foundations of constraint processing, fall 2004 october 3, 2004interchangeability in csps1...

October 3, 2004 Interchangeability in CSPs 1

Foundations of Constraint Processing, Fall 2004

Foundations of Constraint Processing

CSCE421/821, Fall 2004:

www.cse.unl.edu/~choueiry/F04-421-821/

Anagh LalAvery Hall, Room 123D

[email protected]

Interchangeability in CSPs



Outline• Exploiting problem structure• Interchangeability: Introduction• Interchangeability: Types• Search with Interchangeability• Non-binary CSPs: Challenges and

approach• Databases: Challenges and approach



Exploit problem structure• Declared symmetries [Glaisher 1874; Ellman 1993]

– Applies to specific class of problems– Requires that a user or designer know the

structure of the problem• Discover symmetries

– Solver discovers symmetries called interchangeability

– Symmetry detection partitions the domain of variables



Idea

V3

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

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

V2V1

Consider V2

Domain value

Consistent with

c V3(a, b, d)V4(a, b)

d V3(a, b)V4(a, b, c)

e,f V3(a, b, d)V4(a, b, d)



Interchangeability: Basics• Two interchangeable values of a variable

behave in the same way– Globally (likely intractable) or– Locally (tractable)

• Interchangeable values are redundant• Interchangeable values can be placed in a

bundle and treated as a single value

{ c d e, f }V2



Advantages

• Cluster or club together values e and f to form a bundle.

• Search effort reduces– Fewer nodes are visited

• Solutions produced may represent more than one solution– Solution space is compacted.

c e, f d

dV1

V2

S

c e f d

dV1

V2

S



Full Interchangeability• Full Interchangeability (FI)

– All constraints constraints are considered• Even when they do not apply to the variable

– Two values of a variable are Full Interchangeable if on replacing one with the other gives another solution

– Computing FI requires finding all solutions

V3

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

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

V2V1



Neighborhood Interchangeability• Restricted to neighborhood of a variable• Easier to compute NI than FI• Compute bundles and store for use during

search

V3

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

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

V2V1



Computing NI• Discrimination Tree (DT)

– Data structure for computing bundles of a variable.

• Only the constraints that apply to the variable are considered

• Requires an ordering of values to be followed



Building the DT• Let us work with an example on the

board.



Static versus Dynamic NI• Static: Bundles computed prior to search• Dynamic: Bundles computed during

searchS

c d, e, f

dV1

V2

Dynamic bundlingStatic bundling

c e, f d

dV1

V2

S

c e f d

dV1

V2

S



Finding all solutions• Dynamic bundling has strong results

– Guaranteed no more expensive than forward checking

– Guaranteed no more expensive than static bundling

– Guaranteed to produce larger bundles than static bundling



Finding one solution• Traditionally dynamic bundling was

thought to be an overkill for finding one solution

• Theoretical claims can not be made• Show empirically that dynamic bundling is

effective for finding one solution also



Extending to non-binary• Now we start getting into my work, finally!• Non-binary CSP: CSP with at least one

constraint of arity greater than 2.• Many problems modeled more naturally by

non-binary constraints– Theoretically all non-binary CSPs can be

decomposed to binary– Binary version may not be a good choice all

the time



Non-binary CSPs• Informal introduction• Representation

C4

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

{1, 2, 3}

{1, 2, 3}

{1, 2, 3}

{1, 2, 3}

C2

C1

C3

ConstraintVariable



The Challenges• Constraints of different arities cause

problems– Not possible to make one composite DT like

structure– Comparing different length tuples (set of

variable-value pairs) is awkward• Handling “Partially instantiated” constraints



Bundling non-binary CSPs• Data structure used: non-binary

Discrimination Tree (DT)• For bundling nb-CSP variables

– Create a non-binary DT for each constraint that applies to the variable

• Note difference from binary: All constraints were in one DT

– Intersect the bundles from all the nb-DTs to get the bundles



The reason behind the gains• A no-good is a partial

instantiation that does not lead to a solution

• When search with bundling BTs on a no good a larger search space is eliminated

• Of course, the same logic applies to a solution bundle

{1, 2}

{3}

{3}

{1, 2}

{1, 3} {3}

{3} {3}

No-good bundle

V

D

C

A

B

Solution bundle



Some results

0

200

400

600

800

1000

1200

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

# NV in DynBndl# NV in FCFirst Bundle Size

First Bundle Size

Tightness

n = 20, a = 10 p2 = 0.25

c3 = 3 c4 = 2



Some results [2]

0

500

1000

1500

2000

0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7

DynBndlFC

n = 20, a = 10 p2 = 0.25

c3 = 3 c4 = 2

Tightness

Overhead due to large FBS (upto 1200) at low tightness

Maximum gains occur around phase-transition area

CPU

tim

e [m

s]



Moving on to databases• Observations

– The gains due to dynamic bundling are more for larger domain sizes

– For finding all solutions bundling is always better

– Bundling produces compact solution spaces• Deduction

– Lets try it in Databases



Database facts & challenges• Database is about I/O, CPU comparisons

don’t matter• Data does not fit in memory• Memory efficiency is key to database

algorithms• Database query processing algorithms

provide an iterator interface to the query engine

• Dynamic bundling as is, does not fit!!



How did we handle it for DB• A new bundling algorithm

– No DTs– Sort constraints (the tables of the DB), use the

structure introduced by sorting to detect bundles

• Model search with bundling to fit the framework of query processing algorithms (join algorithm, in particular)



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

Current bundleof

R1.A = {1, 5}

Computing a bundle of R1.A

foundations of constraint processing, fall 2004 october 3, 2004interchangeability in csps1...

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