Symmetry Detection in Constraint Satisfaction Problems & Its Application in Databases

Fall 2009, Advanced Constraint Programming

1

Symmetry Detection in Constraint Satisfaction Problems

& Its Application in Databases

Berthe Y. ChoueiryConstraint Systems Laboratory

Department of Computer Science & EngineeringUniversity of Nebraska-Lincoln

Joint work with Amy Beckwith-Davis, Anagh Lal, and Eugene C. Freuder

Supported by NSF CAREER award #0133568


December 9, 2005 2

Outline

• Definitions– CSP– Interchangeability– Bundling

• Bundling in CSPs

• Bundling for join query computation

• Conclusions


December 9, 2005 3

Constraint Satisfaction Problem (CSP)

• Given P = (V, D, C)– V : set of variables– D : set of their domains– C : set of constraints (relations) restricting the

acceptable combination of values for variables– Solution is a consistent assignment of values to variables

• Query: find 1 solution, all solutions, etc.• Examples: SAT, scheduling, product configuration• NP-Complete in general

V3

{d}

{a, b, d} {a, b, c}

{c, d, e, f}

V4

V2V1


December 9, 2005 4

Backtrack search

• DFS + backtracking (linear space) – Variable being instantiated: current variable– Un-instantiated variables: future variables– Instantiated variables: past variables

• + Constraint propagation – Backtrack search with forward checking (FC)

c e f d

dV1

V2

S

V3

Solution

V1 dV2 e

V3 aV4 c

{c,d,e,f}

{a,b,d}

{a,b,c}

V1

V2

V3

V4

d

V3

{d}

{a, b, d} {a, b, c}

{ c, d, e, f }

V4

V2V1


December 9, 2005 5

Interchangeability [Freuder, 91]

• Captures the idea of symmetry between solutions • Functional interchangeability

– Any mapping between two solutions– Including permutation of values across variables, equivalent to

graph isomorphism

• Full interchangeability (FI)– Restricted to values of a single variable– Also, likely intractable

V1 V2 {d, e, f}

V3 V4

In every solutionV1 dV2 c

V3 aV4 b

V1 dV2 c

V3 bV4 a

V3

{d}

{a, b, d} {a, b, c}

{ c, d, e, f }

V4

V2V1


December 9, 2005 6

Value interchangeability [Freuder, 91]

• Full Interchangeability (FI): – d, e, f interchangeable for V2 in any solution

• Neighborhood Interchangeability (NI): – Considers only the neighborhood of the variable – Finds e, f but misses d– Efficiently approximates FI– Discrimination tree DT(V2)

{c, d, e, f }{d}

{a, b, d} {a, b, c} V4

V2V1

V3


December 9, 2005 7

Outline

• Definitions

• Bundling in CSPs– Static bundling– Dynamic bundling– Dynamic bundling for non-binary CSPs


• Conclusions


December 9, 2005 8

• Static bundling

[Haselböck, 93]

– Before search: compute & store NI sets

– During search: • Future variables: remove bundle of equivalent values • Current variable: assign a bundle of equivalent values

• Advantages– Reduces search space

– Creates bundled solutions

Bundling: using NI in search

Static bundling

c e, f d

dV1

V2

S

V3

{d}

{a, b, d} {a, b, c}

{ c, d, e, f }

V4

V2V1

V4 {b,c}

V1 dV2 {e,f}

V3 aV2

{ c, d, e, f }

{ d, c, e, f }

{ c, d, e, f }

V3

V4

V1


December 9, 2005 9

<V4,a> <V3,d><V4,a>

<V4,b><V4,c><V4,b>

<V3,b><V3,a>

• Dynamically identifies NI• Using discrimination tree for forward checking:

– is never less efficient than BT & static bundling

Dynamic bundling (DynBndl) [2001]

Static bundling

S

c d, e, f

dV1

V2

Dynamic bundling

c e, f d

dV1

V2

S

V3

{d}

{a, b, d} {a, b, c}

{ c, d, e, f }

V4

V2V1

V2,{d,e,f} V2,{c}


December 9, 2005 10

Non-binary CSPs

Constraint Variable

C1 C2 C3 C4

V V1 V2 V V3 V2 V3 V4 V1 V4

1 1 3 1 3 1 2 1 1 1

1 3 3 2 3 1 2 2 2 2

2 1 3 3 2 2 2 1 3 1

2 3 3 4 2 2 2 2

3 1 1 4 2 3 1 1

3 2 2 6 1

4 1 1

4 2 2

5 3 2

6 3 2

C4

{1, 2, 3, 4, 5, 6}

{1, 2, 3}

{1, 2, 3}

{1, 2, 3}

{1, 2, 3}

C2

C1

C3 V1

V2

V3

V4

V

• Scope(Cx): the set of variables involved in Cx

• Arity(Cx): size of scope

Computing NI for non-binary CSPs is not a trivial extension from binary CSPs


December 9, 2005 11

1. Building an nb-DT for each constraint– Determines the NI sets of variable given constraint

2. Intersecting partitions from nb-DTs – Yields NI sets of V (partition of DV)

3. Processing paths in nb-DTs– Gives, for free, updates necessary for forward checking

NI for non-binary CSPs [2003,2005]

C4

{1, 2, 3, 4, 5, 6}

C2

C1

C3

V1

V2

V3

V4

V

{1, 2} {5, 6} {3, 4}

Root

nb-DT(V, C1)

Root

{1, 2} {3, 4}{6}

nb-DT(V, C2)

{5}{1, 2} {3, 4} {6}

{5}


December 9, 2005 12

Robust solutions

• Solution bundle– Cartesian product of domain bundles– Compact representation– Robust solutions

• Dynamic bundling finds larger bundles

V1 dV2 e

V3 aV4 c

Single Solution

V1 dV2 {e,f}

V3 aV4 {b,c}

Static bundling

V1 dV2 {d,e,f}

V3 aV4 {b,c}

Dynamic bundling


December 9, 2005 13

DynBndl: worth the effort?

• Finds larger bundles• Enables forward checking at no extra cost• Does not cost more than BT or static bundling

– Cost model: • # nodes visited by search• # constraint checks made

− Theoretical guarantee holds • for finding all solutions• under same variable ordering

¿ Finding first solution ?− Experiments uncover an unexpected benefit


December 9, 2005 14

Bundling of no-goods…

No-good bundle

{1, 2}

{1, 3}

{3}

{3, 4}

{2}

{1}

{1}

{1}

V

V4

V3

V1

V2

Solution bundle

C4

{1, 2, 3, 4, 5, 6}

{1, 2, 3}

{1, 2, 3}

{1, 2, 3}

{1, 2, 3}

C2

C1

C3 V1

V2

V3

V4

V

• … is particularly effective


December 9, 2005 15

• CSP parameters: – n: number of variables {20,30}– a: domain size {10,15}– t: constraint tightness [25%, 75%]– CR: constraint ratio (arity: 2, 3, 4)– 1,000 instances per tightness value

• Phase transition• Performance measures

– Nodes visited (NV)– Constraint checks (CC)– CPU time– First Bundle Size (FBS)

Experimental set-up

Cos

t of

sol

vin

g

Mostly solvable instances

Mostly un-solvable instances

Critical value Order parameter


December 9, 2005 16

Empirical evaluations

• DynBndl versus FC (BT + forward checking)

• Randomly generated problems, Model B• Experiments

– Effect of varying tightness– In the phase-transition region

• Effect of varying domain size • Effect of varying constraint ratio (CR)

• ANOVA to statistically compare performance of DynBndl and FC with varying t

• t-distribution for confidence intervals


December 9, 2005 17

Analysis: Varying tightness• Low tightness

– Large FBS • 33 at t=0.35 • 2254 (Dataset #13, t=0.35)

– Small additional cost

• Phase transition– Multiple solutions present– Maximum no-good bundling

causes max savings in CPU time, NV, & CC

• High tightness– Problems mostly unsolvable– Overhead of bundling minimal

n=20a=15CR=CR3

0

2

4

6

8

10

12

14

16

18

20

0.325 0.35 0.375 0.4 0.425 0.45 0.475 0.5 0.525 0.55 0.575 0.6

TightnessT

ime

[s

ec

]#

NV

, h

un

dre

ds t FBS

0.350 33.44 0.400 10.91 0.425 7.130.437 6.38 0.450 5.620.462 2.370.475 0.660.500 0.03

0.550 0.00 NV

CPU time

DynBndl

FC

DynBndl

FC


December 9, 2005 18

Analysis: Varying domain size• Increasing a in phase-

transition– FBS increases: More

chances for symmetry– CPU time decreases:

more bundling of no-goods

CR Improv (CPU) %

FBS

a=10 a=15 a=10 a=15

CR1 33.3 34.3 5.5 11.9

CR2 28.6 33.0 5.0 5.5

CR3 29.8 31.7 3.6 5.0

CR4 28.4 31.6 1.2 1.4

Increasing a (n=30)

Because the benefits of DynBndl increase with increasing domain size, DynBndl is particularly interesting for database applications where large domains are typical


December 9, 2005 19

Outline

• Definitions


• Bundling for join query computation– Idea– A CSP model for the query join– Sorting-based bundling algorithm– Dynamic-bundling-based join algorithm

• Conclusions


December 9, 2005 20

The join queryJoin query

SELECT R2.A,R2.B,R2.C

FROM R1,R2

WHERE R1.A=R2.A

AND R1.B=R2.B

AND R1.C=R2.C

R1

A B C

1 12 23

1 13 23

1 14 23

2 10 25

3 16 30

4 10 25

5 12 23

5 13 23

5 14 23

6 13 27

6 14 27

7 14 28

R2

A B C

1 12 23

1 13 23

1 14 23

1 15 23

2 10 25

3 17 20

4 10 25

5 12 23

5 13 23

5 14 23

5 15 23

6 13 27

6 14 27

Result: 10 tuples in

3 nested tuples

R1 R2 (compacted)

A B C

{1, 5} {12, 13, 14} {23}

{2, 4} {10} {25}

{6} {13, 14} {27}


December 9, 2005 21

Databases & CSPs

DB terminology CSP terminology

Table, relation Constraint (relational constraint)

Join condition Constraint (join-condition constraint)

Attribute CSP variable

Tuple in a table Tuple in a constraint or allowed by one

Computing a join sequence Finding all solutions to a CSP

• Same computational problems, different cost models– Databases: minimize # I/O operations– CSP community: # CPU operations

• Challenges for using CSP techniques in DB– Use of lighter data structures to minimize memory usage– Fit in the iterator model of database engines

Administrator


December 9, 2005 22

Modeling join query as a CSP

• Attributes of relations CSP variables• Attribute values variable domains• Relations relational constraints• Join conditions join-condition constraintsSELECT R1.A,R1.B,R1.C

FROM R1,R2

WHERE R1.A=R2.A

AND R1.B=R2.B

AND R1.C=R2.C

R1.A R1.B R1.C

R2.A R2.BR2.C

R1 R2


December 9, 2005 23

Join operator

• R1 xy R2– Most expensive operator in terms of I/O is “=” Equi-Join

• x is same as y Natural Join

• Join algorithms– Nested Loop– Sorting-based

• Sort-Merge, Progressive Merge-Join (PMJ)• Partitions relations by sorting, minimizes # scans of relations

– Hashing-based


December 9, 2005 24

Join query• R1 xy R2

– Most expensive operator in terms of I/O is “=” Equi-Join

• x is same as y Natural Join

• CSP model– Attributes of relations CSP variables– Attribute values variable domains– Relations relational constraints– Join conditions join-condition constraints

SELECT R1.A,R1.B,R1.CFROM R1,R2WHERE R1.A=R2.A AND R1.B=R2.BAND R1.C=R2.C

R1.A R1.B R1.C

R2.A R2.BR2.C

R1 R2


December 9, 2005 25

Progressive Merge Join

• PMJ: a sort-merge algorithm

[Dittrich et al. 03]

• Two phases1. Sorting: sorts sub-sets of relations &

2. Merging phase: merges sorted sub-sets

• PMJ produces early results • We use the framework of the PMJ


December 9, 2005 26

New join algorithm

• Sorting & merging phases– Load sub-sets of relations in memory– Compute in-memory join using dynamic

bundling• Uses sorting-based bundling (shown next)• Computes join of in-memory relations using

dynamically computed bundles


December 9, 2005 27

• Heuristic for variable ordering Place variables linked by join conditions as close to each other as possible

R1.A

R2.A

R1.B

R2.B

R1.C

R2.C

R1

R2

• Sort relations using above ordering

• Next: Compute bundles of variable ahead in variable ordering (R1.A)

Sorting-based bundling


December 9, 2005 28

Computing a bundle of R1.A

Partition

Unequalpartitions

Symmetricpartitions

Bundle {1, 5}

R1

A B C

1 12 23

1 13 23

1 14 23

2 10 25

5 12 23

5 13 23

5 14 23

• Partition of a constraint–Tuples of the relation having the same value of R1.A

• Compare projected tuples of first partition with those of another partition

• Compare with every other partition to get complete bundle


December 9, 2005 29

R1

A B C

1 12 23

1 13 23

1 14 23

2 10 25

3 16 30

3 16 24

R2

A B C

1 12 23

1 13 23

1 14 23

1 15 23

2 10 25

3 17 20

{1, 5, x}{1, 5, y, z}

Common {1, 5}1. Compute a bundle

for the attribute 2. Check bundle

validity with future constraints

3. If no common value ‘backtrack’

Assign variable with the surviving values in the bundle

Finding the valid bundle


December 9, 2005 30

Experiments

• XXL library for implementation & evaluation• Data sets

• Random: 2 relations R1, R2 with same schema as example– Each relation: 10,000 tuples– Memory size: 4,000 tuples– Page size 200 tuples

• Real-world problem: 3 relations, 4 attributes

• Compaction rate achieved– Random problem: 1.48

– Savings even with (very) preliminary implementation

– Real-world problem: 2.26 (69 tuples in 32 nested tuples)


December 9, 2005 31

Outline

• Definitions



• Conclusions– Summary– Future research


December 9, 2005 32

Summary

• Dynamic bundling in finite CSPs – Binary and non-binary constraints

– Produces multiple robust solutions

– Significantly reduces cost of search at phase transition

• Application to join-query computation

Constraint Processing inspires innovative solutions to fundamental difficult problems in Databases


December 9, 2005 33

Future research

• CSPs– Only scratched the surface: – interchangeability + decomposition [ECAI 1996],– partial interchangeability [AAAI 1998], – tractable structures

• Databases– Investigate benefit of bundling

• Sampling operator• Main-memory databases• Automatic categorization of query results

• Constraint databases– Design bundling mechanisms for gap & linear constraints over

intervals (spatial databases)

Documents

Symmetry Detection in Constraint Satisfaction Problems & Its Application in Databases