Using interval analysis to generate quad-trees 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. Root [-2, 2] [-2, 2] Grey example : y - x3 0 with x = 0.0625 and y = 0.0625 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, 1/2]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]: 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. Empty node Poorly informed node Informed node Overloaded node 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. Generation of the quad-tree associated to f2 by using interval analyses N1 O I O O Quad-tree generation example: step 1 N2 • Caption : • O : overloaded nodes • I : informed nodes

14. Quad-tree generation example: step 1 w w N2 N1 • 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 I O w N3 O O I O I

15. w w w w I O w w w w w I I G w O I I O I I I I w G I I G w w w I O w w 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. Unitary overloaded node Unitary informed node Illegal node 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. Synthesis : 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 Perspectives : Extension of this method to piecewise constraints with a higher arity 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