Foundations of Constraint Processing

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Lookahead Schemas

• Foundations of Constraint Processing
• CSCE421/821, Spring 2008
• www.cse.unl.edu/choueiry/S08-421-821/
• Berthe Y. Choueiry (Shu-we-ri)
• Avery Hall, Room 123B
• choueiry_at_cse.unl.edu
• Tel [phone]

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Outline

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

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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

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Revising 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)

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Revise(Vi, Vj)

• Revise(Vi, Vj)
• revised ? nil
• ? x ? DVi
• ? y ? DVj
• If ? y ? DVj with (x,y) ? C return()
• ElseIf revised ? t DVi
  ? DVi \ x
• return(revised)
• NOTE only DVi may be updated

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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(failure)

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Look-ahead techniques FC, DAC, MAC

• FC
• DAC
• 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)
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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

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(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)

• Note results depend on
• Variable ordering (static vs. dynamic)
• Problem difficulty (positive relative to
  crossover point)