1. Page 1/21 IntCP 2005 - Sitges Using interval analysis to generate quad-trees of piecewise constraints É. Vareilles, M. Aldanondo, P. Gaborit, K. Hadj-Hamou October, the 1 rst 2005 European project VHT n° G1RD-CT-2002-00835

2. Page 2/21 IntCP 2005 - Sitges 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. Page 3/21 IntCP 2005 - Sitges Need of piecewise constraints Take into account experimental graphs in constraints-based models. Quad-trees were extended to piecewise constraints.

4. Page 4/21 IntCP 2005 - Sitges 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. Page 5/21 IntCP 2005 - Sitges Quad-tree example example : y - x 3  0 with  x = 0.0625 and  y = 0.0625 [-2, 2] Root Grey [-2, 0] [0, 2] NW White [-2, 0] SW Grey [0, 2] [-2, 0] SE Grey [0, 2] NE Grey [-2, -1] [-1, 0] NW White [-2, -1] SW Grey [-1, 0] [-2, -1] SE Grey [-1, 0] NE Grey

6. Page 6/21 IntCP 2005 - Sitges 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 (d n x, d n y ). 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

7. Page 7/21 IntCP 2005 - Sitges Method : –Interval analysis (Moore 1966, Lottaz 2000) : no intersection computations N1 : ([0, 1/2], [1/2, 1]), y - x 3  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 - x 3  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 - x 3  0 = [1, 2]  [1, 2]3  [0, 0] = [1, 2]  [1, 8]  [0, 0] = [-9, 1]  [0, 0]: grey example : y - x 3  0 with  x = 0.0625 and  y = 0.0625 Consistency of the nodes

8. Page 8/21 IntCP 2005 - Sitges 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. Page 9/21 IntCP 2005 - Sitges 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 pieces Closed and bounded outline Uncrossed pieces Piecewise constraint definition

10. Page 10/21 IntCP 2005 - Sitges Empty nodePoorly informed nodeInformed nodeOverloaded 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

11. Page 11/21 IntCP 2005 - Sitges 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

12. Page 12/21 IntCP 2005 - Sitges with  x =  y = 0.125 Quad-tree generation example

13. Page 13/21 IntCP 2005 - Sitges I OO O Caption : O : overloaded nodes I : informed nodes Generation of the quad-tree associated to f2 by using interval analyses N1 N2 Quad-tree generation example: step 1

14. Page 14/21 IntCP 2005 - Sitges ww w G I II 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

15. Page 15/21 IntCP 2005 - Sitges I ww w I O O O Ow w w ww w I I I I I ww ww ww I I IwI 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

16. Page 16/21 IntCP 2005 - Sitges 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

17. Page 17/21 IntCP 2005 - Sitges Propagation from the yellow nodes to their red and green neighbours  Quad-tree generation example: step 2

18. Page 18/21 IntCP 2005 - Sitges  Quad-tree generation example: step 2 Propagation from the blue nodes to their red and green neighbours

19. Page 19/21 IntCP 2005 - Sitges  Quad-tree generation example: step 2 Propagation from the white nodes to their red and green neighbours

20. Page 20/21 IntCP 2005 - Sitges  Quad-tree generation example: step 2 Coloration of the yellow and orange nodes in white

21. Page 21/21 IntCP 2005 - Sitges 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

22. Page 22/21 IntCP 2005 - Sitges Using interval analysis to generate quad-trees of piecewise constraints É. Vareilles, M. Aldanondo, P. Gaborit, K. Hadj-Hamou October, the 1 rst 2005 European project VHT n° G1RD-CT-2002-00835