pattern databases robert holte university of alberta november 6, 2002

Pattern Databases

Robert Holte

University of Alberta

November 6, 2002

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 (< 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))

Pattern Database Successes (2)

Rich Korf (1997)– Rubik’s 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

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

Pattern Database Successes (4) Istvan Hernadvolgyi (2001)

– Macro-operators are concatenated to very quickly construct suboptimal solutions

– For Rubik’s Cube hundreds of macro-operators are needed– Each macro is found by searching in the Rubik’s 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)

Fundamental Questions

How to invent effective heuristics ?

How to use memory to speed up search ?

Create a simplified version of your problem.Use the exact distances in the simplified version as heuristic estimates in the original.

Precompute all distances-to-goal in the simplifiedversion of the problem and store them in alookup table (pattern database).

Example: 8-puzzle

1 2

3 4 5

6 7 8

Domain = blank 1 2 3 4 5 6 7 8

181,440 states

“Patterns”created by domain mapping

1 2

3 4 5

6 7 8

This mapping produces9 patterns

Domain = blank 1 2 3 4 5 6 7 8Abstract = blank

corresponding patternoriginal state

Pattern Database

Pattern

Distance to goal 0 1 1 2 2 2

Pattern

Distance to goal 3 3 4

Calculating h(s)

Given a state in the original problem

Compute the corresponding pattern

and look up the abstract distance-to-goal

8 1 4

3 5

6 7 2

2

Heuristics defined by PDBs are consistent, not just admissible.

Abstract Space

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.

“Pattern” = leave some tiles unique

1 2

3 4 5

6 7 8 6 7 8

3024 patternsDomain = blank 1 2 3 4 5 6 7 8Abstract = blank 6 7 8

Domain Abstraction

1 2

3 4 5

6 7 8 6 7 8

30,240 patternsDomain = blank 1 2 3 4 5 6 7 8Abstract = blank 6 7 8

8-puzzle PDB sizes(with the blank left unique)

9 72 252 504 630 1512 2520 3024 3780 5040 75601008015120

15120 22680 30240 45360 60480 90720181440

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



Problem: Non-surjectivity

1 2

1 2

1 2


1 2

1 2

1 2

Domain = blank 1 2 Abstract = blank 1 blank


1 2

1 2

1 2

1



1 2

1 2

1 2

1

1



??

1 2

1 2

1 2

1

1

1


Pattern Database ExperimentsAim:

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.

8-puzzle: A* vs. PDB size#

node

s ex

pand

ed (

A*)

pattern database size (# of abstract states)

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

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

Korf & Reid – results

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

d prediction avg. error

(all states)

20 376.9 -0.3

21 631.3 -1.4

22 1146.1 -0.4

23 1918.8 -2.7

Korf & Reid – stopping at goald prediction avg. error

(all states)

avg. error

(states d )

20 376.9 -0.3 406

21 631.3 -1.4 500

22 1146.1 -0.4 664

23 1918.8 -2.7 758

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

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

Max of 2 random PDBs

max(h1,h2) worse than h1

Instead of max - interleaveuse

PDB1

usePDB

1

usePDB

1

usePDB

1

usePDB

1

usePDB

1

usePDB

1

usePDB

1

usePDB

1

usePDB

1

usePDB

2

usePDB

2

usePDB

2

usePDB

2

usePDB

2

usePDB

2

usePDB

2

usePDB

2

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 ?

2 random PDBs interleaved

• 93 random pairs (with non-trivial LCA)

• 4 had Max(h1,h2) > h1

• 17 others had Interleave(h1,h2) > h1

• The remaining 72 were “normal”

Max Interleave h1

Relative Performance

Max Interleave h1

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

Future Research

• compression of pattern databases

• understand & avoid non-surjectivity

• alternative methods of abstraction• projection

pattern databases robert holte university of alberta november 6, 2002

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