[PPT] - Chapter 20: Working with Implications Helmut Simonis Cork PowerPoint Presentation

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Problem Program Improvements

Chapter 20: Working with Implications

Helmut Simonis

Cork Constraint Computation Centre Computer Science Department University College Cork Ireland

ECLiPSe ELearning

Overview Helmut Simonis Shikaku 1

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Problem Program Improvements

Licence

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Problem Program Improvements

Outline

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Problem

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Program

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Improvements

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Problem Program Improvements

What we want to introduce

Solving a placement problem without specialized constraints Decomposition into pattern generation and set partitioning Using implications to propagate information

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Problem Program Improvements

Outline

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Problem Program Improvements

Problem Definition

Shikaku The puzzle is played in a given recti-linear grid. Some grid cells contain numbers. The task is to partition the grid area into rectangular rooms satisfying the conditions:

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Each room contains exactly one number.

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The area of the room is equal to the number in it.

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Problem Program Improvements

Solution Approaches

The right way The wrong way

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Problem Program Improvements

The Right Way (I)

Using one geost constraint (Carlsson, Beldiceanu et al.) Each room is an object with fixed surface Shapes defined by Width × Height = Surface Each room has x, y position and size w, h Each room must contain cell with hint Rooms must fit into given space Rooms do not overlap

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Problem Program Improvements

The Right Way (II)

Using regular (Lagerkvist, Pesant) 0/1 Variables xijk cell (i, j) belongs to room k Each cell must belong to exactly one room (equality) Use regular expression to describe possible shapes Disjunction of possible placements One regular constraint for each room Possible to combine regular expressions for multiple rooms

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Problem Program Improvements

The Right Way (III)

SMT?

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Problem Program Improvements

The Wrong Way (Naive)

Each room k has position xk, yk and size wk,hk Room k has fixed surface Sk = wk ∗ hk Room k contains fixed hint at Uk, Vk

xk ≤ Uk < xk + wk yk ≤ Vk < yk + hk

Any rooms k and l do not overlap

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Problem Program Improvements

Expressing Non-Overlap with Disjunctive

X2, Y2 W2 H2 X1, Y1 W1 H1

left X2 ≥ X1 + W1 Y1 ≥ Y2 + H2 above X1 ≥ X2 + W2 right Y2 ≥ Y1 + H1 below

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Problem Program Improvements

Problems

Weak propagation Width and Height not in sync Estimate for room surface much too low Disjunctive does not propagate much

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Problem Program Improvements

Idea

Problem decomposition

Phase 1: Generate possible placement pattern for each room Phase 2: Pick exactly one pattern for each room

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Problem Program Improvements

Phase 1: Pattern Generation

Solve with finite domains For each room, solve

Each room k has position xk, yk and size wk,hk Room k has fixed surface Sk = wk ∗ hk Room k contains fixed hint at Uk, Vk

xk ≤ Uk < xk + wk yk ≤ Vk < yk + hk

All other hints are outside the room

Expressed as negation of inside

Find all solutions

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Problem Program Improvements

Phase 1 Results

Possible placements for each room 1 ≤ p ≤ Pk Position xkp, ykp, size wkp,hkp Predicate q(k,p,i,j) pattern p of room k contains cell (i, j)

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Problem Program Improvements

Phase 2: Set Partitioning

0/1 integer variable zkp if pattern p is used for room k Select one pattern per room Cover every cell with exactly one pattern Constrain all variables which belong to pattern which contain cell

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Problem Program Improvements

Model Phase 2

solve zkp ∈ {0, 1} ∀k ∈ K :

1≤p≤Pk

zkp = 1 ∀(i, j) ∈ Rect :

{k,p|q(k,p,i,j)}

zkp = 1

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Problem Program Improvements

Model Phase 2

solve zkp ∈ {0, 1} ∀k ∈ K :

1≤p≤Pk

zkp = 1 ∀(i, j) ∈ Rect :

{k,p|q(k,p,i,j)}

zkp = 1 The first set of equations is subsumed by the second

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Problem Program Improvements

Intuition Behind Constraints

If only one pattern remains possible for a room, this must be chosen If a pattern is selected for a room, no other pattern can be selected Every cell must be covered by a pattern If a pattern covering some cell is selected, no other pattern covering the same cell may be selected

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Problem Program Improvements

Outline

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Problem

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Program

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Problem Program Improvements

Top Level

:-module(pure). :-export(top/0). :-use_module(structures). :-use_module(data). :-use_module(utility). :-lib(ic). top:- data(Set,Nr,Grade,X,Y,Matrix), solve(Set,Nr,Grade,X,Y,Matrix).

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Problem Program Improvements

Structure Definitions

:-module(structures). :-export struct(hint(i,j,n,d)). :-export struct(rectangle(x,y,w,h, n, % size of rectangle k, % hint nr, links alternative rectangles cnt, % id of rectangle var % 0/1 variable, design used )). :-export struct(overlap(point,cnt,k,var)).

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Problem Program Improvements

Sample Data

:-module(data). :-export(data/6). data(’nikoli-web’ ,0 ,easy,10 ,10 , []( [](x ,8 ,4 ,x ,x ,x ,x ,4 ,x ,x ), [](x ,x ,x ,x ,x ,x ,x ,6 ,x ,x ), [](x ,x ,x ,3 ,3 ,x ,x ,x ,x ,x ), [](x ,x ,x ,x ,x ,x ,2 ,x ,x ,x ), [](5 ,4 ,x ,x ,x ,x ,2 ,x ,x ,x ), [](x ,x ,x ,9 ,x ,x ,x ,x ,6 ,7 ), [](x ,x ,x ,8 ,x ,x ,x ,x ,x ,x ), [](x ,x ,x ,x ,x ,4 ,5 ,x ,x ,x ), [](x ,x ,4 ,x ,x ,x ,x ,x ,x ,x ), [](x ,x ,5 ,x ,x ,x ,x ,5 ,6 ,x ) )).

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Problem Program Improvements

Main Program (I)

solve(Set,Nr,Grade,X,Y,Matrix):- writeln(solving(Set,Nr,Grade)), (multifor([I,J],[1,1],[X,Y]), fromto([],A,A1,Hints), fromto(0,B,B1,_Types), param(Matrix) do hint_cell(Matrix,I,J,A,A1,B,B1) ), create_rectangles(Hints,[],[], Rectangles,X,Y), numbering(Rectangles), extract_vars(Rectangles,var of rectangle,List), List :: 0..1, ...

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Problem Program Improvements

Main Program (II)

... create_overlap(Rectangles,Overlap), group_by(point of overlap,Overlap,Grouped), (foreach(_-Group,Grouped) do extract_vars(Group,var of overlap,List), sum(List) #= 1 ), search(List,0,input_order,indomain, complete,[]), writeln(List).

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Problem Program Improvements

Extracting Hints

hint_cell(Matrix,I,J, A,[hint{i:I,j:J,n:N,d:D1}|A], D,D1):- subscript(Matrix,[I,J],N), integer(N), !, D1 is D+1. hint_cell(_Matrix,_I,_J,A,A,B,B).

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Problem Program Improvements

Creating Pattern

create_rectangles([],_,Rectangles,Rectangles, _Width,_Height). create_rectangles([Hint|Hints],Old,RIn,ROut, Width,Height):- findall(Rectangle, rectangle(Hint,Hints,Old,Width,Height, Rectangle), Rectangles), append(Rectangles,RIn,R1), create_rectangles(Hints,[Hint|Old],R1,ROut, Width,Height).

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Problem Program Improvements

Phase 1 FD Model

rectangle(hint{i:I,j:J,n:N,d:K},Hints,Old, Width,Height, rectangle{x:X,y:Y,w:W,h:H,n:N,k:K}):- X :: 1..Width, Y :: 1..Height, W :: 1..N, H :: 1..N, W*H #= N, X+W-1 #=< Width, Y+H-1 #=< Height, inside(X,Y,W,H,I,J),

utsides(X,Y,W,H,Hints),
utsides(X,Y,W,H,Old),

search([X,Y,W,H],0,input_order,indomain, complete,[]).

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Problem Program Improvements

Phase 1 FD Model Utilities

inside(X,Y,W,H,I,J):- I #>= X, J #>= Y, I #< X+W, J #< Y+H.

utsides(X,Y,W,H,L):-

(foreach(hint{i:I,j:J},L), param(X,Y,W,H) do

utside(X,Y,W,H,I,J)

).

utside(X,Y,W,H,I,J):-

I #< X or J #< Y or I #>=X+W or J #>= Y+H.

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Problem Program Improvements

Creating Overlap Structures

create_overlap(Rectangles,Overlap):- (foreach(Rect,Rectangles), fromto([],A,A1,Overlap) do create_overlap1(Rect,A,A1) ). create_overlap1(rectangle{x:X,y:Y,w:W,h:H, cnt:Cnt,k:K,var:Var},In,Out):- (multifor([I,J],[X,Y],[X+W-1,Y+H-1]), fromto(In,A, [overlap{point:point(I,J), cnt:Cnt,k:K,var:Var}|A],Out), param(Cnt,Var,K) do true ).

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Problem Program Improvements

Utility

extract_vars(Structures,Arg,List):- (foreach(Struct,Structures), param(Arg), foreach(X,List) do arg(Arg,Struct,X) ). numbering(Rectangles):- (foreach(rectangle{cnt:C},Rectangles), count(C,1,_) do true ).

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Problem Program Improvements

Result After Constraint Setup

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Problem Program Improvements

Observation

Only small part of problem filled in Missing information Some cells must belong to a room regardless of pattern used

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Problem Program Improvements

Missing Information, Example

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Problem Program Improvements

Outline

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Problem Program Improvements

Adding View

Variables which indicate which room some cell belongs to 0/1 integer variables wijk, whether cell (i, j) belongs to room k Most wijk entries are zero, as room k does not cover cell (i, j) Each cell must belong to a room

∀(i, j) :

k∈K wijk = 1

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Problem Program Improvements

Channeling Constraints

A cell belongs to a room, if one of the pattern covering the cell is selected Linear constraint

∀(i, j, k) :

{p|q(k,p,i,j)} zkp = wijk

Implications

∀(k, p, i, j) s.t. q(k, p, i, j) : zkp ⇒ wi,j,k

Propagation is equivalent

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Problem Program Improvements

Result After Improvement

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Problem Program Improvements

Not What We Expected

Some cells are marked Others are still missing Cells marked can only be covered by one room Why is this happening?

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Problem Program Improvements

Missing Reasoning: Example

A, B, C ∈ {0, 1} A + B + C = 1 ∧ A + B = 1 ⇒ C = 0 This does not happen! Constraints only see variables, not other constraints We would need global constraint on sets of equations for this

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Problem Program Improvements

Implications Too Weak

We can add redundant constraints If a cell belongs to a room, no pattern for other rooms using that cell can be selected

∀(i, j, k) :

{k′,p′|q(k′,p′,i,j)∧k=k′} zk′p′ + wijk = 1

∀(i, j, k, k′, p′) s.t. q(k′, p′, i, j) : wijk ⇒ ¬zk′p′

If a cell belongs to a room, no pattern for this room which does not cover the cell can be selected

∀(i, j, k) :

{p|¬q(k,p,i,j)} zkp + wijk = 1

∀(i, j, k, p) s.t. ¬q(k, p, i, j) : wijk ⇒ ¬zkp

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Problem Program Improvements

Result After Adding Redundant Constraints

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Problem Program Improvements

Problem Solved?

More to be done Look through example problems Find cases where problem is not solved by propagation Try and find redundant constraints

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Problem Program Improvements

Example

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Problem Program Improvements

Reasoning on Space Required

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Helmut Simonis Shikaku 46

SLIDE 47

Problem Program Improvements

Space required for rooms 76, 94, 102, 107

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Helmut Simonis Shikaku 47

SLIDE 48

Problem Program Improvements

How to model this?

Another view: finite domain variables for room assignment Variables vij state that cell (i, j) belongs to room vij Connection to wijk variables via bool_channel(vij,[wij1,wij2,...,wij|K|]) constraints Each value must occur specified time gcc constraint with fixed counts

Helmut Simonis Shikaku 48

SLIDE 49

Problem Program Improvements

Improvement

gcc reasons on all hints together Just adding

(i,j) wijk = Sk does not do this

Helmut Simonis Shikaku 49

SLIDE 50

Problem Program Improvements

Deduction by gcc (partial)

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Helmut Simonis Shikaku 50

SLIDE 51

Problem Program Improvements

Result

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Helmut Simonis Shikaku 51

SLIDE 52

Problem Program Improvements

Unsolved Part

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Helmut Simonis Shikaku 52

SLIDE 53

Problem Program Improvements

What to do?

Pattern are pair-wise compatible with each other There is enough space But there is only one solution

Helmut Simonis Shikaku 53

SLIDE 54

Problem Program Improvements

Looking more closely

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104 100 104 100 105 100 105 100 104 100 105 104 100 105 100

Helmut Simonis Shikaku 54

SLIDE 55

Problem Program Improvements

Looking more closely

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104 100 104 100 105 100 105 100 104 100 105 104 100 105 100

Helmut Simonis Shikaku 55

SLIDE 56

Problem Program Improvements

Looking more closely

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104 100 104 100 105 100 105 100 104 100 105 104 100 105 100

Helmut Simonis Shikaku 56

SLIDE 57

Problem Program Improvements

Looking more closely

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104 100 104 100 105 100 105 100 104 100 105 104 100 105 100

Helmut Simonis Shikaku 57

SLIDE 58

Problem Program Improvements

As Equations

For one cell we have

a104 + c100 = 1

For the other cell we have

a104 + c100 + a105 = 1 Implies a105 = 0 Similar, a104 = 0 Therefore c100 = 1

Helmut Simonis Shikaku 58

SLIDE 59

Problem Program Improvements

All done? No, One Problem Still Open

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159 153 165 152 159 153 165 159 165 159 152 160 159 153 166 160 159 167 160 174 160 172 165 179 165 165 159 173 166 165 153 167 166 160 174 167 184 171 178 172 171 185 172 165 179 172 165 186 173 172 159 180 173 166 187 180 174 160 181 174 167 184 178 177 198 184 178 171 185 179 178 172

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186 180 179 180 179 173 187 181 180 160 182 181 174 188 182 189 182 191 184 177 198 185 184 171 185 184 178 192 185 184 172 186 185 179 193 187 186 180 193 187 186 180 188 187 181 194 188 187 174 195 188 182 189 188 182 191 184 198 192 178 199 193 192 185

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200 193 186 193 187 201 194 193 195 194 188 195 189 205 199 198 199 192 206 200 199 200 193 207 193 201 193 187 202 201 194 202 189 206 205 199 206 199 206 200 207 206 208 207 209 208 209 189

Helmut Simonis Shikaku 59

SLIDE 60

Problem Program Improvements

Also Open with geost (Image: N. Beldiceanu)

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Problem Program Improvements

Results (I)

Set Nr Grade X Y K Fix Alt Cov Ini Set Red GCC Sub SAC nikoli-web easy 10 10 20 3 80 6 nikoli-web 1 easy 10 10 14 2 81 5 nikoli-web 2 easy 10 10 16 1 83 2 nikoli-web 3 easy 10 10 28 3 96 12 nikoli-web 4 easy 10 18 44 15 78 78 nikoli-web 5 medium 10 18 38 3 115 21 nikoli-web 6 medium 10 18 36 1 210 3 153 50 nikoli-web 7 medium 14 24 68 4 293 16 nikoli-web 8 hard 14 24 70 8 380 17 nikoli-web 9 hard 20 36 128 4 872 8 196 nikoli-web 10 hard 20 36 120 1 986 1 774 627 36 36 giant easy 7 7 12 2 46 2 giant 1

35

21 149 1002 2 995 345 29 7 giant 2

35

21 189 888 5 855 324 giant 3

35

21 209 5 1023 6 830 583 256 256 133 giant 4

35

21 190 2 849 8 495 70 giant 5

35

21 137 1 990 3 929 657 giant 6

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21 160 2 900 5 323 113 giant 7

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21 135 4 1111 4 984 866 7 7 giant 8

35

21 140 1 979 3 943 646 Helmut Simonis Shikaku 61

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Problem Program Improvements

Results (II)

Set Nr Grade X Y K Fix Alt Cov Ini Set Red GCC Sub SAC giant 1701 hard 45 31 255 10 1731 25 617 178 giant 1702 hard 45 31 206 3 1727 11 1497 277 giant 1801 hard 45 31 189 12 1468 35 giant 1802 hard 45 31 276 11 1517 34 769 217 giant 1901 hard 45 31 188 8 1443 36 148 18 giant 1902 hard 45 31 213 4 1813 11 1449 1018 giant 2001 hard 45 31 221 16 1217 92 giant 2002 hard 45 31 278 6 1617 20 1259 758 Helmut Simonis Shikaku 62

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Problem Program Improvements

Why Model the Wrong Way?

We want to mimic human reasoning Human don’t have a built-in geost constraint Most humans use rules to describe solution process

If a cell can only be covered by one room, then it must be assigned to the room

Both views (decide between pattern and assign cell to room) are used by humans At some point people

Either invent special global constraints Or use SAC

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Problem Program Improvements

Conclusions

Shikaku - interesting little puzzle Solved by decomposition and reasoning on implications Could be solved by strong global constraint We use incremental process adding constraints as required Close to human solving method One open problem left, same for geost (M. Carlsson)

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