An Overview of Compiling Constraint Networks into AND/OR Multi-valued

Decision Diagrams (AOMDDs)

William LamCS 275 Project

The slides on AND/OR search and decision diagrams were provided by Rina Dechter.

Papers:Mateescu R., Dechter R. Compiling Constraint Networks into AND/OR Multi-valued Decision Diagrams (AOMDDs). (CP 2006)

Mateescu R., Dechter R., and Marinescu, R. AND/OR Multi-valued Decision Diagrams (AOMDDs) for Graphical Models. (JAIR 2008)

Outline Introduction AND/OR Search Spaces (Quick Review) Decision Diagrams – Binary and Multi-

valued (BDD/MDD) AND/OR MDDs (AOMDDs) Experiments Conclusion

2

Introduction Combining two frameworks

AND/OR Search Spaces Multi-valued Decision Diagrams (MDDs)

Both are more compact ways to represent problems.

Their combination yields an even more compact representation.

Decision Diagrams are known to allow online speed queries.

3

Introduction

4

)()()()()()(

ABCFCDFBDFBCFABEFACFF A

D

B C

E

F

Compiled structure

OFFLINE

ONLINE

Evaluate answer

Polytime? Constant time?

ONCE

MANY

Query

Classic OR Search Space

0 1 0 1 0 1 0 1

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

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

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

A

D

B C

E

F

AND/OR Search Space

AOR

0AND 1

BOR B

0AND 1 0 1

EOR 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 10 1 0 10 1 0 10 1 0 10 1

A

D

B C

E

F

A

D

B

CE

F

Primal graph DFS tree

A

D

B C

E

F

A

D

B C

E

F

AND/OR vs. ORAOR

0AND 1

BOR B

0AND 1 0 1

EOR 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 10 1 0 10 1 0 10 1 0 10 1

E 0 1 0 1 0 1 0 1

0C 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

0B 1 0 1

A 0 1

E 0 1 0 1 0 1 0 1

0C 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

0B 1 0 1

A 0 1

AND/OR

OR

A

D

B C

E

FA

D

B

CE

F

1

1

1

0

1

0

AND/OR size: exp(4), OR size exp(6)

From AND/OR TreeA

D

B C

E

F

A

D

B

CE

F

G H

J

K

G

H

J

KAOR

0AND 1

BOR B

0AND 1 0 1

EOR 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 10 1 0 10 1 0 10 1 0 10 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

To an AND/ORGraph

A

D

B C

E

F

A

D

B

CE

F

G H

J

K

G

H

J

KAOR

0AND 1

BOR B

0AND 1 0 1

EOR 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 10 1 0 10 1 0 10 1 0 10 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

AND/OR Search Graphs Any two nodes that root identical subtrees/subgraphs

(are unifiable) can be merged

Minimal AND/OR search graph: of R relative to tree T is the closure under merge of the AND/OR search tree of R relative to T, where inconsistent subtrees are pruned.

Canonicity: The minimal AND/OR graph relative to T is unique for all constraints equivalent to R.

All Four Search Spaces

Full OR search tree 126 nodes

Full AND/OR search tree54 AND nodes

Context minimal OR search graph28 nodes

Context minimal AND/OR search graph18 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

01 01 0 1 0 10 1 0 1 0 1 0 1 0 1 0 1 0 1 0 10 1 0 1 0 1 0 1 0 1 0 1 0 1 0 10 1 0 1 01 0 1 0 1 0 1 01 010 1 01 0 1 01

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

0 1 0 1

CD

FE

BA 0 1

AOR

0AND

BOR

0AND

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

AOR0ANDBOR

0ANDOR E

OR F FAND 0 1

AND 0 1

C

D D0 1

0 1

1

EC

D D

0 1

1

B

0

E

F F

0 1

C

1

EC

A

E

C

B

F

D

Ordered Binary Decision Diagram

12

A B C f(ABC)0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 01 1 1 1

B = {0,1} f : B3 → B

Table

B

A

C

0 0

C

0 1

B

C

0 1

C

0 1

SimplifyMerge terminal nodes

B

A

C C

0 1

B

C C

1) Merge nodes with identical children

B

A

C

0 1

B

C

2) Remove redundant nodes

B

A

0 1

B

C

B

A

0 1

C

OBDD[Bryant86]

Ordering enables efficient operations

B

A

C

0 0

C

0 1

B

C

0 1

C

0 1

Decision tree

A

B

C

Decision Diagrams

13

f = (A ∨ E) ∧ (B ∨ F)

g = (C ∨ G) ∧ (D ∨ H)

B

A

F

10

E

B

E f ∧ g = ?

f ∧ g =

OBDD

OBDD

D

C

H

10

G

D

G

ANDD

C

H

10

G

D

G

B

A

F

10

E

B

E

0 1

B

A

C

D D

C

B

D D

C

D D

C

D D

E E E E E E E E

F F

H

GG

F F

AOMDDs

14

Point dead-ends to terminal node “0”

Point goods to terminal node “1”

0

A

0 1

B

0 1

1

C

0 1

B

0 1

Decision Diagram

A

B

C

A B C f(ABC)0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 01 1 1 1

A

0 1

B

1

C

1

B

0 1

Minimal AND/OR graph

Removing Redundancy

15

0

A

0 1

B

0 1

1

C

0 1

B

0 1

redundant

Group OR node together with its AND children into a meta-node

0

A

0 1

B

0 1

1

C

0 1

B

0 1

0

A

0 1

B

0 1

1

C

0 1

OBDD(pseudo tree is a chain)

A

B

C

A B C f(ABC)0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 01 1 1 1

0

A

0 1

B

0 1

1

C

0 1

AOBDD

16

A

B

C

A B C f(ABC)0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 01 1 1 1

0

A

0 1

B

0 1

1

C

0 1*

A

B

D

A

B

DC

=

A B D g(ABD)0 0 0 00 0 1 00 1 0 00 1 1 11 0 0 01 0 1 11 1 0 11 1 1 0

0

A

0 1

B

0 1

1

D

0 1

D

0 1

B

0 1

A

0 1

B

0 1

0

C

0 1

D

0 1

1

B

0 1

D

0 1

Apply Operator

17

0

A

0 1

B

0 1

1

C

0 1* =

0

A

0 1

B

0 1

1

D

0 1

D

0 1

B

0 1

A

B

DC

A1

A2

A3

A4

A5

B1

B2

B3

B4

B5

B6

B7

A1B1

A

0 1A1B1

A2B2

A4B6

B

0 1A2B2

A3B3

A4B4

0

Apply Operator

18

0

A

0 1

B

0 1

1

C

0 1* =

0

A

0 1

B

0 1

1

D

0 1

D

0 1

B

0 1

A

B

DC

A1

A2

A3

A4

A5

B1

B2

B3

B4

B5

B6

B7

A

0 1A1B1

A4B6

B

0 1A2B2

A4B4

0

A4

B4

Apply Operator

19

0

A

0 1

B

0 1

1

C

0 1* =

0

A

0 1

B

0 1

1

D

0 1

D

0 1

B

0 1

A

B

DC

A1

A2

A3

A4

A5

B1

B2

B3

B4

B5

B6

B7

A

0 1A1B1

A4B6

B

0 1A2B2

0

A4

B4

C

0 1A4

D

0 1B4

1

Apply Operator

20

0

A

0 1

B

0 1

1

C

0 1* =

0

A

0 1

B

0 1

1

D

0 1

D

0 1

B

0 1

A

B

DC

A1

A2

A3

A4

A5

B1

B2

B3

B4

B5

B6

B7

A

0 1A1B1

A4B6

B

0 1A2B2

0

C

0 1A4

D

0 1B4

1

B

0 1A4B6

A4B4

A4B7

Apply Operator

21

0

A

0 1

B

0 1

1

C

0 1* =

0

A

0 1

B

0 1

1

D

0 1

D

0 1

B

0 1

A

B

DC

A1

A2

A3

A4

A5

B1

B2

B3

B4

B5

B6

B7

A

0 1A1B1

B

0 1A2B2

0

C

0 1A4

D

0 1B4

1

B

0 1A4B6

A4B7

D

0 1B7

And/Or Multi-Valued Decision Diagrams

AOMDDs are:

AND/OR search graphs

canonical representations, given a pseudo tree

Defined by two rules: All isomorphic subgraphs are merged There are no redundant (meta) nodes

22

23

Example:

HG

ED

C F

B

A

F0 1

H0 1

0 1C1

A0 1

H0 1

0 1C2

A0 1

B0 1

G0 1

G0 1

B0 1

0 1C3

F0 1

G0 1

0 1C4

A0 1

E0 1

0 1C6

C0 1

E0 1

0 1C7

C0 1

D0 1

D0 1

0 1C8

B0 1

C0 1

0 1C9

B0 1

F0 1

0 1C5

m1

m3

m7

m6

m4 m 2m 5

A0 1

F0 1

H0 1

F0 1

H0 1

0 1m1

A0 1

B0 1

G0 1

F0 1

B0 1

G0 1

0 1m2

A0 1

E0 1

C0 1

0 1m4

C0 1

D0 1

D0 1

0 1m5

A0 1

B0 1

F0 1

G0 1

H0 1

F0 1

G0 1

B0 1

F0 1

F0 1

H0 1

0 1m3

A0 1

B0 1

C0 1

C0 1

D0 1

E0 1

D0 1

B0 1

C0 1

C0 1

0 1m6

m7

A

F

H

D

C B F

AE

G

H

primal graph

A

B

C F

D E G H

pseudo tree

A

B

F

G H

C

D

A

B

C

D E

A

B

C F

D E G H

A

B

F

G

C

E

A

(f h) (a∨ ∧ ∨ !h) (a#b#g) (f g) ∧ ∧ ∨ ∧(b f) (a e) (c e) (c#d) (b c)∨ ∧ ∨ ∧ ∨ ∧ ∧ ∨

Example (continued)

24

B

A

m7

A0 1

B0 1

F0 1

G0 1

H0 1

F0 1

G0 1

B0 1

F0 1

F0 1

H0 1

0 1m3

A0 1

B0 1

C0 1

C0 1

D0 1

E0 1

D0 1

B0 1

C0 1

C0 1

0 1m6

A

B

C F

D E G H

m7

A0 1

B0 1

C0 1

0

D0 1

1

F0 1

G0 1

H0 1

C0 1

D0 1

E0 1

F0 1

G0 1

B0 1

C0 1

F0 1

C0 1

F0 1

H0 1

AOBDD vs. OBDD

25

D

C

B

F

A

E

G

H

1 0

B

CC C

D D D D D

E E

F F F

G G G G

H

AOBDD18 nonterminals47 arcs

OBDD27 nonterminals54 arcs

A0 1

B0 1

C0 1

0

D0 1

1

F0 1

G0 1

H0 1

C0 1

D0 1

E0 1

F0 1

G0 1

B0 1

C0 1

F0 1

C0 1

F0 1

H0 1

A

B

C F

D E G H

A

B

C

F

D

E

G

H

D

C B F

AE

G

H

primal graph

Complexity of Compilation The size of the AOMDD is O(n kw*)

The compilation time is also bounded by O(n kw*)

26

k = domain sizen = number of variablesw*= treewidth

Semantic Treewidth Given a network, there may exist a sparser

equivalent network. Challenges the idea of using induced width

to measure the difficulty of the problem AOMDD sizes are much smaller than the

bound

27

Semantic Treewidth With respect to a pseudo tree, this is the

smallest treewidth over all equivalent networks that can have that pseudo tree

With respect to the network, this is the smallest semantic treewidth over all pseudo trees that can express the set of solutions

Instead of the induced width bounding AOMDD size, we can use semantic treewidth.

28

Experimental Results

29

Langford(2,k) problem as a CSP

Experimental Results

30

Langford(2,k) problem as a SAT problem

Experimental Results

31

Uniform Random 3-SAT instances

Experimental Results

32

Equality constraint network results

Conclusion AOMDDs allow us to encode the same

information about a problem in an exponentially more compact form

Canonicity aids in revealing whether the induced width is overestimating the difficulty of a problem

It may be interesting to perform more experiments on real-world CSPs.

33

Acknowledgements Rina Dechter

Pointing me to additional literature Providing me many of these slides

Radu Marinescu Providing me the code to compile AOMDDs

34