FC for Non-Binary CSPs
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
[email protected]
Tel: +1(402)472-5444
Foundations of Constraint Processing, Spring 2008
April 9, 2008
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Recommended reading
• (short, preliminary version) On forward
checking for non-binary constraint
satisfaction, by Christian Bessière, Pedro Meseguer,
Eugene C Freuder, Javier Larrosa, CP 1999
• (longer, more complete version) On forward
checking for non-binary constraint
satisfaction, by Christian Bessière, Pedro Meseguer,
Eugene C Freuder, Javier Larrosa, AIJ 2002
Foundations of Constraint Processing, Spring 2008
April 9, 2008
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Non-binary FC
• Definitions
• Example
• Properties
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nFC0 and nFC1
Cc,1: Constraints involving the current
variable and exaclty one future variable
Cc,1: Set of constraint projections involving
the current variable and exactly one
future variable
nFC0: Apply AC on each constraint in Cc,1
(one pass)
nFC1: Apply AC to each constraint in Cc,1
and Cc,1 (one pass)
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nFC2, nFC3, nFC4, nFC5
Cc,f: set of constraints involving the current
variable and at least one future variable
Cp,f: set of constraints involving at least one
past variable and at least one future
variable
nFC2: Apply AC to each constraint in Cc,f
nFC3: Make Cc,f arc-consistent
nFC4: Apply AC to each constraint in Cp,f
nFC5: Make Cp,f arc-consistent
Foundations of Constraint Processing, Spring 2008
April 9, 2008
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Non-binary FC
• Definitions
• Example
• Properties
Foundations of Constraint Processing, Spring 2008
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Filtering for x  a
c1
a, b, c
a, b, c
c3
c2
a, b, c
a, b, c
a, b, c
a, b, c
Xa
x
y
z
u
v
w
c1
c2
c3
x y z x y z x y z
a a a a a a a a a
a b c a b b a b c
a c b c c c
• nFC0 does no filtering
• nFC1 applies AC on c1 {x,y}, {x.z}, and
and c3 on {x,y} and {x,w}. It removes c
from D(y), b from D(w)
• nFC2 applies AC on c1, then on c3.
Same pruning as nFC1. A different
ordering of constraints yields different
filtering
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Filtering for x  a
c1
a, b, c
a, b, c
c3
c2
a, b, c
a, b, c
a, b, c
a, b, c
Xa
x
y
z
u
v
w
c1
c2
c3
x y z x y z x y z
a a a a a a a a a
a b c a b b a b c
a c b c c c
• nFC3 applies AC on {c1,c3}. Same
filtering as nfC2, but also, removes b
from D(z).
• nFC4 applied AC on c1 then c3 (like
nFC2). Same filtering as nFC2 because
x is the first variable.
• nfC5 yields the same filtering as nFC3
because x is the first variable.
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Filtering for u  a
c1
a, b, c
a, b, c
c3
c2
a, b, c
a, b, c
a, b, c
a, b, c
Xa
x
y
z
u
v
w
c1
c2
c3
x y z x y z x y z
a a a a a a a a a
a b c a b b a b c
a c b c c c
• nFC0 does no filtering
• nFC1 applies AC on c1 on {u,v}, {u,w}.
It removes c from D(v), c from D(w)
• nFC2 applies AC on c2. It removes b
and c from D(v) and c from D(w)
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Filtering for u  a
c1
a, b, c
a, b, c
c3
c2
a, b, c
a, b, c
a, b, c
a, b, c
ua
x
y
z
u
v
w
c1
c2
c3
x y z x y z x y z
a a a a a a a a a
a b c a b b a b c
a c b c c c
• nFC3 applies AC on {c2}. Same filtering
as nFC2
• nFC4 applies AC on c1, c2, then c3. It
removes b from D(y) and D(z), b and c
from D(v) and c from D(w).
• nfC5 does AC on {c1, c2, c3}. It removes
b from D(y), c from D(z), b and c from
D(v), and c from D(w)
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Non-binary FC
• Definitions
• Example
• Properties
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Filtering effectiveness
Let (nFCi, k) be the set of (x,a) where a is
removed from D(x)
• (nFC0,k)  (nFC1,k)  (nFC2,k)
• (nFC2,k)  (nFC3,k)  (nFC5,k)
• (nFC2,k)  (nFC4,k)  (nFC5,k)
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Nodes visited by nFCi
•
•
•
•
nodes(nFC2,k)  nodes(nFC1,k)  nodes(nFC0,k)
nodes(nFC5k)  nodes(nFC3,k)  nodes(nFC2,k)
nodes(nFC5,k)  nodes(nFC4,k)  nodes(nFC2,k)
nFC1 visits exactly the same nodes as FC+ on the
hidden variable representation
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