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1 Constraint Satisfaction Problems

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2 Intro Example: 8-Queens Generate-and-test: 8 8 combinations

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3 Intro Example: 8-Queens

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4 Constraint Satisfaction Problem Set of variables {X 1, X 2, …, X n } Each variable X i has a domain D i of possible values Usually D i is discrete and finite Set of constraints {C 1, C 2, …, C p } Each constraint C k 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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5 Example: 8-Queens Problem 8 variables X i, i = 1 to 8 Domain for each variable {1,2,…,8} Constraints are of the forms: X i = k X j k for all j = 1 to 8, j i X i = k i, X j = k j |i-j| | k i - k j | for all j = 1 to 8, j i

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6 Example: 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: WA NT, WA SA, NT SA, NT Q, SA Q, SA NSW, SA V,Q NSW, NSW V WA NT SA Q NSW V T WA NT SA Q NSW V T

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7 Constraint Graph Binary constraints T WA NT SA Q NSW V Two variables are adjacent or neighbors if they are connected by an edge or an arc

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8 Map Coloring {} WA=redWA=greenWA=blue WA=red NT=green WA=red NT=blue WA=red NT=green Q=red WA=red NT=green Q=blue WA NT SA Q NSW V T

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9 Backtracking 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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10 Questions 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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11 Choice of Variable Map coloring WA NT SA Q NSW V T WA NTSA

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12 Choice of Variable 8-queen

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13 Choice of Variable Most-constrained-variable heuristic: Select a variable with the fewest remaining values = Fail First Principle

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14 Choice 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 NSW V T SA

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15 {} Choice of Value WA NT SA Q NSW V T WA NT

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16 Choice 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 NSW V T WA NT

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17 Choice of Constraint to Test Most-constraining-Constraint: Prefer testing constraints that are more difficult to satisfy = Fail First Principle

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18 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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19 Forward 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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20 Map Coloring WANTQNSWVSAT RGBRGBRGBRGBRGBRGBRGBRGBRGBRGBRGBRGBRGBRGB T WA NT SA Q NSW V

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21 Map Coloring WANTQNSWVSAT RGBRGBRGBRGBRGBRGBRGBRGBRGBRGBRGBRGBRGBRGB RGBGBRGBRGBRGBRGBRGBRGBGBGBRGBRGB T WA NT SA Q NSW V

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22 WANTQNSWVSAT RGBRGBRGBRGBRGBRGBRGBRGBRGBRGBRGBRGBRGBRGB RGBGBRGBRGBRGBRGBRGBRGBGBGBRGBRGB RBGRBRBRGBRGBBRGBRGB Map Coloring T WA NT SA Q NSW V

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23 Map Coloring WANTQNSWVSAT RGBRGBRGBRGBRGBRGBRGBRGBRGBRGBRGBRGBRGBRGB RGBGBRGBRGBRGBRGBRGBRGBGBGBRGBRGB RBGRBRBRGBRGBBRGBRGB RBGRBRGBRGB Impossible assignments that forward checking do not detect T WA NT SA Q NSW V

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24 Example: 4-Queens Problem 1 3 2 4 3241 X1 {1,2,3,4} X3 {1,2,3,4} X4 {1,2,3,4} X2 {1,2,3,4}

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25 Example: 4-Queens Problem 1 3 2 4 3241 X1 {1,2,3,4} X3 {1,2,3,4} X4 {1,2,3,4} X2 {1,2,3,4}

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26 Example: 4-Queens Problem 1 3 2 4 3241 X1 {1,2,3,4} X3 {,2,,4} X4 {,2,3, } X2 {,,3,4}

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27 Example: 4-Queens Problem 1 3 2 4 3241 X1 {1,2,3,4} X3 {,2,,4} X4 {,2,3, } X2 {,,3,4}

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28 Example: 4-Queens Problem 1 3 2 4 3241 X1 {1,2,3,4} X3 {,,, } X4 {,2,3, } X2 {,,3,4}

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29 Example: 4-Queens Problem 1 3 2 4 3241 X1 {,2,3,4} X3 {1,2,3,4} X4 {1,2,3,4} X2 {1,2,3,4} BT

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30 Example: 4-Queens Problem 1 3 2 4 3241 X1 {,2,3,4} X3 {1,,3, } X4 {1,,3,4} X2 {,,,4} BT

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31 Example: 4-Queens Problem 1 3 2 4 3241 X1 {,2,3,4} X3 {1,,3, } X4 {1,,3,4} X2 {,,,4} BT

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32 Example: 4-Queens Problem 1 3 2 4 3241 X1 {,2,3,4} X3 {1,,, } X4 {1,,3, } X2 {,,,4} BT

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33 Example: 4-Queens Problem 1 3 2 4 3241 X1 {,2,3,4} X3 {1,,, } X4 {1,,3, } X2 {,,,4} BT

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34 Example: 4-Queens Problem 1 3 2 4 3241 X1 {,2,3,4} X3 {1,,, } X4 {,,3, } X2 {,,,4} BT

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35 Example: 4-Queens Problem 1 3 2 4 3241 X1 {,2,3,4} X3 {1,,, } X4 {,,3, } X2 {,,,4} BT

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36 Edge Labeling in Computer Vision

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37 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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38 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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39 Junction Label Sets ++ - - - -- ++ ++ + + + - - - - - + (Waltz, 1975; Mackworth, 1977)

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40 Edge Labeling

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41 Edge Labeling + + + + + + + + + + - -

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42 Edge Labeling as a CSP A variable is associated with each junction The domain of a variable is the label set of the corresponding junction Each constraint imposes that the values given to two adjacent junctions give the same label to the joining edge

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43 Edge Labeling + - + - + - - + +

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44 Edge Labeling + + + + - - - - - - +

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45 Edge Labeling + + + + + + - -- ++ ++

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46 Edge Labeling + + + -- ++ ++ - -

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47 Removal of Arc Inconsistencies REMOVE-ARC-INCONSISTENCIES(J,K) removed false X label set of J Y label set of K For 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 Y Return removed

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48 CP Algorithm for Edge Labeling 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)

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