1 / 55

InterBlock-Backtracking : Exploiting Structure in Numerical CSPs Solving

Christophe Jermann LINA/CNRS, University of Nantes Joint work with: Bertrand Neveu, Gilles Trombettoni I3S/CNRS-INRIA, University of Nice. InterBlock-Backtracking : Exploiting Structure in Numerical CSPs Solving. Outline. IBB: a general framework for solving decomposed NCSPs Principle

alyn
Download Presentation

InterBlock-Backtracking : Exploiting Structure in Numerical CSPs Solving

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Christophe Jermann LINA/CNRS, University of Nantes Joint work with: Bertrand Neveu, Gilles Trombettoni I3S/CNRS-INRIA, University of Nice InterBlock-Backtracking: Exploiting Structure in Numerical CSPs Solving

  2. Outline • IBB: a general framework for solving decomposed NCSPs • Principle • Inputs • Process • IBB+Interval: an instance of IBB • Interval solving • Interblock Filtering • Experiments • Conclusion & Future directions

  3. Outline • IBB: a general framework for solving decomposed NCSPs • Principle • Inputs/Parameters • Process • IBB+Interval: an instance of IBB • Interval solving • Interblock Filtering • Experiments • Conclusion & Future directions

  4. IBB: general sight • Main ideas in [Bliek et al, CP98] • Since then, made a general framework • Parameters: fix to obtain one instance of IBB • a (set of) solving method(s) • a backtracking process • optional: a set of “add-ons” • Inputs: • a decomposed numerical CSP (NCSP) • Output: • One (or all the) solution(s) of the NCSP

  5. Numerical CSPs • Defined by (X,D,C) where: • X: a set of variables • D: a set of continuous domains, one for each xX • C: a set of constraints (equations, inequalities, …) on X • Applications: • Physics: forces, electrical measurements… • Design & Geometry: distances, angles, … • Program verification: instructions with floats, … • Robotics: kinematic constraints, … • …

  6. Decomposed NCSPs • Decomposed(S) = ({S1, S2, …, S3}, <, +) • Si = well-constrained sub-NCSP • < = partial order for Si’s solving • + = partial solutions combination operator Such that Sol(S1)+Sol(S2)+…+Sol(S3) = Sol(S) • Why decompose ? • Divide & conquer => reduce solving complexity • Several kinds of decompositions : • Equational [Michelucci et al. 1996], [Bliek et al. 1998], … • Geometric [Hoffmann et al 1995], [Jermann et al. 2000], … • …

  7. Solving methods • NCSP  system of (non-linear) equations/inequalities • Symbolic: Groebner basis, … • Numeric: Local (Newton-Raphson, …), Homotopy, Interval, …  Generally, the structure is under-exploited IBB allows to generically exploit the structure identified by decomposition

  8. Exploiting structure • Sub-NCSP = block • A subset of constraints • All the induced variables • Input variables (computed in another block) • Output variables (computed in the block) • Partial order  DAG A C B E F D H I G J

  9. Exploiting structure • Sub-NCSP = block • A subset of constraints • All the induced variables • Input variables (computed in another block) • Output variables (computed in the block) • Partial order  DAG A C B E F D H I G J

  10. Exploiting structure xA,yA xC,yC xJ,yJ • Sub-NCSP = block • A subset of constraints • All the induced variables • Input variables (computed in another block) • Output variables (computed in the block) • Partial order  DAG A C B E F D H I G J

  11. Exploiting structure xA,yA xC,yC xJ,yJ • Sub-NCSP = block • A subset of constraints • All the induced variables • Input variables (computed in another block) • Output variables (computed in the block) • Partial order  DAG A C B E F D H I G J

  12. Exploiting structure xA,yA xC,yC xJ,yJ • Sub-NCSP = block • A subset of constraints • All the induced variables • Input variables (computed in another block) • Output variables (computed in the block) • Partial order  DAG A C B xB,yB E F D H I G J

  13. Exploiting structure xA,yA xC,yC xJ,yJ • Sub-NCSP = block • A subset of constraints • All the induced variables • Input variables (computed in another block) • Output variables (computed in the block) • Partial order  DAG A C B xB,yB E F D H I G J

  14. Exploiting structure xA,yA xC,yC xJ,yJ • Sub-NCSP = block • A subset of constraints • All the induced variables • Input variables (computed in another block) • Output variables (computed in the block) • Partial order  DAG A C B xB,yB E F xE,yE D H I G J

  15. Exploiting structure xA,yA xC,yC xJ,yJ • Sub-NCSP = block • A subset of constraints • All the induced variables • Input variables (computed in another block) • Output variables (computed in the block) • Partial order  DAG A C B xB,yB E F xE,yE xF,yF D H I G J

  16. Exploiting structure xA,yA xC,yC xJ,yJ • Sub-NCSP = block • A subset of constraints • All the induced variables • Input variables (computed in another block) • Output variables (computed in the block) • Partial order  DAG A C B xB,yB E F xE,yE xF,yF D H I G YD J

  17. Exploiting structure xA,yA xC,yC xJ,yJ • Sub-NCSP = block • A subset of constraints • All the induced variables • Input variables (computed in another block) • Output variables (computed in the block) • Partial order  DAG A C B xB,yB E F xE,yE xF,yF D H I G YD xG,yG xH,yH xI,yI J

  18. Solving process xA,yA xC,yC xJ,yJ • Choosing a total order: • Compatible with the partial order • Fixes a static block ordering for backtracking purpose xB,yB xE,yE xF,yF YD xG,yG xH,yH xI,yI

  19. Solving a DAG of Blocks xA,yA xC,yC xJ,yJ • Choosing a total order: • Compatible with the partial order • Fixes a static block ordering for backtracking purpose xB,yB 1 xE,yE xF,yF 2 3 YD xG,yG xH,yH xI,yI 5 4

  20. Solving a DAG of Blocks xA,yA xC,yC xJ,yJ • Solving each block in sequence • with a solving method which can differ from block to block Search Tree Block 1 xB,yB 1 Solving Method 1 => Internal Search Tree 1 => 2 solutions: B1 and B2 xE,yE xF,yF 2 3 YD xG,yG xH,yH xI,yI 5 4

  21. Solving a DAG of Blocks xA,yA xC,yC xJ,yJ • Solving each block in sequence • with a solving method which can differ from block to block Search Tree Block 1 xB,yB B1 Solving Method 2 => Internal Search Tree 2 => 2 solutions: E1 and E2 Depending on B1 1 Block 2 xE,yE xF,yF 2 3 YD xG,yG xH,yH xI,yI 5 4

  22. Solving a DAG of Blocks xA,yA xC,yC xJ,yJ • Solving each block in sequence • with a solving method which can differ from block to block Search Tree Block 1 xB,yB B1 1 Block 2 xE,yE xF,yF E1 2 3 YD xG,yG xH,yH xI,yI 5 4 No Solution in Block 5

  23. Solving a DAG of Blocks xA,yA xC,yC xJ,yJ • Backtracking: • On “no solution” Search Tree Block 1 xB,yB B1 1 Block 2 xE,yE xF,yF E1 2 3 Backtracking YD xG,yG xH,yH xI,yI 5 4 No Solution in Block 5

  24. Solving a DAG of Blocks xA,yA xC,yC xJ,yJ • Backtracking: • On “no solution” => block solving should be complete for this purpose Search Tree Block 1 xB,yB B1 1 Block 2 xE,yE xF,yF E1 E2 2 3 F1,D1,G1,H1,I1 YD xG,yG xH,yH xI,yI 5 4 One global solution B1,E2,F1,D1,G1,H1,I1

  25. Solving a DAG of Blocks xA,yA xC,yC xJ,yJ • Backtracking: • On “no solution” • Or on “next solution” => block solving should be complete for this purpose Search Tree Block 1 xB,yB B1 Backtracking for completion 1 Block 2 xE,yE xF,yF E1 E2 2 3 F1,D1,G1,H1,I1 YD xG,yG xH,yH xI,yI 5 4 One global solution B1,E2,F1,D1,G1,H1,I1

  26. Solving a DAG of Blocks xA,yA xC,yC xJ,yJ • Backtracking => solving several times the same block … but not the same problem ! • Input variables = parameters => Blocks = parametric NCSPs Search Tree Block 1 xB,yB B2 B1 1 Block 2 Block 2 The constraints have changed depending on B2 xE,yE xF,yF E1 E2 2 3 YD xG,yG xH,yH xI,yI 5 4 One global solution B1,E2,F1,D1,G1,H1,I1

  27. Solving a DAG of Blocks xA,yA xC,yC xJ,yJ • Backtracking => solving several times the same block … but not the same problem ! • Input variables = parameters => Blocks = parametric NCSPs Search Tree Block 1 xB,yB B2 B1 Solving Method 3 => Internal Search Tree 3 => 2 solutions: E3 and E4 Depending on B2 1 Block 2 Block 2 xE,yE xF,yF E1 E2 2 3 YD xG,yG xH,yH xI,yI 5 4 One global solution B1,E2,F1,D1,G1,H1,I1

  28. Solving a DAG of Blocks xA,yA xC,yC xJ,yJ • Backtracking => solving several times the same block … but not the same problem ! • Input variables = parameters => Blocks = parametric NCSPs Search Tree Block 1 xB,yB B2 B1 1 Block 2 Block 2 xE,yE xF,yF E1 E2 E3 E4 2 3 YD xG,yG xH,yH xI,yI 5 4 All the global solutions

  29. Intelligent Backtracking xB,yB xE,yE xF,yF yD xG,yG xH,yH xI,yI • Possible InterBlock Backtracking • BT: Chronological Backtracking • GBJ: Graph-based Backjumping [Dechter, 1990] • GPB: Generalized Partial-Order BT [Bliek, 1998] xC,yC xA,yA 1 xJ,yJ 2 3 5 4

  30. Intelligent Backtracking xB,yB xE,yE xF,yF yD xG,yG xH,yH xI,yI • Possible InterBlock Backtracking • BT: Chronological Backtracking • GBJ: Graph-based Backjumping [Dechter, 1990] • GPB: Generalized Partial-Order BT [Bliek, 1998] xC,yC xA,yA 1 xJ,yJ 2 3 BT 5 4

  31. Intelligent Backtracking xB,yB xE,yE xF,yF yD xG,yG xH,yH xI,yI • Possible InterBlock Backtracking • BT: Chronological Backtracking • GBJ: Graph-based Backjumping [Dechter, 1990] • GPB: Generalized Partial-Order BT [Bliek, 1998] xC,yC xA,yA 1 xJ,yJ 2 3 GBJ GPB BT 5 4

  32. Intelligent Backtracking xB,yB xE,yE xF,yF yD xG,yG xH,yH xI,yI • Possible InterBlock Backtracking • BT: Chronological Backtracking • GBJ: Graph-based Backjumping [Dechter, 1990] • GPB: Generalized Partial-Order BT [Bliek, 1998] xC,yC xA,yA 1 GPB xJ,yJ 2 3 GBJ GBJ GPB BT 5 4

  33. Intelligent Backtracking xB,yB xE,yE xF,yF yD xG,yG xH,yH xI,yI • Possible InterBlock Backtracking • BT: Chronological Backtracking • GBJ: Graph-based Backjumping [Dechter, 1990] • GPB: Generalized Partial-Order BT [Bliek, 1998] xC,yC xA,yA 1 + = with the recompute condition GPB xJ,yJ 2 3 GBJ GBJ+ GBJ GPB BT 5 4

  34. IBB: extending the framework • Specific treatments as add-ons: • Generally, depending on the solving methods • Choose the strategy of application (before/after/during a block solving, …) • E.g.: • Pre-conditioners, • Propagators, • Inequalities checkers, • …

  35. Outline • IBB: a general framework for solving decomposed NCSPs • Principle • Inputs • Process • IBB+Interval: an instance of IBB • Interval solving • Interblock Filtering • Experiments • Conclusion & Future directions

  36. IBB + Interval • Solving method = Interval constraint programming techniques • Backtracking: BT, GBJ & GPB => 3 instances in fact • An interesting add-on: Inter-Block Filtering (IBF) • Propagates domain reductions in following blocks

  37. Interval solving • 3 operations: Search space = cross-product of the domains (intervals) (x-1)2 -3=y y y<x/3 x -x2+3=y

  38. Interval solving • 3 operations: • Filtering: reduces the bounds of the domain of each variable using a local consistency (x-1)2 -3=y y y<x/3 x -x2+3=y

  39. Interval solving • 3 operations: • Filtering: reduces the bounds of the domain of each variable using a local consistency (x-1)2 -3=y y y<x/3 x -x2+3=y

  40. Interval solving • 3 operations: • Filtering: reduces the bounds of the domain of each variable using a local consistency • Splitting: splits search space into parts to be explored individually => a search-tree appears (x-1)2 -3=y y y<x/3 x -x2+3=y

  41. Interval solving • 3 operations: • Filtering: reduces the bounds of the domain of each variable using a local consistency • Splitting: splits search space into parts to be explored individually => a search-tree appears • Existence: checks if a unique solution exists in the current sub-search-space (x-1)2 -3=y y y<x/3 x -x2+3=y

  42. InterBlock Filtering A C B Block 1 Principle: Use local consistency to propagate the reductions during the solving of current blocks in related blocs

  43. InterBlock Filtering E1 A C A C B B E2 Block 2 Block 1

  44. InterBlock Filtering E1 A C A C B B E2 Block 2 Block 1 E1 A C B E2 F Block 3

  45. InterBlock Filtering E1 A C A C B B E2 Block 2 Block 1 E1 E1 A C A C B B E2 E2 F F Block 3 Block 4 D??

  46. InterBlock Filtering E1 A C A C B B E2 Block 2 Block 1 E1 E1 A C A C B B E2 E2 F F Block 3 Block 4 D?? Using IBF, E1 incompatibility can be detected in Block 2 by propagating on Block 4

  47. Benchmarking • Implemented as a C++ prototype • Run on a PIII 935 • Experimental protocol: • Parameters: ~12 instances of IBB+Interval • Interval Solving: (uses ILOG Solver 5.0) • Filtering: 2B, 3B, Box, Bound, 2B+Box, 3B+Bound; best choice per problem (usually 2B+Box or 3B) • Splitting:classical interval bisection • Existence:always and only with Box and Bound • Backtracking: BT, GBJ, GPB, with or without + • InterBlock Filtering: with or without • Inputs: • NCSPs: 8 from CAD, 4 domain sizes • Decomposition:best among 4 methods

  48. Test set

  49. Overall comparison

  50. Overall comparison

More Related