1. Lookahead Schemas Foundations of Constraint Processing
CSCE496/896, Fall 2011
All questions to
Berthe Y. Choueiry (Shu-we-ri)
Avery Hall, Room 360
Tel: +1(402)

2. Outline Looking ahead Schemas Forward checking (FC)
Directional Arc Consistency (DAC)
Maintaining Arc Consistency (a.k.a. full arc-consistency)

3. Looking ahead Rationale: Techniques
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

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

5. Revise(Vi, Vj) NOTE: only DVi may be updated
revised  nil
 x  DVi
found  nil
 y  DVj
If Check((Vi,x),(Vj,y)) Then Begin
found  t
Break
End
If found=nil Then Begin
revised  t
DVi  DVi \ {x}
Return (revised)

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

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)

8. 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)

9. 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

10. Look-ahead techniques: FC, DAC, MAC
assumes a fixed variable ordering d
MAC:
does more pruning (search may visit fewer nodes) at the cost of more consistency checks
FC(Vc)
FC(Vc);
While not failure: For the next Vf in the ordering d, FC(Vf)
FC(Vc); AC({Vf})
FC(Vc);
Repeat until quiescence or failure
 Vf1,Vf2  {Vf}, Revise(Vf1,Vf2)

11. 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

12. (BT Search +) MAC vs. FC
Reference: [Sabin & Freuder, ECAI94], [Bessière & Régin, CP97], [Sabin & Freuder, CP97], [Gent & Prosser, APES ], [Experiments by Lin XU, 2001], [Yang, MS thesis 2003]
Results: (sketchy)
Low tightness
High tightness
Low density (sparse) FC
MAC
High density (dense)
Note: Results depend on
Variable ordering (static vs. dynamic)
Problem difficulty (positive relative to crossover point)