Using interval analysis to generate quadtrees of piecewise constraints

1
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

2
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

3
Need of piecewise constraints
Take into account experimental graphs in
constraints-based models.
Quad-trees were extended to piecewise constraints.
4
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

5
Quad-tree example
NW
SW
SE
NE
-2, 0 0, 2
-2, 0 -2, 0
0, 2 -2, 0
0, 2 0, 2
White
Grey
Grey
Grey
NW
SW
SE
NE
-2, -1 -1, 0
-2, -1 -2, -1
-1, 0 -2, -1
-1, 0 -1, 0
White
Grey
Grey
Grey
6
Definition of a quad-tree

• 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 (dnx, 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.

7
Consistency of the nodes

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

N1 (0, 1/2, 1/2, 1), y - x3 ? 0 1/2, 1
? 0, 1/23 ? 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, 23 ? 0,
0 -1, 0 ? 1, 8 ? 0, 0 -9, -1 ?
0, 0 blue N3 (1, 2, 1, 2), y - x3 ? 0
1, 2 ? 1, 23 ? 0, 0 1, 2 ? 1, 8 ?
0, 0 -9, 1 ? 0, 0 grey
example y - x3 ? 0 with ?x 0.0625 and ?y
0.0625
8
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

9
Piecewise constraint definition

• 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

Uncrossed pieces
Consistent pieces
Closed and bounded outline
10
Information degrees definition
Information degrees determine by two types of
intersection node ? Dci(x,y) node ?
ci(x,y) (Moore 1966)
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
11
Quad-tree generation algorithm

• 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)

12
Quad-tree generation example
with ?x ?y 0.125
13
Quad-tree generation example step 1
N2

• Caption
• O overloaded nodes
• I informed nodes

14
Quad-tree generation example step 1
w
w
N1
N2

• Caption
• O overloaded nodes
• I Informed nodes
• w legal nodes
• G nodes which have to be decomposed
• red empty nodes
• green poorly informed nodes

G
w
I
O
N3
O
O
I
I
O
15
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

16
Quad-tree generation example step 1
Precision reached

• Caption
• red empty nodes
• green poorly informed nodes
• blue illegal nodes
• yellow unitary informed nodes
• orange unitary overloaded nodes

17
Quad-tree generation example step 2
Propagation from the yellow nodes to their red
and green neighbours
?
18
Quad-tree generation example step 2
Propagation from the blue nodes to their red and
green neighbours
?
19
Quad-tree generation example step 2
Propagation from the white nodes to their red and
green neighbours
?
20
Quad-tree generation example step 2
Coloration of the yellow and orange nodes in white
?
21
Conclusion
22
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