Download - Foundations of Constraint Processing CSCE496/896, Fall 2011 cse.unl/~choueiry/F11-496-896

Problem Solving with Constraints

Lookahead Schemas 1

Foundations of Constraint Processing

CSCE496/896, Fall 2011

www.cse.unl.edu/~choueiry/F11-496-896/

All questions to [email protected]

Berthe Y. Choueiry (Shu-we-ri)

Avery Hall, Room 360

Tel: +1(402)472-5444

Lookahead Schemas


Lookahead Schemas 2

Outline

• Looking ahead Schemas– Forward checking (FC)– Directional Arc Consistency (DAC)– Maintaining Arc Consistency (a.k.a. full arc-

consistency)


Lookahead Schemas 3

Looking ahead• Rationale:

– As decisions are made (conditioning), • Revise the domain of future variables to propagate the

effects of decisions • i.e., eliminate inconsistent choices in future sub-problem

– Domain annihilation of a future variable avoids expansion of useless portions of the tree

• Techniques– Partial: forward-checking (FC), directional arc-

consistency (DAC)– Full: Maintaining arc-consistency (MAC)– Use: Revise(Vf, Vc), Vf future variable, Vc current

variable


Lookahead Schemas 4

Revise the domain of Vi

• Revising the domain of Vi given a constraint CVi,Vj on Vi (i.e., Vi Scope(C))

• General notation: Revise(Vi,CVi,Vj)

• In a binary CSP:

Revise(Vi,CVi,Vj)=Revise(Vi, Vj)


Lookahead Schemas 5

Revise(Vi, Vj) NOTE: only DVi may be

updated

Revise(Vi, Vj)

1. revised nil

2. x DVi

3. found nil

4. y DVj

5. If Check((Vi,x),(Vj,y)) Then Begin

6. found t

7. Break8. End

9. If found=nil Then Begin

10. revised t

11. DVi DVi \ {x}

12. End

13. Return (revised)


Lookahead Schemas 6

Revise(Vi, Vj) Revise(Vi, Vj)

– revised nil x Dvi

– y DVj

– If Check((Vi,x),(Vj,y)) Then Break()

– revised t

– DVi DVi \ {x}

– Return (revised)

Simpler, equivalent code but not as obvious as the previous one


Lookahead Schemas 7

Domain filtering in lookahead

• Vc current variable

• Vf future variable

• {Vf} all future variables

• Revise(Vf, Vc)

• FC(Vc):

– Vf {Vf} connected to Vc

– Revise(Vf,Vc)

– If DVf ={} then return(nil)


Lookahead Schemas

Directional Arc Consistency

• Choose an ordering d, stick to it

• After instantiating a variable at level i, do the following– For k from i to (n-1) in the ordering d Do

– If FC(Vk)=nil then Return(nil)

8


Lookahead Schemas

Maintaining Arc Consistency• First, FC(Vc),

• If it does not fail, then, form a queue with all constraints (Vi,Vj) and (Vj,Vi) between future variables, and run AC

AC-1• Q {(Vi,Vj),(Vj,Vi), …, (Vk,Vm), (Vm,VK)}

• Change true• While Change Do• Change false

• For all (Va,Vb) in Q Do

• If Revise(Va,Vb)

• Then If Dom(Va) =

• Then Return (nil) • Else Change true

9


Lookahead Schemas 10

• FC:

• DAC:

– assumes a fixed variable ordering d

• MAC:

– does more pruning (search may visit fewer nodes) at the cost of more consistency checks

Look-ahead techniques: FC, DAC, MAC

FC(Vc); While not failure: For the next Vf in the ordering d, FC(Vf)

FC(Vc); AC({Vf})

FC(Vc)

FC(Vc);

Repeat until quiescence or failure

Vf1,Vf2 {Vf}, Revise(Vf1,Vf2)



Terminology overload alert: FC

• FC is used to denote any of the following:– a partial look-ahead schema – a specific chronological backtrack search

algorithm that uses the partial look-ahead schema

• Meaning is inferred from context• Not a healthy situation, but a fact of reality• Advice: state upfront the meaning of your terms

and stick to them throughout your paper



(BT Search +) MAC vs. FC

• Reference: [Sabin & Freuder, ECAI94], [Bessière & Régin, CP97], [Sabin & Freuder, CP97], [Gent & Prosser, APES-20-2000], [Experiments by Lin XU, 2001], [Yang, MS thesis 2003]

• Results: (sketchy)Low tightness High tightness

Low density (sparse) FC MAC

High density (dense) FC FC

Note: Results depend on

• Variable ordering (static vs. dynamic)

• Problem difficulty (positive relative to crossover point)