Page 1/21 IntCP 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

Page 2/21 IntCP 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

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

Page 4/21 IntCP 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

Page 5/21 IntCP Sitges Quad-tree example example : y - x 3  0 with  x = and  y = [-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

Page 6/21 IntCP 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

Page 7/21 IntCP 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 = and  y = Consistency of the nodes

Page 8/21 IntCP 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

Page 9/21 IntCP 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

Page 10/21 IntCP 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

Page 11/21 IntCP 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

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

Page 13/21 IntCP 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

Page 14/21 IntCP 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

Page 15/21 IntCP 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

Page 16/21 IntCP 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

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

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

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

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

Page 21/21 IntCP 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

Page 22/21 IntCP 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