Compressing Pattern Databases

1
Compressing Pattern Databases

• Ariel Felner
• Bar-Ilan University.
• felner_at_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-first search 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
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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

• Rubiks Cube. Korf 1997
• The best way to combine different non-additive
  pattern databases is to take their maximum!

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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)

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Statically-partitioned additive databases

• 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.

7
8
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

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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.

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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 4n states and 2n
  bits suffice to store each state.
• The largest databases that we stored was of size
  14 which needed 414256MB.

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TOH4 results
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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.

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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.

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Cliques in the pattern space

• The values in a PDB for a clique are d or d1
• In permutation puzzles cliques exist when only
  one object moves to another location.
• Usually they have
• nearby entries
• in the PDB

d
G
d1
d
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Storing cliques

• Assume a clique of size K with values d or d1
• 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
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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 4P
• This can be generalized to a set of nodes with
  diameter D. (for cliques D1)
• For TOH4, we fix the position of the largest P-2
  disks and compress all the 4216 entries of the
  smallest 2 disks.
• In general, compressing any block will work, not
  necessarily cliques.

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TOH4 results 16 disks (142)

• 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.

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TOH4 larger versions
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.

• 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!!

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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 ?
  
• A1616161616 size1.04Mb
• (Packed mapping) One dimensional array with ?
• A1615141312 size 0.52Mb.
• The time and memory tradeoff is straightforward!!

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15 puzzle results

• A clique in the tile puzzle is of size 2.
• We compressed the last index by two ?
• A161616168

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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.

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Ongoing and future work

• An item for the PDB of tiles (a,b,c,d) is in the
  form 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

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Trie pruninig
Simple (lossless) pruning Fold leaves with
exactly the same values.
No data will be lost.
2
2
2
2
2
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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
2
4
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Trie Initial Results
A 5-5-5 partitioning stored in a trie with simple
folding
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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 dy21
• A NN can learn
• these rules

2
1
dy1 2 dy20
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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.

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

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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.