1. Constraint Satisfaction Problems & Its Application in Databases
Symmetry Detection in
Constraint Satisfaction Problems
& Its Application in Databases
Berthe Y. Choueiry
Constraint Systems Laboratory
Department of Computer Science & Engineering
University of Nebraska-Lincoln
Joint work with Amy Beckwith-Davis, Anagh Lal, and Eugene C. Freuder
Supported by NSF CAREER award #

2. Outline Definitions Bundling in CSPs
Interchangeability
Bundling
Bundling in CSPs
Bundling for join query computation
Conclusions
December 9, 2005

3. Constraint Satisfaction Problem (CSP)V3
{d}
{a, b, d}
{a, b, c}
{c, d, e, f} V4 V2 V1
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
Graphical representation, binary CSP
December 9, 2005

4. Backtrack search DFS + backtracking (linear space)
Solution
V1  d
V2  e
V3  a
V4  c S
{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 V2 V1 V1 d V2 c e f d V3
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)
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 solution V3
{d}
{a, b, d}
{a, b, c}
{ c, d, e, f } V4 V2 V1
V1  d
V2  c
V3  a
V4  b
V1  d
V2  c
V3  b
V4  a
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 } V1
{d} V2 V3
{a, b, d}
{a, b, c} V4
December 9, 2005

7. Outline Definitions Bundling in CSPs
Static bundling
Dynamic bundling
Dynamic bundling for non-binary CSPs
Bundling for join query computation
Conclusions
December 9, 2005

8. Bundling: using NI in searchV1 V3
{d}
{a, b, d}
{a, b, c}
{ c, d, e, f } V4 V2 V1 c
e, f d V1 V2 S
V1  d
V2  {e,f}
V3  a
{ c, d, e, f } V2
{ c, d, e, f }
{ d, c, e, f }
V4  {b,c} V3
Static bundling V4
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
7min
December 9, 2005

9. Dynamic bundling (DynBndl) [2001]V3
{d}
{a, b, d}
{a, b, c}
{ c, d, e, f } V4 V2 V1 c
e, f d V1 V2 S S c
d, e, f d V1 V2
<V3,a>
<V3,b>
<V4,a>
<V3,d>
<V4,b>
<V4,a>
<V4,c>
<V4,b>
V2,{d,e,f}
V2,{c}
Static bundling
Dynamic bundling
Dynamically identifies NI
Using discrimination tree for forward checking:
is never less efficient than BT & static bundling
7min
December 9, 2005

10. Non-binary CSPs Scope(Cx): the set of variables involved in Cx
{1, 2, 3,
4, 5, 6}
{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
Constraint
Variable C1 C2 C3 C4 V V1 V2 V3 V4 1 3 2 4 6 5
Computing NI for non-binary CSPs is not a trivial extension from binary CSPs
DT compares neighbohood that have the same size
It is difficult to compare neighborhoods that have different sizes
December 9, 2005

11. NI for non-binary CSPs [2003,2005]
Building an nb-DT for each constraint
Determines the NI sets of variable given constraint
Intersecting partitions from nb-DTs
Yields NI sets of V (partition of DV)
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
Root
Root
{5}
{1, 2}
{5, 6}
{3, 4}
{1, 2}
{6}
{3, 4}
nb-DT(V, C1)
nb-DT(V, C2)
For each one of the cnstraints, we build a discrimination tree that is appropriate for non-binary constraints.
For example, V is involved in 2 constraints, C1 and C2.
So, we build one nb-dt for C1 and another one for C2.
In the next slide, I will explain in detail how we build this tree.
What is important to notice here is that, in each tree, we have these boxes that we call annotations.
the domain of V is partitioned in these annotations.
Each partition is a set of equivalent values for V given the constraint.
Now, intersecting the partitions from the constraints that apply to the variable gives us the setS of equivalent values for V given the constraints. Of course, these equivalence sets partition the domain of V.
Furthermore/Moreover, we collect the paths from every root to every annotation in every nb-DT.
The information in these paths is important during search, when V is being instantiated.
It will allow to determine the effect of forward checking on the future variables for free , w/o running checking mechanism.
{5}
{1, 2}
{3, 4}
{6}
December 9, 2005

12. Robust solutions Solution bundle Dynamic bundling finds larger bundles
Single Solution
Static bundling
Dynamic bundling
V1  d
V2  e
V3  a
V4  c
V1  d
V2  {e,f}
V3  a
V4  {b,c}
V1  d
V2  {d,e,f}
V3  a
V4  {b,c}
Solution bundle
Cartesian product of domain bundles
Compact representation
Robust solutions
Dynamic bundling finds larger bundles
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… … is particularly effective No-good bundle
{3, 4}
{2}
{1} V V4 V3 V1 V2
{1, 2}
{1, 3}
{3} C4
{1, 2, 3,
4, 5, 6}
{1, 2, 3} C2 C1 C3 V1 V2 V3 V4 V
No-good bundle
Solution bundle
… is particularly effective
December 9, 2005

15. Experimental set-up CSP parameters: Phase transition
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)
Cost of solving
Mostly
solvable instances
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 FC 20
n=20
t FBS 18
a=15
Time [sec]
DynBndl
CR=CR3
#NV, hundreds 16 14 12 FC 10 NV 8 6
DynBndl 4 2
CPU time
19min
0.325
0.35
0.375
0.4
0.425
0.45
0.475
0.5
0.525
0.55
0.575
0.6
Tightness
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
CR1
33.3
34.3
5.5
11.9
CR2
28.6
33.0
5.0
CR3
29.8
31.7
3.6
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 in CSPs
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 query Result: 10 tuples in 3 nested tuples Join 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 R2
(compacted)
Result:
10 tuples in
3 nested tuples A B C
{1, 5}
{12, 13, 14}
{23}
{2, 4}
{10}
{25}
{6}
{13, 14}
{27}
December 9, 2005

21. Databases & CSPs 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
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
21min
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 constraints
SELECT 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.B
R2.C R1 R2 24
December 9, 2005

23. Join operator R1 xy R2 Join algorithms
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
22min
December 9, 2005

24. Join query R2 R1 R1.A R1.B R1.C R2.A R2.B R2.C R1 xy R2 CSP model
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
R1.A
R1.B
R1.C
R2.A
R2.B
R2.C R1 R2
SELECT R1.A,R1.B,R1.C
FROM R1,R2
WHERE R1.A=R2.A
AND R1.B=R2.B
AND R1.C=R2.C
December 9, 2005

25. Progressive Merge Join
PMJ: a sort-merge algorithm [Dittrich et al. 03]
Two phases
Sorting: sorts sub-sets of relations &
Merging phase: merges sorted sub-sets
PMJ produces early results
We use the framework of the PMJ
32min
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. 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
R1.B
R2.B R2
R1.C
R2.C
Sort relations using above ordering
Next: Compute bundles of variable ahead in variable ordering (R1.A)
Shrink figure lift up bottom lines
December 9, 2005

28. Computing a bundle of R1.A
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 R1 A B C 1 12 23
Partition 1 13 23 1 14 23
Unequal
partitions 2 10 25
Symmetric
partitions 5 12 23 26 5 13 23 5 14 23
Bundle
{1, 5}
December 9, 2005

29. Finding the valid bundle
Common
{1, 5}
{1, 5, x}
Compute a bundle for the attribute
Check bundle validity with future constraints
If no common value ‘backtrack’
 Assign variable with the surviving values in the bundle
{1, 5, y, z}
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 Bundling in CSPs
Bundling for join query computation
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 35
Constraint Processing inspires innovative solutions to fundamental difficult problems in Databases
December 9, 2005

33. Future research CSPs Databases Constraint databases
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)
Query-size estimation:
December 9, 2005