Constraint Systems Laboratory April 21, 2005Lal: M.S. thesis defense1 Neighborhood Interchangeability (NI) for Non-Binary CSPs & Application to Databases

April 21, 2005 Lal: M.S. thesis defense 1

Constraint Systems Laboratory

Neighborhood Interchangeability (NI) for Non-Binary CSPs

& Application to Databases

Anagh Lal

Constraint Systems LaboratoryComputer Science & Engineering

University of Nebraska-Lincoln

Research supported by NSF CAREER award #0133568 and by Maude Hammond Fling Faculty Research Fellowship.



Main contributions

CSPs1. Interchangeability: An algorithm for neighborhood

interchangeability (NI) in non-binary CSPs2. Dynamic bundling: Integrating NI + backtrack search

for solving non-binary CSPs3. Exploratory: Towards detecting substitutabilityDatabases1. A new model of the join query as a CSP2. A new sorting-based bundling algorithm3. A new sort-merge join algorithm that produces

bundled tuples4. Exploratory: Application to materialized views



Outline

• Background

• Neighborhood Interchangeability (NI) for non-binary CSPs

• Empirical evaluations

• Database algorithms based on dynamic bundling

• Conclusions & future work

Administrator



Constraint Satisfaction Problem

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



Systematic search• Basic mechanism

– DFS & backtracking (BT)– Variable being instantiated: current variable– Uninstantiated variables: future variables– Instantiated variables: past variables

• Constraint propagation– Remove values inconsistent with constraints– Forward checking filters domains of future

variables given the instantiation of current variable



Value interchangeability [Freuder, ‘91]

Equivalent values in the domain of a variable

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

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

V4

V2V1

V3

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

• Neighborhood Interchangeability (NI): – Efficiently approximates FI– Finds e, f but misses d– Discrimination tree DT(Vx)



• Dynamic bundling [Our group, ‘01]

– Dynamically identifies NI– Finds fatter solution than BT & static bundling– Never less efficient than BT & static bundling

Bundling: using NI in search

BT Static bundling

S

c d, e, f

dV1

V2

Dynamic bundling

c e, f d

dV1

V2

S

c e f d

dV1

V2

S

V3

{d}

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

{ c, d, e, f }

V4

V2V1

• Static bundling [Haselböck, ‘93]



Robust solutionsSingle solution• V1 d• V2 e• V3 a• V4 c

Robust solution

• V1 {d}

• V2 {d, e, f}

• V3 {a}

• V4 {b, c}

V3

{d}

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

{c, d, e, f}

V4

V2V1• Solution bundle: Cartesian

product of bundles of variables• Solution-bundle size

= 1 3 1 2 = 6



Phase transition [Cheeseman et al. ‘91]

• Significant increase of cost around critical value• In CSPs, order parameter is constraint tightness & ratio• Algorithms compared around phase transition

Cos

t of

sol

ving

Mostly solvable problems

Mostly un-solvable problems

Critical value of order parameter

Order parameter



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



CSP parameters

• n number of variables

• a domain size

• t constraint tightnessratio of number of disallowed tuples over all possible tuples

• deg degree of a variable

• ck number of constraints of arity k

• pk = ck / (nk) constraint ratio



Outline

• Background

• Neighborhood Interchangeability (NI) for non-binary CSPs– Non-binary discrimination tree (nb-DT)




Administrator



NI for non-binary CSPs1. 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

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}

{5}

nb-DT(V, C2)



Building an nb-DT: nb-DT(V, C1)

(<V1 1>, <V2 3>)

(<V1 3>, <V2 3>)

{1, 2}

Root

C1

V V1 V2

1 1 3

1 3 3

2 1 3

2 3 3

3 1 1

3 2 2

4 1 1

4 2 2

5 3 2

6 3 2

(<V1 3>, <V2 2>)

Annotation

Path

{1}

Domain of V

5 62 3 41

O (deg . a (k+1) . (1 - t))

(<V1 2>, <V2 2>)

{3, 4}

(<V1 1>, <V2 1>)

{5, 6}



Bundling = Search + NI• Benefits of bundling

1. Bundles solutions

2. Bundles no-goods

• Dynamic bundling (DynBndl)– Re-computes NI during search– Yields larger bundles,boosts effects

of bundling

• Skeptics’ objection to DynBndl – Costly & not worthwhile

• We show that the converse holds

{3, 4}

{2}

{1}

{1, 2}

{1, 3}{1}

{3}{1}

No-good

bundle

V

V4

V3

V1

V2

Solution bundle



Advantages of DynBndl

• We exploit nb-DTs for forward checking• DynBndl versus FC (BT+ forward checking)

– Finding all solutions: theoretically best– Finding first solution: empirical evidence

DynBndl yields multiple, robust for less cost



Outline

• Background





Administrator



Empirical evaluations

• DynBndl versus FC (BT+forward checking)

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

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

• Randomly generated problems, Model B• ANOVA to statistically compare performance of

DynBndl and FC with varying t• t-distribution for confidence intervals



Experimental set-up

• Generated 16 data sets– n = {20,30} a = {10,15} {CR1,CR2,CR3,CR4}– 9—12 values for t [25%,75%] – 1,000 instances per tightness value

• Performance measurements– FBS, size of the first solution bundle– NV, number of nodes visited in the search tree– CC, number of constraints checked– CPU time



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



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



Outline

• Background• Neighborhood Interchangeability (NI) for

non-binary CSPs• Empirical evaluations• Database algorithms based on

dynamic bundling– Sorting-based bundling algorithm– Dynamic-bundling-based join algorithm


Administrator



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

A sequence of natural joins 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



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



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}



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



Progressive Merge Join• PMJ: a sort-merge algorithm by [Dittrich

et al. ‘03]

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

produces early results

2. Merging phase: merges sorted sub-sets

• We use the framework of the PMJ for our external join



New join algorithm

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

bundling

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

dynamically computed bundles Cool animation upon request



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



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)



Outline

• Background





Administrator



Conclusions

• Algorithm for computing NI sets in non-binary CSPs

• DynBndl – produces multiple robust solutions – significantly reduces cost of search at phase

transition

• New dynamic-bundling-based join algorithm

Constraint Processing inspires innovative solutions to fundamental difficult problems in Databases



Future work

• Sort constraint definitions to improve CSP techniques

• Design bundling mechanisms for gap & linear constraints in Constraint Databases

• Explore benefits of bundling in Databases– Sampling operator– Main-memory databases– Automatic categorization of query results



Thanks!!



Related work

• Join algorithms– Well established algorithms– Do not focus on exploiting symmetry

• Database compression– Output results are not compressed– Compression at value level, not tuple level



Related work (contd)

• [Mamoulis & Papadias 1998] – Join using FC for spatial DB – Restricted to binary constraints– No compaction of solution space

• [Bayardo et al. 1996]– Reduce the number of the intermediate tuples of a sequence of

joins

• [Rich et al. 1993]– Do not compact join attribute values– Does not detect redundancy present in the grouped sub-relations



Analysis of overheads

• For Bundling– Additional data structures: 2 arrays, 1 pointer– Only 1 array (Processed values) may become

cumbersome

• Array size is largest – when all the values of a variable are in one

bundle – But, this case also leads to best savings!



Sorting-based bundling

• 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)



R1 J oin R2 (Compacted)

A B C

Join using bundling

Computing bundle for R1.A

A B C

A B C

Processed values R1

Processed values R2

Select partition to compare for R1.ASymmetric partitions, Adding to bundle of R1.A, Current bundle of R1.A = {1, 5}

Computing bundle for R2.ASelect partition to compareSymmetric partitions,

Adding to bundle of R2.A, Current bundle of R2.A = {1, 5}

Update processed values for R1.A

5

5

Update processed values for R2.A

R1

R2

R2.C

R1.A

R2.A

R1.B

R2.B

R1.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



R1 J oin R2 (Compacted)

A B C

Join using bundling

5

1, 55

Current bundle of R1.A = {1, 5}

Current bundle of R2.A = {1, 5}

Common(R1.A, R2.A) = {1, 5}

Compute current constraint of R1

Assign {1, 5} to R1.A

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

A B C

A B C

Processed values R1

Processed values R2

R1

R2

R2.C

R1.A

R2.A

R1.B

R2.B

R1.C

1, 5

1, 5

Assign {1, 5} to R2.A

Compute current constraint of R2

Next variable R1.B

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

Documents

Constraint Systems Laboratory April 21, 2005Lal: M.S. thesis defense1 Neighborhood Interchangeability (NI) for Non-Binary CSPs & Application to Databases