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/21IntCP 2005 - Sitges

Using interval analysis to generate quad-trees of piecewise constraints

É. Vareilles, M. Aldanondo, P. Gaborit, K. Hadj-Hamou

October, the 1rst 2005

European project VHT n° G1RD-CT-2002-00835


Summary

• Need of piecewise constraints• General definition of a quad-tree

• Definition• Example

• Generation of quad-tree of piecewise constraints• Definition of a piecewise constraint• Definition of particular information degrees• Algorithm of generation• Example


Need of piecewise constraints

Take into account experimental graphs in constraints-based models.

Quad-trees were extended to piecewise constraints.


Summary

• Need of piecewise constraint• General definition of a quad-tree


• Generation of quad-tree of piecewise constraints• Definition of a piecewise constraint• Definition of the information degrees• Algorithm of generation• Example


Quad-tree example

example : y - x3 0 with x = 0.0625 and y = 0.0625

[-2, 2]

[-2, 2]

Root

Grey

[-2, 0]

[0, 2]

NW

White[-2, 0]

[-2, 0]

SW

Grey[0, 2]

[-2, 0]

SE

Grey[0, 2]

[0, 2]

NE

Grey

[-2, -1]

[-1, 0]

NW

White[-2, -1]

[-2, -1]

SW

Grey[-1, 0]

[-2, -1]

SE

Grey[-1, 0]

[-1, 0]

NE

Grey


Quad-tree principle : (Sam-Haroud, 1995)

– Hierarchical data structure– Based on a recursive decomposition of the search area in coherent and incoherent

regions

Quad-tree definition : (Sam-Haroud, 1995)

– Quad-tree associated to the constraint C(x,y) defined on (Dx, Dy):• Each node is defined on a sub-region (dn

x, dny).

• Each node is constrained by C(x,y).

• The consistency of each node is determined and coloured : white, blue, grey

• Each grey node has four children (NW, NE, SW, SE)

• Each variable has a decomposition precision (x for x and y for y) which defines the size of the unitary nodes.

• When one of the decomposition precision is reached, unitary grey nodes turn white.

Definition of a quad-tree


Method :– Interval analysis (Moore 1966, Lottaz 2000) : no intersection computations

N1 : ([0, 1/2], [1/2, 1]), y - x3 0

= [1/2, 1] [0, 1/2]3 [0, 0]

= [1/2, 1] [0, 1/8] [0, 0]

= [3/8, 1] [0, 0] : white

N2 : ([1, 2], [-1, 0]), y - x3 0

= [-1, 0] [1, 2]3 [0, 0]

= [-1, 0] [1, 8] [0, 0]

= [-9, -1] [0, 0]: blue

N3 : ([1, 2], [1, 2]), y - x3 0

= [1, 2] [1, 2]3 [0, 0]

= [1, 2] [1, 8] [0, 0]

= [-9, 1] [0, 0]: greyexample : y - x3 0 with x = 0.0625 and y = 0.0625

Consistency of the nodes


Summary

• Need of piecewise constraint• General definition of a quad-tree


• Generation of quad-tree of piecewise constraints• Definition of a piecewise constraint• Definition of the information degrees• Algorithm of generation• Example


• Definition : (Vareilles et al., 2005)

C(x,y) : collection of k number of single numerical constraints called pieces and notated ci(x,y) covering a specific part of the serach area (dx, dy) such as

dx Dx and dy Dy.The pieces ci(x,y) are either equality or inequality constraints.

• Hypothesis on the general outline:

Consistent piecesClosed and bounded outline Uncrossed pieces

Piecewise constraint definition


Empty node Poorly informed node Informed node Overloaded node

Information degrees determine by two types of intersection:

node Dci(x,y)

node ci(x,y) (Moore 1966)

Information degrees definition

n Dci(x,y) = ø

n ci(x,y) = ø

n Dci(x,y) ø

n ci(x,y) = ø

n Dci(x,y) ø

n ci(x,y) ø

n Dci(x,y) ø

n ci(x,y) ø


• Principle : Recursive decomposition of the search area in coherent and incoherent regions :

• 2 steps : – Step 1 : Detection and marking of the information degree of each node with

specific colours

– Step 2 : Propagation of legal and illegal regions from the nodes which know their consistence to those which are ignorant (empty and poorly informed nodes)

Quad-tree generation algorithm


with x = y = 0.125

Quad-tree generation example


I

OO

O

Caption :

• O : overloaded nodes

• I : informed nodes

Generation of the quad-tree associated to f2

by using interval analyses

N1N2

Quad-tree generation example: step 1


w w

wGI

I I

O

O

O

O

Caption :


• I : Informed nodes

• w: legal nodes

• G : nodes which have to be decomposed

• red : empty nodes

• green : poorly informed nodes

N1N2

N3



I

w w

w

I

O

O

O

O w

w

w

ww

w

I

I

I

I

I

w w

w w

ww

I

I

Iw I

I

G

GG


Caption :


• I : Informed nodes

• w: legal nodes

• G : nodes which have to be decomposed




Precision reached

Caption :



• blue : illegal nodes

• yellow : unitary informed nodes

• orange : unitary overloaded nodes

Unitary informed node

Unitary overloaded node

Illegal node



Propagation from the yellow nodes to their red and green neighbours




Propagation from the blue nodes to their red and green neighbours



Propagation from the white nodes to their red and green neighbours



Coloration of the yellow and orange nodes in white


Relevant neighbours are found thanks to an encoding following Peano’s filled path, arranged with Morton’s order (Bridge et Peat, 1991)

Taking into account of piecewise constraints in CSP models, for instance to model experimental graphs

Quad-trees filtering techniques can be applied (Sam 1995)

Development of a mock-up

Synthesis :

Extension of this method to piecewise constraints with a higher arity Perspectives :

Conclusion


Using interval analysis to generate quad-trees of piecewise constraints

É. Vareilles, M. Aldanondo, P. Gaborit, K. Hadj-Hamou

October, the 1rst 2005

European project VHT n° G1RD-CT-2002-00835

page 1/21 intcp 2005 - sitges using interval analysis to generate quad-trees of piecewise...

Documents

sitges definition

quadtree slide

grey example

sitges quadtree principle

se grey

ne grey

sw grey

root grey