Pattern Databases

1
Pattern Databases

• Robert Holte
• University of Alberta

November 6, 2002
2
Pattern Database Successes (1)

• Joe Culberson Jonathan Schaeffer (1994).
• 15-puzzle (1013 states).
• 2 hand-crafted patterns (fringe (FR) and
  corner (CO))
• Each PDB contains over 500 million entries (lt 109
  abstract states).
• Used symmetries to compress and enhance the use
  of PDBs
• Used in conjunction with Manhattan Distance (MD)
• Reduction in size of search tree
• MD 346 max(MD,FR)
• MD 437 max(MD,CO)
• MD 1038 max(MD, interleave(FR,CO))

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Pattern Database Successes (2)

• Rich Korf (1997)
• Rubiks Cube (1019 states).
• 3 hand-crafted patterns, all used together (max)
• Each PDB contains over 42 million entries
• took 1 hour to build all the PDBs
• Results
• First time random instances had been solved
  optimally
• Hardest (solution length 18) took 17 days
• Best known MD-like heuristic would have taken a
  century

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Pattern Database Successes (3)

• Stefan Edelkamp (2001)
• Planning benchmarks e.g. logistics, Blocks world
• Automatically generated PDBs (not domain
  abstraction)
• Additive pattern databases (in some cases)
• Results
• PDB competitive with the best planners
• logistics domain (weighted A), PDB run-time 100
  times smaller than FF heuristic

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Pattern Database Successes (4)

• Istvan Hernadvolgyi (2001)
• Macro-operators are concatenated to very quickly
  construct suboptimal solutions
• For Rubiks Cube hundreds of macro-operators are
  needed
• Each macro is found by searching in the Rubiks
  Cube state space with a macro-specific subgoal
  and start state
• For every one of these searches, a PDB was
  generated automatically (domain abstraction) so
  that an optimal-length macro could be found
  quickly
• Results
• Optimal-length macros for all subgoals found for
  the first time
• So quick that it permitted subgoals to be merged
• This shortened solutions from 90 moves to 50
  (optimal is 18)

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Fundamental Questions

• How to invent effective heuristics ?

Create a simplified version of your problem. Use
the exact distances in the simplified version
as heuristic estimates in the original.
How to use memory to speed up search ?
Precompute all distances-to-goal in the
simplified version of the problem and store them
in a lookup table (pattern database).
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Example 8-puzzle
181,440 states
Domain blank 1 2 3 4 5 6 7 8
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Patternscreated by domain mapping
This mapping produces 9 patterns
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Pattern Database
Pattern Distance to goal 0 1 1
2 2 2
Pattern Distance to goal 3 3 4
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Calculating h(s)

• Given a state in the original problem
• Compute the corresponding pattern
• and look up the abstract distance-to-goal

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Heuristics defined by PDBs are consistent, not
just admissible.
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Abstract Space
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Efficiency

• Time for the preprocessing to create a PDB is
  usually negligible compared to the time to solve
  one problem-instance with no heuristic.
• Memory is the limiting factor.

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Pattern leave some tiles unique
3024 patterns
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Domain Abstraction
Domain blank 1 2 3 4 5 6 7 8 Abstract
blank 6 7 8
30,240 patterns
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8-puzzle PDB sizes(with the blank left unique)
9 72 252 504 630
1512 2520 3024 3780 5040
7560 10080 15120
15120 22680 30240 45360 60480
90720 181440
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Automatic Creation of Domain Abstractions

• Easy to enumerate all possible domain
  abstractions
• They form a lattice, e.g.
• is more abstract than the domain abstraction
  above

Domain blank 1 2 3 4 5 6 7 8 Abstract
blank
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Problem Non-surjectivity

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Problem Non-surjectivity
Domain blank 1 2 Abstract blank
1 blank

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Problem Non-surjectivity
Domain blank 1 2 Abstract blank
1 blank

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Problem Non-surjectivity
Domain blank 1 2 Abstract blank
1 blank

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Problem Non-surjectivity
Domain blank 1 2 Abstract blank
1 blank

• ??

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Pattern Database Experiments

• Aim
• To understand how search performance using PDBs
  is related to easily measurable characteristics
  of the PDBs
• e.g. size, average value
• Basic Method
• Choose a variety of state spaces.
• For each state space generate thousands of PDBs.
• For each PDB, measure its characteristics and the
  performance of A (IDA etc.) using it.

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8-puzzle A vs. PDB size
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Korf Reid (1998)

• When the depth bound is d, node n at level j will
  be expanded by IDA iff
• a parent(n) was expanded
• b g(n)h(n) ? d, in other words
  h(n) ? d-j
• b ? a if the heuristic is consistent
• Total nodes expanded ?N(j)P(j,d-j)
• N(j) nodes at level j in the brute-force tree
• P(j,x) percentage of nodes at level j with h()
  ? x

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Korf Reid experiment

• In their 8-puzzle experiment
• Use exact N(j)
• Approximate P(j,x) by EQ(x) limit (j??)
  P(j,x)
• IDA, but complete enumeration of last level
• Run all 181,400 start states to all depths

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Korf Reid results

• Seems the ideal tool for choosing which of two
  PDBs is better

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Korf Reid stopping at goal

• For choosing which of two PDBs is better in a
  practical setting, adaptations are needed.

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Using Multiple Abstractions

• Given 2 consistent heuristics, max(h1(s),h2(s))
  is also consistent.
• In some circumstances, can add them.
• How good is max ?
• hope it is at least 2x because it takes 2x the
  space

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Max of 2 random PDBs
max(h1,h2) worse than h1
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Instead of max - interleave
use PDB1
use PDB2
use PDB2
use PDB1
use PDB1
use PDB1
use PDB2
use PDB2
use PDB1
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Interleaved Pattern Databases

• The hope almost as good as max, but only half
  the memory.
• Intuitively, strict alternation between PDBs
  expected to be almost as good as max.
• How to generalize this to any abstraction of any
  space ?

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2 random PDBs interleaved

• 93 random pairs (with non-trivial LCA)
• 4 had Max(h1,h2) gt h1
• 17 others had Interleave(h1,h2) gt h1
• The remaining 72 were normal

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Relative Performance
Max Interleave
h1
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Current Research

• Istvan Hernadvolgyi (Ph.D. student, U. Ottawa)
• automatic creation of good pattern databases
• adaptation to weighted graphs
• Project Students (U of A)
• Jack Newton
• max of two pattern databases
• interleaved pattern databases
• Daniel Neilson - additive abstractions
• Ajit Singh predicting IDA performance

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Future Research

• compression of pattern databases
• understand avoid non-surjectivity
• alternative methods of abstraction
• projection