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Constraint Satisfaction Constraint Satisfaction ProblemsProblems
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Intro Example: 8-QueensIntro Example: 8-Queens
Generate-and-test: 88 combinations
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Intro Example: 8-QueensIntro Example: 8-Queens
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Constraint Satisfaction Constraint Satisfaction ProblemProblem
Set of variables {X1, X2, …, Xn}Each variable Xi has a domain Di of possible values Usually Di is discrete and finite
Set of constraints {C1, C2, …, Cp} Each constraint Ck involves a subset of
variables and specifies the allowable combinations of values of these variables
Assign a value to every variable such that all constraints are satisfied
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Example: 8-Queens Example: 8-Queens ProblemProblem
8 variables Xi, i = 1 to 8 Domain for each variable {1,2,…,8} Constraints are of the forms: Xi = k Xj k for all j = 1 to 8, ji Xi = ki, Xj = kj |i-j| | ki - kj|
for all j = 1 to 8, ji
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Example: Map ColoringExample: Map Coloring
• 7 variables {WA,NT,SA,Q,NSW,V,T}• Each variable has the same domain {red, green, blue}• No two adjacent variables have the same value: WANT, WASA, NTSA, NTQ, SAQ, SANSW, SAV,QNSW, NSWV
WA
NT
SA
Q
NSWV
T
WA
NT
SA
Q
NSWV
T
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Constraint GraphConstraint Graph
Binary constraints
T
WA
NT
SA
Q
NSW
V
Two variables are adjacent or neighbors if theyare connected by an edge or an arc
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Map ColoringMap Coloring
{}
WA=red WA=green WA=blue
WA=redNT=green
WA=redNT=blue
WA=redNT=greenQ=red
WA=redNT=greenQ=blue
WA
NT
SA
Q
NSWV
T
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Backtracking AlgorithmBacktracking Algorithm
CSP-BACKTRACKING(PartialAssignment a) If a is complete then return a X select an unassigned variable D select an ordering for the domain of X For each value v in D do
If v is consistent with a then Add (X= v) to a result CSP-BACKTRACKING(a) If result failure then return result
Return failure
CSP-BACKTRACKING({})
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QuestionsQuestions
1. Which variable X should be assigned a value next?
2. In which order should its domain D be sorted?
3. In which order should constraints be verified?
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Choice of VariableChoice of Variable
Map coloring
WA
NT
SA
Q
NSWV
T
WA
NT
SA
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Choice of VariableChoice of Variable
8-queen
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Choice of VariableChoice of Variable
Most-constrained-variable heuristic: Select a variable with the fewest
remaining values
= Fail First Principle
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Choice of VariableChoice of Variable
Most-constraining-variable heuristic: Select the variable that is involved in the largest
number of constraints on other unassigned variables= Fail First Principle again
WA
NT
SA
Q
NSWV
T
SA
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{}
Choice of ValueChoice of Value
WA
NT
SA
Q
NSWV
T
WA
NT
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Choice of ValueChoice of Value
Least-constraining-value heuristic: Prefer the value that leaves the largest subset
of legal values for other unassigned variables
{blue}
WA
NT
SA
Q
NSWV
T
WA
NT
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Choice of Constraint to Choice of Constraint to TestTest
Most-constraining-Constraint: Prefer testing constraints that are
more difficult to satisfy= Fail First Principle
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Constraint Propagation …Constraint Propagation …
… is the process of determining how the possible values of one variable affect the possible values of other variables
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Forward CheckingForward Checking After a variable X is assigned a value v,
look at each unassigned variable Y that is connected to X by a constraint and deletes from Y’s domain any value that is inconsistent with v
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Map ColoringMap Coloring
WA NT Q NSW V SA T
RGB RGB RGB RGB RGB RGB RGB
TWA
NT
SA
Q
NSW
V
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Map ColoringMap Coloring
WA NT Q NSW V SA T
RGB RGB RGB RGB RGB RGB RGB
R GB RGB RGB RGB GB RGB
TWA
NT
SA
Q
NSW
V
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WA NT Q NSW V SA T
RGB RGB RGB RGB RGB RGB RGB
R GB RGB RGB RGB GB RGB
R B G RB RGB B RGB
Map ColoringMap Coloring
TWA
NT
SA
Q
NSW
V
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Map ColoringMap Coloring
WA NT Q NSW V SA T
RGB RGB RGB RGB RGB RGB RGB
R GB RGB RGB RGB GB RGB
R B G RB RGB B RGB
R B G R B RGB
Impossible assignments that forward checking do not detect
TWA
NT
SA
Q
NSW
V
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Example: 4-Queens Problem
1
3
2
4
32 41
X1{1,2,3,4}
X3{1,2,3,4}
X4{1,2,3,4}
X2{1,2,3,4}
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Example: 4-Queens Problem
1
3
2
4
32 41
X1{1,2,3,4}
X3{1,2,3,4}
X4{1,2,3,4}
X2{1,2,3,4}
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Example: 4-Queens Problem
1
3
2
4
32 41
X1{1,2,3,4}
X3{ ,2, ,4}
X4{ ,2,3, }
X2{ , ,3,4}
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Example: 4-Queens Problem
1
3
2
4
32 41
X1{1,2,3,4}
X3{ ,2, ,4}
X4{ ,2,3, }
X2{ , ,3,4}
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Example: 4-Queens Problem
1
3
2
4
32 41
X1{1,2,3,4}
X3{ , , , }
X4{ ,2,3, }
X2{ , ,3,4}
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Example: 4-Queens Problem
1
3
2
4
32 41
X1{ ,2,3,4}
X3{1,2,3,4}
X4{1,2,3,4}
X2{1,2,3,4}
BTBT
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Example: 4-Queens Problem
1
3
2
4
32 41
X1{ ,2,3,4}
X3{1, ,3, }
X4{1, ,3,4}
X2{ , , ,4}
BTBT
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Example: 4-Queens Problem
1
3
2
4
32 41
X1{ ,2,3,4}
X3{1, ,3, }
X4{1, ,3,4}
X2{ , , ,4}
BTBT
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Example: 4-Queens Problem
1
3
2
4
32 41
X1{ ,2,3,4}
X3{1, , , }
X4{1, ,3, }
X2{ , , ,4}
BTBT
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Example: 4-Queens Problem
1
3
2
4
32 41
X1{ ,2,3,4}
X3{1, , , }
X4{1, ,3, }
X2{ , , ,4}
BTBT
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Example: 4-Queens Problem
1
3
2
4
32 41
X1{ ,2,3,4}
X3{1, , , }
X4{ , ,3, }
X2{ , , ,4}
BTBT
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Example: 4-Queens Problem
1
3
2
4
32 41
X1{ ,2,3,4}
X3{1, , , }
X4{ , ,3, }
X2{ , , ,4}
BTBT
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Edge Labeling Edge Labeling in Computer in Computer VisionVision
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Trihedral Objects
Objects in which exactly three plane surfaces come together at each vertex. Goal: label a 2-D object to produce a 3-D object
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Labels of Edges
Convex edge: two surfaces intersecting at an angle greater than
180°
Concave edge two surfaces intersecting at an angle less than 180°
+ convex edge, both surfaces visible− concave edge, both surfaces visible convex edge, only one surface is visible and it is on the right side of
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Junction Label SetsJunction Label Sets
+ + --
-- - + +
++ +
+
+
--
--
-+
(Waltz, 1975; Mackworth, 1977)
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Edge LabelingEdge Labeling
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Edge LabelingEdge Labeling
+
++
+
+
+
+
+
++
--
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Edge Labeling as a CSPEdge Labeling as a CSP
A variable is associated with each junctionThe domain of a variable is the label set of the corresponding junctionEach constraint imposes that the values given to two adjacent junctions give the same label to the joining edge
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Edge LabelingEdge Labeling
+ -
+
-+- -
++
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Edge LabelingEdge Labeling +
+
+
+---
-- -
+
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Edge LabelingEdge Labeling
+
+
+
++
+
-- - + +
++
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Edge LabelingEdge Labeling
++
+
- -++
+ + --
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Removal of Arc Removal of Arc InconsistenciesInconsistencies
REMOVE-ARC-INCONSISTENCIES(J,K)removed falseX label set of JY label set of KFor every label y in Y do If there exists no label x in X such that the
constraint (x,y) is satisfied then Remove y from Y If Y is empty then contradiction true removed true
Label set of K YReturn removed
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CP Algorithm for Edge CP Algorithm for Edge LabelingLabeling
Associate with every junction its label set Q stack of all junctions while Q is not empty do J UNSTACK(Q) For every junction K adjacent to J do
If REMOVE-ARC-INCONSISTENCIES(J,K) then If K’s domain is non-empty then
STACK(K,Q) Else return false
(Waltz, 1975; Mackworth, 1977)