1 Constraint Satisfaction Problems. 2 Intro Example: 8-Queens Generate-and-test: 8 8 combinations

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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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8-queen

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Most-constrained-variable heuristic: Select a variable with the fewest

remaining values

= Fail First Principle

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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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WA NT Q NSW V SA T

RGB RGB RGB RGB RGB RGB RGB

TWA

NT

SA

Q

NSW

V

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WA NT Q NSW V SA T


R GB RGB RGB RGB GB RGB

TWA

NT

SA

Q

NSW

V

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WA NT Q NSW V SA T



R B G RB RGB B RGB


TWA

NT

SA

Q

NSW

V

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WA NT Q NSW V SA T



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

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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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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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1

3

2

4

32 41

X1{1,2,3,4}

X3{ ,2, ,4}

X4{ ,2,3, }

X2{ , ,3,4}

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1

3

2

4

32 41

X1{1,2,3,4}

X3{ ,2, ,4}

X4{ ,2,3, }

X2{ , ,3,4}

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1

3

2

4

32 41

X1{1,2,3,4}

X3{ , , , }

X4{ ,2,3, }

X2{ , ,3,4}

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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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1

3

2

4

32 41

X1{ ,2,3,4}

X3{1, ,3, }

X4{1, ,3,4}

X2{ , , ,4}

BTBT

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1

3

2

4

32 41

X1{ ,2,3,4}

X3{1, ,3, }

X4{1, ,3,4}

X2{ , , ,4}

BTBT

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1

3

2

4

32 41

X1{ ,2,3,4}

X3{1, , , }

X4{1, ,3, }

X2{ , , ,4}

BTBT

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1

3

2

4

32 41

X1{ ,2,3,4}

X3{1, , , }

X4{1, ,3, }

X2{ , , ,4}

BTBT

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1

3

2

4

32 41

X1{ ,2,3,4}

X3{1, , , }

X4{ , ,3, }

X2{ , , ,4}

BTBT

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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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+

++

+

+

+

+

+

++

--

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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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+ -

+

-+- -

++

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Edge LabelingEdge Labeling +

+

+

+---

-- -

+

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+

+

+

++

+

-- - + +

++

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++

+

- -++

+ + --

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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)

Documents

1 Constraint Satisfaction Problems. 2 Intro Example: 8-Queens Generate-and-test: 8 8 combinations