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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

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Overview

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

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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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

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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.

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Domains

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

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

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

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

Domains

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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 10^13 states• The (20,4) version has 10^18 states

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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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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.

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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: 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].

Patternspace

Search space

Mapping/Projection

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Example - 15 puzzle

2 3 6 7

2 3 6 7

• 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

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Disjoint Additive PDBs (DADB)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

• Values of disjoint databases can be added and are still admissible [Korf & Felner: AIJ-02,

Felner, Korf & Hanan: JAIR-04]

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•If you have many PDBS, take their maximum

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DADB:Tile puzzles

PuzzleHeuristicValueNodes TimeMemory15Breadth-FS10^1328 days3-tera-bytes15Manhattan36.942401,189,63053.4240155-5-541.5623,090,4050.5413,145156-6-342.924617,5550.16333,554157-845.63236,7100.034576,575246-6-6-6360,892,479,6712 days242,000

5-5-5 6-6-3 7-8 6-6-6-6[Korf, AAAI 2005]

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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.– Rubik’s 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

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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

2 3 6 7

• 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: PDB[8,12,13,14]• Dual lookup: PDB[9,5,12,15]

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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]

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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, Rubik’s cube• Bad example: Towers of Hanoi• Problematic example: Tile Puzzles

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Inconsistency of Dual lookups

Example: Top-Spin

c(b,c)=1

Consistency of heuristics: |h(a)-h(b)| <= c(a,b)

Regular Dual

b00c12

• 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

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Traditional Pathmax• children inherit f-value from their parents if it

makes them larger

g=1h=4f=5

g=2h=2f=4

g=2h=3f=5

Inconsistency

Pathmax

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Bidirectional pathmax (BPMX)

• 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!!

h-values

2

5 1

4

5 3

h-values

BPMX

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Results: (17,4) TopSpin puzzleregulardualBPMXnodestime

10----40,019,42967.7601no7,618,80515.7201yes1,397,6142.9344yes82,6060.94

1717yes27,5751.34

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

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Results: Rubik’s cube

regulardualBPMXnodestime10----90,930,66228.1801no19,653,3867.3801yes8,315,1163.2444yes615,5630.51

2424yes362,9270.90• Nodes improvement (24r+24d) : 250• Time improvement (4r+4d) : 55

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

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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.

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Results: tile puzzles

HeuristicBPMXValuenodestimeManhattan----36.94401,189,63053.424R----44.75136,2890.081R+R*----45.6336,7100.034R+R*+D+D*yes46.1218,6010.022

• 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 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

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Summary

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

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