR

F

AND/OR

A

D

B C

E

F A

D

B

C E

F

E 0 1 0 1 0 1

C 1 1 0 1 0 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

D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

0 B 1 0

A 0 1

A OR

0 AND 1

B OR B

0 AND 1 0

E OR C E C E C

OR D F D F D F D F AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

AND 1 0 1 0 1 0 1 1 0 1

E 0 1 0 1 0 1

C 1 1 0 1 0 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

D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

0 B 1 0

A 0 1

E 0 1 0 1 0 1

C 1 1 0 1 0 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

D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

0 B 1 0

A 0 1

OR

(A=1,B=1) (B=0,C=0)

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AND/OR vs. OR

F

AND/OR

A

D

B C

E

F A

D

B

C E

F

E 0 1 0 1 0 1

C 1 1 0 1 0 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

D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

0 B 1 0

A 0 1

A OR

0 AND 1

B OR B

0 AND 1 0

E OR C E C E C

OR D F D F D F D F AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

AND 1 0 1 0 1 0 1 1 0 1

E 0 1 0 1 0 1

C 1 1 0 1 0 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

D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

0 B 1 0

A 0 1

E 0 1 0 1 0 1

C 1 1 0 1 0 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

D 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

0 B 1 0

A 0 1

OR

(A=1,B=1) (B=0,C=0)

Space: linear Time: O(exp(m)) O(w* log n)

Linear space, Time: O(exp(n))

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AND/OR vs. OR Spaces

width depth OR space AND/OR space

Time (sec.) Nodes Time (sec.) AND nodes OR nodes

5 10 3.15 2,097,150 0.03 10,494 5,247

4 9 3.13 2,097,150 0.01 5,102 2,551

5 10 3.12 2,097,150 0.03 8,926 4,463

4 10 3.12 2,097,150 0.02 7,806 3,903

5 13 3.11 2,097,150 0.10 36,510 18,255

Random graphs with 20 nodes, 20 edges and 2 values per node

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#CSP – AND/OR Search Tree

A

E

C

B

F

D

A

D

B

E C

F

A B C RABC 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0

A B E RABE 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0

A E F RAEF 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0

B C D RBCD 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 1

A OR

0 AND

B OR

0 AND

OR E

OR F F

AND 0 1 0 1

AND 0 1

C

D D

0 1 0 1

0 1

1

E

F F

0 1 0 1

0 1

C

D D

0 1 0 1

0 1

1

B

0

E

F F

0 1 0 1

0 1

C

D D

0 1 0 1

0 1

1

E

F F

0 1 0 1

0 1

C

D D

0 1 0 1

0 1

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#CSP – AND/OR Tree DFS

A

E

C

B

F

D

A

D

B

E C

F

A B C RABC 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0

A B E RABE 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0

A E F RAEF 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0

B C D RBCD 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 1

A OR

0 AND

B OR

0 AND

OR E

OR F

AND 0 1

AND 0 1

C

D D

0 1 0 1

0 1

1 1 1 0 0 1

2 1 1

2 1 1 0

3 1

3

9

9

1

E

F F

0 1 0 1

0 1

C

D

0 1

0 1

1

B

0

E

F

0 1

0 1

C

D D

0 1 0 1

0 1

1

E

F

0 1

0 1

C

D

0 1

0 1

1 1 0 1 1 1

2 1 2

2 1 2 0

2 3

6

1 1 1 0 1 0 1 0 1 1

2 1 1 1 2

3 1

2 1 1 0 0 1 0 2

1 2

3 2

5

5

14

OR node: Marginalization operator (summation)

AND node: Combination operator (product)

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Pseudo-Trees (Freuder 85, Bayardo 95, Bodlaender and Gilbert, 91)

(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

m

a

1

2

b c

3

a c

4

OR

AND

OR

AND

OR

AND

AND

OR

c b

4

b

3

a b

4

c b

4

c

7

b c

5

a c

6

c b

6

b

5

a b

6

c b

6

c

b

2

a c

3

b c

4

c a

4

a

3

a b

4

c

4

c a

7

a c

5

b c

6

c a

6

a

5

a b

6

c

6

c a

c

2

a b

3

b c

4

a

4

b a

3

a c

4

b

4

b a

7

a b

5

b c

6

a

6

b a

5

a c

6

b

6

b a

a

1

3

a

4

b c

OR

AND

OR

AND

OR

AND

2

b c

b

4

c

2

c

c

4

b

2

b

5

a

7

b c

6

b c

b

7

c

6

c

c

7

b

6

b

b

3

b

4

a c

2

a c

a

4

c

2

c

c

4

a

2

a

c

3

c

4

a b

2

a b

a

4

b

2

b

b

4

a

2

a

5

b

7

a c

6

a c

a

7

c

6

c

c

7

a

6

a

5

c

7

a b

6

a b

a

7

b

6

b

b

7

a

6

a

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AND/OR search tree for graphical models

n  The AND/OR search tree of R relative to a tree, T, has: n  Alternating levels of: OR nodes (variables) and AND nodes (values)

n  Successor function: n  The successors of OR nodes X are all its consistent values along its path n  The successors of AND are all X child variables in T

n  A solution is a consistent subtree n  Task: compute the value of the root node

0

A

B

0

E

OR

AND

OR

AND

OR

AND

C

0

OR

AND

D

0 1

F

0 1

1

D

0 1

F

0 1

0 1

1

E C

0

D

0 1

F

0 1

1

D

0 1

F

0 1

0 1

1

B

0

E C

0

D

0 1

F

0 1

1

D

0 1

F

0 1

0 1

1

E C

0

D

0 1

F

0 1

1

D

0 1

F

0 1

0 1

A

D

B C

E

F

A

D

B

C E

F

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From Search A/O Trees to Search A/O Graphs

n  Any two nodes that root identical subtrees/subgraphs can be merged

n  Minimal AND/OR search graph: closure under merge of the AND/OR search tree

n  Inconsistent sub-trees can be pruned too. n  Some portions can be collapsed or reduced.

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From search trees to search graphs

n  Any two nodes that root identical subtrees (subgraphs) can be merged

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From search trees to search graphs

n  Any two nodes that root identical subtrees (subgraphs) can be merged

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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

0 AND 1

B OR B

0 AND 1 0 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 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

OR

OR

AND

AND

0

G

H H

0 1 0 1

0 1

1

G

H H

0 1 0 1

0 1

0

J

K K

0 1 0 1

0 1

1

J

K K

0 1 0 1

0 1

0

G

H H

0 1 0 1

0 1

1

G

H H

0 1 0 1

0 1

0

J

K K

0 1 0 1

0 1

1

J

K K

0 1 0 1

0 1

0

G

H H

0 1 0 1

0 1

1

G

H H

0 1 0 1

0 1

0

J

K K

0 1 0 1

0 1

1

J

K K

0 1 0 1

0 1

0

G

H H

0 1 0 1

0 1

1

G

H H

0 1 0 1

0 1

0

J

K K

0 1 0 1

0 1

1

J

K K

0 1 0 1

0 1

0

G

H H

0 1 0 1

0 1

1

G

H H

0 1 0 1

0 1

0

J

K K

0 1 0 1

0 1

1

J

K K

0 1 0 1

0 1

0

G

H H

0 1 0 1

0 1

1

G

H H

0 1 0 1

0 1

0

J

K K

0 1 0 1

0 1

1

J

K K

0 1 0 1

0 1

0

G

H H

0 1 0 1

0 1

1

G

H H

0 1 0 1

0 1

0

J

K K

0 1 0 1

0 1

1

J

K K

0 1 0 1

0 1

0

G

H H

0 1 0 1

0 1

1

G

H H

0 1 0 1

0 1

0

J

K K

0 1 0 1

0 1

1

J

K K

0 1 0 1

0 1

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An AND/OR Graph: Caching Goods

A

D

B C

E

F

A

D

B

C E

F

G H

J

K

G

H

J

K A OR

0 AND 1

B OR B

0 AND 1 0 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 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

OR

OR

AND

AND

0

G

H H

0 1 0 1

0 1

1

G

H H

0 1 0 1

0 1

0

J

K K

0 1 0 1

0 1

1

J

K K

0 1 0 1

0 1

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Context-based Caching

n  Caching is possible when context is the same

n  context = parent-separator set in induced pseudo-graph = current variable + parents connected to subtree below

A

D

B C

E

F

A

D

B

C E

F

G H

J

K

G

H

J

K

context(B) = {A, B}

context(c) = {A,B,C}

context(D) = {D}

context(F) = {F}

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Complexity of AND/OR Graph

n  Theorem: Traversing the AND/OR search graph is time and space exponential in the induced width/tree-width.

n  If applied to the OR graph complexity is time and space exponential in the path-width.

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#CSP – AND/OR Tree DFS

A

E

C

B

F

D

A

D

B

E C

F

A B C RABC 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0

A B E RABE 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0

A E F RAEF 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0

B C D RBCD 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 1

A OR

0 AND

B OR

0 AND

OR E

OR F

AND 0 1

AND 0 1

C

D D

0 1 0 1

0 1

1 1 1 0 0 1

2 1 1

2 1 1 0

3 1

3

9

9

1

E

F F

0 1 0 1

0 1

C

D

0 1

0 1

1

B

0

E

F

0 1

0 1

C

D D

0 1 0 1

0 1

1

E

F

0 1

0 1

C

D

0 1

0 1

1 1 0 1 1 1

2 1 2

2 1 2 0

2 3

6

1 1 1 0 1 0 1 0 1 1

2 1 1 1 2

3 1

2 1 1 0 0 1 0 2

1 2

3 2

5

5

14

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#CSP – AND/OR Search Graph (Caching Goods)

A

E

C

B

F

D

A

D

B

E C

F

A B C RABC 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0

A B E RABE 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0

A E F RAEF 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0

B C D RBCD 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 1

A OR

0 AND

B OR

0 AND

OR E

OR F F

AND 0 1

AND 0 1

C

D D

0 1

0 1

1

E C

D D

0 1

1

B

0

E

F F

0 1

C

1

E C

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#CSP – AND/OR Search Graph (Caching Goods)

A

E

C

B

F

D

A

D

B

E C

F

A B C RABC 0 0 0 1 0 0 1 1 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0

A B E RABE 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 1 1 0 0 0 1 0 1 1 1 1 0 1 1 1 1 0

A E F RAEF 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0

B C D RBCD 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 1

A OR

0 AND

B OR

0 AND

OR E

OR F F

AND 0 1

AND 0 1

C

D D

0 1

0 1

1

E C

D D

0 1

1

B

0

E

F F

0 1

C

1

E C

Time and Space O(exp(w*))

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All Four Search Spaces

Full OR search tree

126 nodes

Full AND/OR search tree

54 AND nodes

Context minimal OR search graph

28 nodes

Context minimal AND/OR search graph

18 AND nodes

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 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

C D

F E

B A 0 1

A OR 0 AND B OR

0 AND

OR E

OR F F AND 0 1 0 1

AND 0 1

C

D D

0 1 0 1

0 1

1

E

F F

0 1 0 1

0 1

C

D D

0 1 0 1

0 1

1

B

0

E

F F

0 1 0 1

0 1

C

D D

0 1 0 1

0 1

1

E

F F

0 1 0 1

0 1

C

D D

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

C D

F E

B A 0 1

A OR 0 AND B OR

0 AND OR E

OR F F AND 0 1

AND 0 1 C

D D 0 1

0 1

1

E C

D D

0 1

1

B

0

E

F F

0 1

C

1

E C

A

E

C

B

F

D

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AND/OR vs. OR DFS Algorithms

n  AND/OR tree n  Space: O(n) n  Time: O(n km)

O(n kw* log n) (Freuder85; Bayardo95; Darwiche01)

n  AND/OR graph n  Space: O(n kw*) n  Time: O(n kw*)

k = domain size m = tree depth n = # of variables w*= induced width pw*= path width l  OR tree

•  Space: O(n) •  Time: O(kn)

l  OR graph •  Space: O(n kpw*) •  Time: O(n kpw*)

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Road Map

n  Graphical models n  Constraint networks Model n  Inference

n  Variable elimination: n  Tree-clustering n  Constraint propagation

n  Search n  Probabilistic Networks

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