Advances in Pattern Databases

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Advances in Pattern Databases

• Ariel Felner,
• Ben-Gurion University
• Israel
• email felner_at_bgu.ac.il

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Overview

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

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optimal path search algorithms

• For small graphs provided explicitly, algorithm
  such as Dijkstras shortest path, Bellman-Ford or
  Floyd-Warshal. Complexity O(n2).
• For very large graphs , which are implicitly
  defined, the A algorithm which is a best-first
  search algorithm.

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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..

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Open-List
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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.
• ExamplesAir distance
• Manhattan Dinstance
• Different cost combinations of g and h
• f(x)level(x) Breadth-First Search.
• f(x)g(x) Dijkstras algorithms.
• f(x)h(x) Pure Heuristic Search (PHS).
• f(x)g(x)h(x) The A algorithm (1968).

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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
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Domains

• 15 puzzle
• 1013 states
• First solved by Korf 85 with IDA and Manhattan
  distance
• Takes 53 seconds
• 24 puzzle
• 1024 states
• First solved by Korf 96
• Takes 2 days

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Domains

• Rubiks cube
• 1019 states
• First solved by Korf 97
• Takes 2 days to solve

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(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 1013 states
• The (20,4) version has 1018 states

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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.
• Size 4k

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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.

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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

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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 Rubiks cube Korf 97
• Problem ? 3x3x3 Rubiks cube
• Subproblem ? 2x2x2 Corner cubies.

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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 Rubiks cube.
• Has 1019 States.
• The corner cubies subproblem has 88 Million
  states
• A table with 88 Million entries fits in memory
  Korf 97.

Search space
Pattern space
Mapping/Projection
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Non-additive pattern databases

• Fringe pattern database Culberson Schaeffer
  1996.
• Has only 259 Million states.
• Improvement of a factor of 100 over Manhattan
  Distance

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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
• PDB812131418

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Disjoint Additive PDBs (DADB)

• If you have many PDBS, take their maximum

• Values of disjoint databases 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

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DADBTile puzzles
5-5-5
6-6-3
7-8
6-6-6-6
Korf, AAAI 2005
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Heuristics for the TOH

• Infinite peg heuristic (INP) Each disk moves to
  its own temporary peg.
• Additive pattern databases
• Felner, Korf Hanan, JAIR-04

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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 4n states
• The largest database that we stored was of 14
  disks which needed 414256MB.

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10
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TOH4 results

• The difference between static and dynamic is
  covered in Felner, Korf Hanan JAIR-04

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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 showed that 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.

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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.

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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.

d
G
d1
d

• Usually they have nearby entries in the PDB
• A44444

A clique in TOH4
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Compressing cliques

• Assume a clique of size K with values d or d1
• Store only one entry (instead of K) for the
  clique with the minimum d. Lose at most 1.
• A44444 A44441
• Instead of 4p we need only 4(p-1) entries.
• This can be generalized to a set of nodes with
  diameter D. (for cliques D1)
• A44444 A44411
• In general compressing by k disks reduces memory
  requirements from 4p to 4(p-k)

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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.

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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
Goal State
Clique

• Storing PDBs for the tile puzzle
• (Simple mapping) A multi dimensional array ?
  
• A1616161616 size1.04Mb
• (Packed mapping) One dimensional array ?
  A1615141312 size 0.52Mb.
• Time versus memory tradeoff !!

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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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• Dual lookups in pattern databases Felner et al,
  IJCAI-04

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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.
• Rubiks cube rotate the cube
• We can take the maximum among the different
  lookups
• These are all geometrical symmetries
• We suggest a new type of symmetry!!

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8
8
7
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Regular and dual representation

• Regular representation of a problem
• Variables objects (tiles, cubies etc,)
• Values locations
• Dual representation
• Variables locations
• Values objects

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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?

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Regular and dual lookups

• Regular lookup PDB8,12,13,14
• Dual lookup PDB9,5,12,15

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Regular and dual in TopSpin

• Regular lookup for C PDB1,2,3,7,6
• Dual lookup for C PDB1,2,3,8,9

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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, Rubiks cube
• Bad example Towers of Hanoi
• Problematic example Tile Puzzles

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Inconsistency of Dual lookups
Consistency of heuristics h(a)-h(b) lt
c(a,b)

• Example Top-Spin
• c(b,c)1

• Both lookups for B
• PDB1,2,3,4,50
• Regular lookup for C PDB1,2,3,7,61
• Dual lookup for C PDB1,2,3,8,92

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Traditional Pathmax

• children inherit f-value from their parents if
  it makes them larger

g1 h4 f5
Inconsistency
g2 h2 f4
g2 h3 f5
Pathmax
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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!!

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Results (17,4) TopSpin puzzle

• Nodes improvement (17r17d) 1451
• Time improvement (4r4d) 72
• We also solved the (20,4) TopSpin version.

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Results Rubiks cube

• Data on 1000 states with 14 random moves
• PDB of 7-edges cubies

• Nodes improvement (24r24d) 250
• Time improvement (4r4d) 55

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Results Rubiks cube

• With duals we improved Korfs results on random
  instances by a factor of 1.5 using exactly the
  same PDBs.

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Results tile puzzles

• With duals, the time for the 24 puzzle drops
  from 2 days to 1 day.

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Discussion

• Results for the TopSpin and Rubiks 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

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Summary

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

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Future work

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

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Ongoing and future work compressing PDBs

• An item for the PDB of tiles (a,b,c,d) is in the
  form ltLa, Lb, Lc, Ldgtd
• 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 gt 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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Ongoing and Future Work on Duality

• Definition 1
• Let O be a sequence of operators such that s ? O
  ? G
• We apply O to G and get sd
• G ? O ? sd

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Duality continued

• Definition 2 For a state S we flip the roles of
  variables and objects

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Duality
The two definitions are equivalent!! Therefor
e Copt(S,G) Copt(Sd,G)
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Duality

• We can use heuristics from Sd for S
• We can search from both ways.
• Dual A (DA) An algorithm that searches and
  jumps if the heuristic is better. Results in
  progress.

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Workshop

• You are all welcome to the workshop on
• Heuristic Search, Memory-based Heuristics and
  Their application
• To be held in AAAI-06
• See www.ise.bgu.ac.il/faculty/felner