Compressing Pattern Databases

1. Compressing Pattern Databases Ariel Felner Bar-Ilan University. felner@cs.biu.ac.il March 2004 Joint work with Ram Meshulam, Robert Holte and Richard E. Korf Submitted to AAAI04. Available at: http://www.cs.biu.ac.il/~felner

2. A* and its variants • A* (and IDA*) is a best-firstsearch algorithm that uses f(n)=g(n)+h(n)as its cost function. Nodes are sorted in an open-list according to their f-value. • g(n)is the shortest known path between the initial node and the current node n. • h(n)is an admissible (lower bound) heuristic estimation from n to the goal node • Recently, the attention has shifted towards creating more accurate heuristic functions.

3. Pattern databases • Many problems can be decomposed into subproblems (patterns) that must be also solved. • The pattern space is a domain abstraction of the original space • The cost of a solution to a subproblem is a lower-bound on the cost of the complete solution • Instead of calculating the lower bounds on the fly, we expand the whole pattern-space and store the solution to each pattern configuration in a pattern database Search space Mapping function Pattern space

4. Non-additive pattern databases • Fringe database for the 15 puzzle by [Culberson and Schaeffer 1996]. • Stores the number of moves including tiles not in the pattern • Rubik’s Cube. [Korf 1997] • The best way to combine different non-additive pattern databases is to take their maximum!

5. Additive pattern databases • We can add values from different pattern databases if they are disjoint (and count their own moves) • There are two ways to build additive databases: • Statically-partitioned additive databases (they were also called disjoint pattern databases) • Dynamically-partitioned additive databases. • Applications of additive pattern databases • Tile puzzles • 4-peg Towers of Hanoi Puzzle (TOH4)

6. Statically-partitioned additive databases 7 8 • These were created for the 15 and 24 puzzles [Korf & Felner 2002] • We statically partition the tiles into disjoint patterns and compute the cost of moving only these tiles into their goal states. 6 6 6 6 • For the 15 puzzle: • 36,710 nodes. • 0.027 seconds. • 575 MB • For the 24 puzzle: • 360,892,479,671 • 2 days • 242 MB

7. 4-peg Towers of Hanoi (TOH4) • There is a conjecture about the length of optimal path but it was not proven. • Systematic search is the only way to solve this problem or to verify the conjecture. • There are too many cycles. IDA* as a DFS will not prune these cycle. Therefore, A* (actually frontier A* [Korf & Zhang 2000]) was used.

8. Additive PDBS for TOH4 • Partition the disks into disjoint sets (patterns) . For example, 10 and 6 for the 16-disk problem. • Store the cost of the complete pattern space of each set in a pattern database. (There are many enhancements) • The n-disk problem contains 4^n states and 2n bits suffice to store each state. • The largest databases that we stored was of size 14 which needed 4^14=256MB.

9. TOH4: results

10. How to best use the memory • The speed of the search is directly related to the size of the pattern database. • We usually omit the computation time of the PDBs but cannot ignore the memory requirements • [Holte, Newton, Felner, Mushulam and Furcy 2004] showed that it is better to use many small databases and take their maximum instead of one large database. • We limit the discussion to 1 Giga bytes.

11. Compressing pattern databases • Traditionally, each configuration of the pattern had a unique entry in the PDB. • Our main cliam  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 cab be used speedup in search is obtained.

12. 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. • Usually they have nearby entries in the PDB d G d+1 d

13. Storing cliques • Assume a clique of size K with values d or d+1 • Lossy compression  Store only one entry for the clique with the minimum d.Loose at most 1. • Lossless compression  Store the minimum d. Also store K additional bits, one per entry. A clique in TOH4

14. Compressing PDBs in TOH4 • If we compress the last index of smallest disk then a PDB with P disks can now be stored in only 4^(P-1) entries instead of 4^P • This can be generalized to a set of nodes with diameter D. (for cliques D=1) • For TOH4, we fix the position of the largest P-2 disks and compress all the 4^2=16 entries of the smallest 2 disks. • In general, compressing any block will work, not necessarily cliques.

15. 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. • Lossless compressing is not efficient in this domain.

16. 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!!

17. Tile Puzzles 0 0 3 • We can take advantage of the simple heuristics. We can store only the addition above the Manhattan distance heuristic Storing PDBs for the tile puzzle 0 0 6 7 0 0 10 11 0 0 2 2 • (Simple mapping) A multi dimensional array  A[16][16][16][16][16] size=1.04Mb • (Packed mapping) One dimensional array with  A[16*15*14*13*12 ] size = 0.52Mb. • The time and memory tradeoff is straightforward!!

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

19. 24 puzzle • The same tendencies were obtained for the 24 puzzle. • The 6-6-6-6 partitioning is so good that adding another set of 6-6-6-6 did not speedup the search. • We have also tried a 7-7-5-5 partitioning but it did not speedup the search.

20. Ongoing and future work • 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

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

22. 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

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

24. Neural Networks (NN) • We can feed a PDB into a neural network engine. Especially, Addition above MD • For each tile we focus on its dx and dy from its goal position. (i.e. MD) • Linear conflict : • dx1= dx2 = 0 • dy1 > dy2+1 • A NN can learn these rules 2 1 dy1 =2dy2=0

25. Neural network • We train the NN by feeding the entire (or part of the) pattern space. • For example for a pattern of 5 tiles we have 10 features, 2 for each tile. • During the search, given the locations of the tiles we look them up in the NN.

26. Neural network example dx4 Layout for the pattern of the tiles 4, 5 and 6 dy4 dx5 4 dy5 dx6 dy6

27. Neural Network: problems • We face the problem of overestimating and will have to bias the results towards underestimating. • We keep the overestimating values in a separate hash table • Results are encouraging!!

28. Selective Pattern Database • Only part of the pattern space is queried for a single problem instance. • If we can identify that part we can only generate that part.