Sweep as a Generic Pruning Technique Applied to Constraint - - PowerPoint PPT Presentation

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Sep 06, 2023 157 likes •338 views

Sweep as a Generic Pruning Technique Applied to Constraint Relaxation Nicolas Beldiceanu and Mats Carlsson SICS Lgerhyddsvgen 18 75237 Uppsala email: nicolas@sics.se Soft01, Paphos, Cyprus Outline of the Presentation

SLIDE 1

Soft’01, Paphos, Cyprus

Sweep as a Generic Pruning Technique Applied to Constraint Relaxation Nicolas Beldiceanu and Mats Carlsson SICS

Lägerhyddsvägen 18 75237 Uppsala email: nicolas@sics.se

SLIDE 2

Outline of the Presentation

INTRODUCTION

Sweep Algorithms in Computational Geometry
Main Ideas of Sweep Algorithms

CARDINALITY OPERATOR

Representing an Elementary Constraint: Forbidden and Safe Regions
The Sweep Algorithm

CONCLUSION

SLIDE 3

Sweep Algorithms in Computational Geometry

Standard technique in the design of efficient algorithms, described in:

Computational geometry, an introduction

[Preparata & Shamos, 1985]

Computational Geometry, Algorithms and Applications

[Berg, Kreveld, Overmars & Schwarzkopf, 1997]

Géométrie algorithmique

[Boissonnat & Yvinec, 1995]

SLIDE 4

Applications of Sweep Algorithms

Within the Geometry Literature Database, more than 100 references:

Voronoi diagram
Map overlay
Nearest objects
Triangulations
Hidden surface removals
Rectangles intersection
Shortest path

But not yet common within constraint programming !

SLIDE 5

Applications of Sweep Algorithms within Constraint Programming

Pruning for the following constraint patterns:

A conjunction of constraints with two shared variables

TRICS, CP’01 [Beldiceanu,Carlsson]

The cardinality operator with two shared variables

Soft’01 [Beldiceanu,Carlsson]

The non-overlapping constraint between polygons

CP’01 [Beldiceanu,Guo,Thiel]

A multi-resource cumulatives constraint

http://www.sics.se/libindex.html [Beldiceanu,Carlsson]

Filtering algorithms for tabular constraints

CICLOPS’01 [Bartak]

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Main Ideas of Sweep Algorithms

(in the context of line segment intersection) y x

event point sweep line sweep line status (1) (2)

Steps of the sweep algorithm:

(1) Move to the next event point (2) Update the sweep line status:

start events:
end events:

GOAL: the worst case complexity should also depend of the number of intersections

SLIDE 7

A Restricted Case of the cardinality Operator

The cardinality operator, Van Hentenryck, P., Deville, Y. (ICLP 1991): : C = ∑ #CTRj(V1,.., Vm ) A restricted case of the cardinality operator :

j=1 n

Definition
Pruning
Restriction
Pruning

: 2 variables X and Y occur in each constraint CTRj

j

considers interaction between constraints (through the shared variables) : : based on counting entailment

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Forbidden and Safe Regions

Two intervals inf_x..sup_x and inf_y..sup_y such that: ∀ x ∈ inf_x..sup_x, ∀ y ∈ inf_y..sup_y: Ctr with the assignment X=x and Y=y is false. DEFINITION forbidden region according to a constraint Ctr and 2 variables X,Y of Ctr : Two intervals inf_x..sup_x and inf_y..sup_y such that: ∀ x ∈ inf_x..sup_x, ∀ y ∈ inf_y..sup_y: Ctr with the assignment X=x and Y=y is true. DEFINITION safe region according to a constraint Ctr and 2 variables X,Y of Ctr :

X Y

0 1 2 3 4 1 2 3 4

0 ≤ X ≤ 4 0 ≤ Y ≤ 4 1 ≤ S ≤ 6 X + 2Y ≤ S

EXAMPLE

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Examples of Forbidden and Safe Regions

(A) (B) (C) (D) (E)

0≤X≤4 0≤Y≤4 0≤R≤9 X Y 0 1 2 3 4 1 2 3 4 X Y 0 1 2 3 4 1 2 3 4 X Y 0 1 2 3 4 1 2 3 4 0≤X≤4 0≤Y≤4 2≤Z≤3 0≤X≤4 0≤Y≤4 0≤T≤0 1≤U≤2 X Y 0 1 2 3 4 1 2 3 4 0≤X≤4 0≤Y≤4 alldifferent( |X-Y|>Z X Y 0 1 2 3 4 1 2 3 4 0≤X≤4 0≤Y≤4 1≤S≤6 X+2Y≤S X+1≤T ∨ T+1≤X ∨ Y+1≤U ∨ U+4≤Y X+Y≡0 (mod 2) {X,Y,4-Y,R})

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0 1 2 3 4 1 2 3 4

get_first_regions(X,Y,Ctr) get_next_regions(X,Y,Ctr,Previous) get_last_regions(X,Y,Ctr) get_prev_regions(X,Y,Ctr,Previous)

Primitives for Getting Forbidden/Safe Regions on Request

X Y get_first_regions X Y get_next_regions X Y get_next_regions max(Y) X Y min(X) max(X) min(Y) X+Y≡0 (mod 2) 0 1 2 3 4 1 2 3 4 0 1 2 3 4 1 2 3 4

0 1 2 3 4

1 2 3 4

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Sweep Line Status

For each y ∈ dom(Y):

number of safe regions
number of forbidden regions

containing the point (Δ,y) Y X

Δ

sweep line status Y

1,0 1,1 0,1 0,1 1,0 1,0

X Remove a value Δ ∈ dom(X) if for all y ∈ dom(Y): nsafe[y]..nctr−nforbidden[y] ∩ C = ∅ Forbidden regions Safe regions

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alldifferent({X,Y,4-Y,R})

An Example

0≤X≤4 0≤Y≤4 2≤Z≤3 1≤S≤6 0≤T≤0 1≤U≤2 alldifferent({X,Y,4-Y, R}) |X-Y|>Z X+2Y≤S X+1≤T ∨ T+1≤X ∨ Y+1≤U ∨ U+4≤Y X+Y≡0 (mod 2)

PROBLEM:

Adjust minimum of X according to Y and to the fact that 4 or 5 constraints should hold: |X-Y|>Z X+2Y≤S X+1≤T ∨ T+1≤X ∨ Y+1≤U ∨ U+4≤Y X+Y≡0 (mod 2) Y X=0

1 2 3 4

Y

1 2 3 4

X=2 Y

1 2 3 4

X=1

Deduction: X>1

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Possible Utilisations

Feasibility check
Adjusting bounds
Pruning values

X Y X Y X Y

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Typical Constraint Structure for Applying Sweep

A R G U M E N T(S)
R E S T R I C T I O N (S)
V E R T E X I N P U T
V E R T E X G E N E R A T O R
E D G E I N P U T
E D G E G E N E R A T O R
E D G E A R I T Y
E D G E C O N S T R A I N T
G R A P H P R O P E R T Y

: COUNT : dvar : OBJECTS: collection(X-dvar,Y-dvar,...) : required(OBJECTS.X,OBJECTS.Y) : OBJECTS : IDENTITY : OBJECTS : CLIQUE(≠) : 2 : : NEDGE = COUNT

X1,Y1 X2,Y2 X3,Y3 X4,Y4

Non-overlapping Scheduling with set-up Cyclic scheduling

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Summary

Conjunction

f constraints

Cardinality

perator

Sweep axis Event points Sweep status

domain of the

variable to prune

start of forbidden regions
end of forbidden regions
# of forbidden regions
# of forbidden regions
# of safe regions
contradiction
start of forbidden regions
end of forbidden regions
start of safe regions
end of safe regions
domain of the

variable to prune

SLIDE 16

Conclusion

Yet another use of sweep algorithms

(sweep algorithms were already used for a lot of different purpose)

Combine generality (forbidden, safe regions) with a specific algorithm (sweep)

(lead to a generic class of propagation algorithms and to open global constraints)

Worst case complexity of the algorithm could perhaps be improved