page 1/21 intcp 2005 - sitges using interval analysis to generate quad-trees of piecewise...
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Page 1/21IntCP 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
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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
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Need of piecewise constraints
Take into account experimental graphs in constraints-based models.
Quad-trees were extended to piecewise constraints.
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Summary
• Need of piecewise constraint• General definition of a quad-tree
• Definition• Example
• Generation of quad-tree of piecewise constraints• Definition of a piecewise constraint• Definition of the information degrees• Algorithm of generation• Example
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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
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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
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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
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Summary
• Need of piecewise constraint• General definition of a quad-tree
• Definition• Example
• Generation of quad-tree of piecewise constraints• Definition of a piecewise constraint• Definition of the information degrees• Algorithm of generation• Example
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• 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
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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
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• 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
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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
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w w
wGI
I I
O
O
O
O
Caption :
• O : overloaded nodes
• I : Informed nodes
• w: legal nodes
• G : nodes which have to be decomposed
• red : empty nodes
• green : poorly informed nodes
N1N2
N3
Quad-tree generation example: step 1
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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
Quad-tree generation example: step 1
Caption :
• O : overloaded nodes
• I : Informed nodes
• w: legal nodes
• G : nodes which have to be decomposed
• red : empty nodes
• green : poorly informed nodes
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Precision reached
Caption :
• red : empty nodes
• green : poorly informed nodes
• blue : illegal nodes
• yellow : unitary informed nodes
• orange : unitary overloaded nodes
Unitary informed node
Unitary overloaded node
Illegal node
Quad-tree generation example: step 1
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Propagation from the yellow nodes to their red and green neighbours
Quad-tree generation example: step 2
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Quad-tree generation example: step 2
Propagation from the blue nodes to their red and green neighbours
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Quad-tree generation example: step 2
Propagation from the white nodes to their red and green neighbours
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Quad-tree generation example: step 2
Coloration of the yellow and orange nodes in white
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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