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Advances in Pattern Databases

Advances in Pattern Databases. Ariel Felner, Ben-Gurion University Israel email: felner@bgu.ac.il. Overview. Heuristic search and pattern databases Disjoint pattern databases Compressed pattern databases

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Advances in Pattern Databases

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  1. Advances in Pattern Databases Ariel Felner, Ben-Gurion University Israel email: felner@bgu.ac.il

  2. Overview • Heuristic search and pattern databases • Disjoint pattern databases • Compressed pattern databases • Dual lookups in pattern databases • Current and future work

  3. optimal path search algorithms • For small graphs: provided explicitly, algorithm such as Dijkstra’s shortest path, Bellman-Ford or Floyd-Warshal. Complexity O(n^2). • For very large graphs , which are implicitly defined, the A* algorithm which is a best-first search algorithm.

  4. Best-first search schema • sorts all generated nodes in an OPEN-LIST and chooses the node with the best cost value for expansion. • generate(x): insert x into OPEN_LIST. • expand(x): delete x from OPEN_LIST and generate its children. • BFS depends on its cost (heuristic) function. Different functions cause BFS to expand different nodes.. 20 25 30 30 35 35 35 40 Open-List

  5. Best-first search: Cost functions • g(x):Real distance from the initial state to x • h(x): The estimated remained distance from x to the goal state. • Examples:Air distance Manhattan Dinstance Different cost combinations of g and h • f(x)=level(x) Breadth-First Search. • f(x)=g(x) Dijkstra’s algorithms. • f(x)=h’(x) Pure Heuristic Search (PHS). • f(x)=g(x)+h’(x) The A* algorithm (1968).

  6. A* (and IDA*) • A* is a best-first search algorithm that uses f(n)=g(n)+h(n)as its cost function. • f(x) in A* is an estimation of the shortest path to the goal via x. • A* is admissible, complete and optimally effective. [Pearl 84] • Result: any other optimal search algorithm will expand at least all the nodes expanded by A* Breadth First Search A*

  7. 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

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

  9. (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

  10. 4-peg Towers of Hanoi (TOH4) • There is a conjecture about the length of optimal path but it was not proven. • Size 4^k

  11. 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.

  12. Better heuristics • In the 3rd Millennium we have very large memories. We can build large tables. • For enhanced algorithms: large open-lists or transposition tables. They store nodes explicitly. • A more intelligent way is to store general knowledge. We can do this with heuristics

  13. 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.

  14. 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

  15. Non-additive pattern databases • Fringe pattern database[Culberson & Schaeffer 1996]. • Has only 259 Million states. • Improvement of a factor of 100 over Manhattan Distance

  16. 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

  17. 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] • Additivity can be applied if the cost of a subproblem is composed from costs of objects from corresponding pattern only

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

  19. Heuristics for the TOH • Infinite peg heuristic (INP): Each disk moves to its own temporary peg. • Additive pattern databases [Felner, Korf & Hanan, JAIR-04]

  20. Additive PDBS for TOH4 • Partition the disks into disjoint sets • Store the cost of the complete pattern space of each set in a pattern database. • Add values from these PDBs for the heuristic value. • The n-disk problem contains 4^n states • The largest database that we stored was of 14 disks which needed 4^14=256MB. 6 10

  21. TOH4: results • The difference between static and dynamic is covered in [Felner, Korf & Hanan: JAIR-04]

  22. Best Usage of Memory • Given 1 giga byte of memory, how do we best use it with pattern databases? • [Holte, Newton, Felner, Meshulam and Furcy, ICAPS-2004] showedthat it is better to use many small databases and take their maximum instead of one large database. • We will present a different (orthogonal) method [Felner, Mushlam & Holte: AAAI-04].

  23. Compressing pattern database Felner et al AAAI-04]] • Traditionally, each configuration of the pattern had a unique entry in the PDB. • Our main claim  Nearby entries in PDBs are highly correlated !! • We propose to compress nearby entries by storing their minimum in one entry. • We show that  most of the knowledge is preserved • Consequences: Memory is saved, larger patterns can be used speedup in search is obtained.

  24. Cliques in the pattern space • The values in a PDB for a clique are d or d+1 • In permutation puzzles cliques exist when only one object moves to another location. d G d+1 d • Usually they have nearby entries in the PDB • A[4][4][4][4][4] A clique in TOH4

  25. Compressing cliques • Assume a clique of size K with values d or d+1 • Store only one entry (instead of K) for the clique with the minimum d.Lose at most 1. • A[4][4][4][4][4] A[4][4][4][4][1] • Instead of 4^p we need only 4^(p-1) entries. • This can be generalized to a set of nodes with diameter D. (for cliques D=1) • A[4][4][4][4][4] A[4][4][4][1][1] • In general: compressing by k disks reduces memory requirements from 4^pto4^(p-k)

  26. TOH4 results: 16 disks (14+2) • Memory was reduced by a factor of 1000!!! at a cost of only a factor of 2 in the search effort.

  27. Memory was reduced by a factor of 1000!!! At a cost of only a factor of 2 in the search effort. Lossless compressing is noe efficient in this domain. TOH4: larger versions • For the 17 disks problem a speed up of 3 orders of magnitude is obtained!!! • The 18 disks problem can be solved in 5 minutes!!

  28. Tile Puzzles Goal State Clique • Storing PDBs for the tile puzzle • (Simple mapping) A multi dimensional array  A[16][16][16][16][16] size=1.04Mb • (Packed mapping) One dimensional array  A[16*15*14*13*12 ] size = 0.52Mb. • Time versus memory tradeoff !!

  29. 15 puzzle results • A clique in the tile puzzle is of size 2. • We compressed the last index by two  A[16][16][16][16][8]

  30. Dual lookups in pattern databases[Felner et al, IJCAI-04]

  31. 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

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

  33. 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?

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

  35. 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]

  36. 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

  37. 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

  38. 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

  39. 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!!

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

  41. 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

  42. 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.

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

  44. 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

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

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

  47. Ongoing and future work compressing PDBs • An item for the PDB of tiles (a,b,c,d) is in the form: <La, Lb, Lc, Ld>=d • Store the PDBs in a Trie • A PDB of 5 tiles will have a level in the trie for each tile. The values will be in the leaves of the trie. • This data-structure will enable flexibility and will save memory as subtrees of the trie can be pruned

  48. Trie pruninig Simple (lossless) pruning: Fold leaves with exactly the same values. No data will be lost. 2 2 2 2 2

  49. Trie pruninig Intelligent (lossy)pruning: Fold leaves/subtrees with are correlated to each other (many option for this!!) Some data will be lost. Admissibility is still kept. 2 2 2 4 2

  50. Trie: Initial Results A 5-5-5 partitioning stored in a trie with simple folding

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