foundations of constraint processing csce421/821, fall 2004:

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 search

S

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

CP

U t

ime

[ms]



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

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