Foundations of Constraint Processing 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@cse.unl.edu Tel: +1(402)472-5444

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

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

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)

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

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)

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)

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