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Integrating CSP Decomposition Techniques and BDDs for

Compiling Configuration Problems

Sathiamoorthy SubbarayanIT University of Copenhagen

Denmark

CP-AI-OR 2005

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Configuration

An arrangement of parts

Valid Configuration- a valid arrangement

Valid according to some specified rules

NP-Hard problem

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Examples

• Bike• Car• PC • Power-Backup Systems• Controllers• Business Software

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Problem Representation

Constraint Satisfaction Problem (CSP)

• A set of variables and their domains• A set of rules/constraints on variable

values

Configuration- A legal assignment to all variables

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Interactive Configurator

• Interactively guides the user• Shows only valid extensions• User always in a consistent state

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Required Functionalities

• Propagate the user selections• Display the valid choices• Explain reasons for unselectable or

implied options

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Binary Decision Diagrams

A compressed representation of solutions

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Previous work

Compile into a Binary Decision Diagram

rulesBDD

generation BDD lookup

offline online

configurator

Com

pila

tion

NP-hard P

BDD

Problem! BDD size could explode

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This work

• Configuration rules have hierarchies• Hierarchies close to tree-shape

Tree Decomposition of CSP?

Space reduction? (upto 96%)

Response time reduction? (upto 200x)

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CSP to BDD compilation

C1

C2

C3 C4

C5

C6C7

bdd

CSP BDD

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Tree Decomposition of CSP

C1

C2

C3 C4

C5

C6C7

C1

C2C3

C4

C5C6

C7

C3

C6

CSP

CSP’

b1

b2

b3 Tree-of-BDDs

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Benefit of decomposition

BDD

b1

b2

b3

bdd

Tree-of-BDDs

|b1|+|b2|+|b3| < |bdd|

Potential Space Reduction

-Online Configurators

-Embedded Configurators

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Required Functionalities

• Propagate the user selections• Display the valid choices• Explain reasons for unselectable or

implied options

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Explanations

• A partial assignment by the user might cause some options unselectable /implied

• Explanation for such an option is the subset of the partial assignment that makes the option unselectable/implied

• When user assignments are given priorities, we might need minimum explanations

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Explanations in single BDD case

Variant of Shortest path problem in DAG

• Assigned variable (v=val) – 0 for ’val’ edges– 1 for other edges

• Other variables– 0 for their edges

• Shortest path from root to 1-node corresponds to an explanation

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Example

Assign a=0Assign b=1 e=1 is implied

Explain e=0?

1

1

Explanation b=1

Can also handle priorities

and non-Boolean variables

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Don’t Care variables for Explain

• In the example, necessary variables are: a,b, and e

• Variables c and d are hence don’t cares for Explain

Abstraction !

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Explanations in Tree-of-BDDs

b1

b2

b3

b1’ b2’

b3’

Abstraction

Join

b’

ExplainExplanation

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Experiments

• Four Benchmarks– PC– Renault– Two PSR instances

• 10000 random interactions• Comparison of:

– Space usage– Average and Worst time for Responses– Average and Worst time for Explanations

• Tools– CLab: an existing single BDD-based tool– iCoDE: implementation of the tree-of-BDDs

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Compilation Details

BenchmarkTree(sec)

TreeNodes

Compile (sec) BDD-nodes Peak BDD-nodes

CLab iCoDE CLab iCoDE CLab iCoDE

PC 0.04 16 0.11 0.19 16494 4458 80K 40K

Renault 0.25 73 119 77 455796 17602 2500K 80K

psr-1-32-1 2 114 0.46 4 56923 8917 600K 600K

psr-1-32-1 2 114 2 9 246775 22101 1200K 1200K

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Time Comparisons

• Response (Propagate+Display)

BenchmarkAverage(sec) Worst(sec)

CLab iCoDE CLab iCoDE

PC 0.004 0.0001 0.050 0.006

Renault 0.070 0.0003 0.452 0.020

psr-1-32-1 0.016 0.0010 0.057 0.038

psr-1-32-2 0.037 0.0002 0.618 0.107

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Time Comparisons

• Explanation

BenchmarkAverage(sec) Worst(sec)

CLab iCoDE CLab iCoDE

PC 0.004 0.010 0.010 0.030

Renault 0.160 0.088 0.440 0.921

psr-1-32-1 0.031 1.208 0.067 2.155

psr-1-32-2 0.122 5.38 0.329 10.78

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Conclusion

• Decomposition results in– Space reduction– Shorter responses

• Tree-of-BDDs can scale higher than the single BDD approach

• Future work include:– BDD variable orderings– Multi-valued Decision Diagrams– different decomposition techniques