Dual lookups in Pattern Databases

1. Dual lookups in Pattern Databases Ariel Felner,Ben-Gurion Univ. Israel Uzi Zahavi,Bar-Ilan Univ. Israel Jonathan Schaeffer, Univ.of Alberta, Canada Robert C. Holte,Univ. of Alberta,Canada

2. Overview • Heuristic search and A* • Test domains • Pattern databases • Dual lookups in pattern databases • Experimental results

3. A* • Best-first search with a cost function of f(n)=g(n)+h(n) • g(n):actual distance from the initial state to n • h(n): estimated remained distance from n to the goal state. • h(n) is admissible if it is always underestimating • Examples of h: Air distance, Manhattan Distance

4. Attributes of A* • If h(n)is admissible then A* is guaranteed to return an optimal (shortest) solution • A* is admissible, complete and optimally effective. [Pearl & Dechter 84] • IDA* is a linear space version of A* [Korf 85] which uses DFS iterations.

5. Domains 15 puzzle • 10^13 states • First solved by [Korf 85] with IDA* and Manhattan distance • Takes 53 seconds 24 puzzle • 10^24 states • First solved by [Korf 96] • Takes 2 days

6. Domains • Rubik’s cube • 10^19 states • First solved by [Korf 97] • Takes 2 days to solve

7. (n,k) Top Spin Puzzle • n tokens arranged in a ring • States: any possible permutation of the tokens • Operators: Any k consecutive tokens can be reversed • The (17,4) version has 10^13 states • The (20,4) version has 10^18 states

8. How to improve search? • Enhanced algorithms: • Perimeter-search [Delinberg and Nilson 95] • RBFS [Korf 93] • Frontier-search [Korf and Zang 2003] • Breadth-first heuristic search [Zhou and Hansen 04]  They all try to better explore the search tree. • Better heuristics:more parts of the search tree will be pruned.

9. Subproblems-Abstractions • Many problems can be abstracted into subproblems that must be also solved. • A solution to the subproblem is a lower bound on the entire problem. • Example: Rubik’s cube [Korf 97] • Problem:  3x3x3 Rubik’s cube Subproblem:  2x2x2 Corner cubies.

10. Pattern Databases heuristics • A pattern database[Culbreson & Schaeffer 96] is a lookup table that stores solutions to all configurations of the subproblem / abstraction / pattern. • This table is used as a heuristic during the search. • Example: Rubik’s cube. • Has 10^19 States. • The corner cubies subproblem has 88 Million states • A table with 88 Million entries fits in memory [Korf 97]. Search space Mapping/Projection Pattern space

11. Example - 15 puzzle • How many moves do we need to move tiles 2,3,6,7 from locations 8,12,13,14 to their goal locations • The solution to this is located in PDB[8][12][13][14]=18

12. 7-8 Disjoint Additive PDBs (DADB) • If you have many PDBS, take their maximum • Values of disjointdatabases can be added and are still admissible [Korf & Felner: AIJ-02, Felner, Korf & Hanan: JAIR-04]

13. DADB:Tile puzzles 5-5-5 6-6-3 7-8 6-6-6-6 [Korf, AAAI 2005]

14. Symmetries in PDBs • Symmetric lookups were already performed by the first PDB paper of [Culberson & Schaeffer 96] • examples • Tile puzzles: reflect the tiles about the main diagonal. • Rubik’s cube: rotate the cube • We can take the maximum among the different lookups • These are all geometricalsymmetries • We suggest a new type of symmetry!! 7 8 7 8

15. Regular and dual representation • Regular representation of a problem: • Variables – objects (tiles, cubies etc,) • Values – locations • Dualrepresentation: • Variables – locations • Values – objects

16. Regular vs. Dual lookups in PDBs • Regular question: Where are tiles {2,3,6,7} and how many moves are needed to gather them to their goal locations? • Dual question: Who are the tiles in locations {2,3,6,7} and how many moves are needed to distribute them to their goal locations?

17. Regular and dual lookups • Regular lookup: PDB[8,12,13,14] • Dual lookup: PDB[9,5,12,15]

18. Regular and dual in TopSpin • Regular lookup for C : PDB[1,2,3,7,6] • Dual lookup for C: PDB[1,2,3,8,9]

19. Dual lookups • Dual lookups are possible when there is a symmetry between locations and objects: • Each object is in only one location and each location occupies only one object. • Good examples: TopSpin, Rubik’s cube • Bad example: Towers of Hanoi • Problematic example: Tile Puzzles

20. Inconsistency of Dual lookups • Consistency of heuristics: • |h(a)-h(b)| <= c(a,b) Example:Top-Spin c(b,c)=1 • Both lookups for B PDB[1,2,3,4,5]=0 • Regular lookup for C PDB[1,2,3,7,6]=1 • Dual lookup for C PDB[1,2,3,8,9]=2

21. Traditional Pathmax • children inherit f-value from their parents if it makes them larger g=1 h=4 f=5 Inconsistency g=2 h=2 f=4 g=2 h=3 f=5 Pathmax

22. Bidirectional pathmax (BPMX) h-values h-values 2 4 BPMX 5 1 5 3 • Bidirectional pathmax: h-values are propagated in both directions decreasing by 1 in each edge. • If the IDA* threshold is 2 then with BPMX the right child will not even be generated!!

23. Results: (17,4) TopSpin puzzle • Nodes improvement (17r+17d) : 1451 • Time improvement (4r+4d) : 72 • We also solved the (20,4) TopSpin version.

24. Results: Rubik’s cube • Data on 1000 states with 14 random moves • PDB of 7-edges cubies • Nodes improvement (24r+24d) : 250 • Time improvement (4r+4d) : 55

25. Results: Rubik’s cube • With duals we improved Korf’s results on random instances by a factor of 1.5 using exactly the same PDBs.

26. Results: tile puzzles • With duals, the time for the 24 puzzle drops from 2 days to 1 day.

27. Discussion • Results for the TopSpin and Rubik’s cube are better than those of the tile puzzles • Dual PDB lookups and BPMX cutoffs are more effective if each operators changes larger part of the states. • This is because the identity of the objects being queried in consecutive states are dramatically changed

28. Summary • Dual PDB lookups • BPMX cutoffs for inconsistent heuristics • State of the art solvers.

29. Future work • Duality in search spaces • Which and how many symmetries to use • Other sources of inconsistencies • Better ways for propagating inconsistencies