On Variable Dependencies and Compressed Pattern Databases Malte - - PowerPoint PPT Presentation

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On Variable Dependencies and Compressed Pattern Databases Malte - - PowerPoint PPT Presentation

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Introduction Good News Bad News Conclusion On Variable Dependencies and Compressed Pattern Databases Malte Helmert 1 Nathan Sturtevant 2 Ariel Felner 3 1 University of Basel, Switzerland 2 University of Denver, USA 3 Ben Gurion University,

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Introduction Good News Bad News Conclusion

On Variable Dependencies and Compressed Pattern Databases

Malte Helmert1 Nathan Sturtevant2 Ariel Felner3

1University of Basel, Switzerland 2University of Denver, USA 3Ben Gurion University, Israel

SoCS 2017

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Introduction Good News Bad News Conclusion

Introduction

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Introduction Good News Bad News Conclusion

Quotation

previous work on compressed pattern databases: Sturtevant, Felner and Helmert (SoCS 2014) “This approach worked very well for the 4-peg Towers of Hanoi, for instance, but its success for the sliding tile puzzles was limited and no significant advantage was reported for the Top-Spin domain (Felner et al., 2007).” this paper: try to understand why

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Introduction Good News Bad News Conclusion

Compressed PDBs

A B C D E F G H I J K L

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Introduction Good News Bad News Conclusion

Compressed PDBs

A B C D E F G H I J K L

h∗(A) = 6

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Introduction Good News Bad News Conclusion

Compressed PDBs

A B C D E F G H I J K L

h∗(A) = 6 ⇒

A B C D E F G H I J K L

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Introduction Good News Bad News Conclusion

Compressed PDBs

A B C D E F G H I J K L

h∗(A) = 6 ⇒

A B C D E F G H I J K L

⇓

AB CD EF GH IJ KL 4 3 2 3 1

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Introduction Good News Bad News Conclusion

Compressed PDBs

A B C D E F G H I J K L

h∗(A) = 6 hPDB(A) = 4 ⇒

A B C D E F G H I J K L

⇓

AB CD EF GH IJ KL 4 3 2 3 1

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SLIDE 9

Introduction Good News Bad News Conclusion

Compressed PDBs

A B C D E F G H I J K L

h∗(A) = 6 hPDB(A) = 4 hcomp

PDB (A) = 3

⇒

A B C D E F G H I J K L

⇓

AB CD EF GH IJ KL 4 3 2 3 1 3 2

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Introduction Good News Bad News Conclusion

Compressed PDBs

A B C D E F G H I J K L

h∗(A) = 6 hPDB(A) = 4 hcomp

PDB (A) = 3

⇒

A B C D E F G H I J K L

⇓

AB CD EF GH IJ KL 4 3 2 3 1 3 2

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Introduction Good News Bad News Conclusion

Comparing PDBs to Compressed PDBs

Assume we have N units of memory. Consider three heuristics: hF: fine-grained PDB (M ≫ N entries) hcomp

F

: compressed fine-grained PDB (N entries) hC: coarse-grained PDB (N entries) Which one should we use, hcomp

F

  • r hC?
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Introduction Good News Bad News Conclusion

Experimental Results

hcomp

F

State Space M/N hF MOD DIV random hC Hanoi 4 104.32 87.04 103.76 90.08 87.04 Sliding Tiles A 10 34.99 29.89 32.08 26.38 32.08 Sliding Tiles B 10 34.99 30.50 32.84 26.38 15.29 TopSpin 12 10.78 9.29 9.59 8.73 9.59 Hanoi: 4 pegs and 16 disks; pattern with 15 disks Sliding Tiles A: 4 × 4 puzzle; pattern blank, 1, 2, 3, 4, 5, 6 Sliding Tiles B: 4 × 4 puzzle; pattern 6, 5, 4, 3, 2, 1, blank TopSpin: 18 tokens and turnstile size 4; pattern with 7 tokens all use lexicographic ranking

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Introduction Good News Bad News Conclusion

Experimental Results

hcomp

F

State Space M/N hF MOD DIV random hC Hanoi 4 104.32 87.04 103.76 90.08 87.04 Sliding Tiles A 10 34.99 29.89 32.08 26.38 32.08 Sliding Tiles B 10 34.99 30.50 32.84 26.38 15.29 TopSpin 12 10.78 9.29 9.59 8.73 9.59 hcomp

F

better than hC on average Hanoi: 4 pegs and 16 disks; pattern with 15 disks Sliding Tiles A: 4 × 4 puzzle; pattern blank, 1, 2, 3, 4, 5, 6 Sliding Tiles B: 4 × 4 puzzle; pattern 6, 5, 4, 3, 2, 1, blank TopSpin: 18 tokens and turnstile size 4; pattern with 7 tokens all use lexicographic ranking

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Introduction Good News Bad News Conclusion

Experimental Results

hcomp

F

State Space M/N hF MOD DIV random hC Hanoi 4 104.32 87.04 103.76 90.08 87.04 Sliding Tiles A 10 34.99 29.89 32.08 26.38 32.08 Sliding Tiles B 10 34.99 30.50 32.84 26.38 15.29 TopSpin 12 10.78 9.29 9.59 8.73 9.59 hcomp

F

worse than hC on average Hanoi: 4 pegs and 16 disks; pattern with 15 disks Sliding Tiles A: 4 × 4 puzzle; pattern blank, 1, 2, 3, 4, 5, 6 Sliding Tiles B: 4 × 4 puzzle; pattern 6, 5, 4, 3, 2, 1, blank TopSpin: 18 tokens and turnstile size 4; pattern with 7 tokens all use lexicographic ranking

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Introduction Good News Bad News Conclusion

Experimental Results

hcomp

F

State Space M/N hF MOD DIV random hC Hanoi 4 104.32 87.04 103.76 90.08 87.04 Sliding Tiles A 10 34.99 29.89 32.08 26.38 32.08 Sliding Tiles B 10 34.99 30.50 32.84 26.38 15.29 TopSpin 12 10.78 9.29 9.59 8.73 9.59 hcomp

F

equal to hC on average Hanoi: 4 pegs and 16 disks; pattern with 15 disks Sliding Tiles A: 4 × 4 puzzle; pattern blank, 1, 2, 3, 4, 5, 6 Sliding Tiles B: 4 × 4 puzzle; pattern 6, 5, 4, 3, 2, 1, blank TopSpin: 18 tokens and turnstile size 4; pattern with 7 tokens all use lexicographic ranking

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Introduction Good News Bad News Conclusion

Good News

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Introduction Good News Bad News Conclusion

Dominance of Compressed PDBs

Theorem (dominance of compressed PDBs) Let hF and hC be heuristics such that hF is a refinement of hC. Consider compressed heuristics with a compression regime that is compatible with hF and hC. Then hcomp

F

(s) ≥ hC(s) for all states s. informally: compression step applies further abstraction

  • n top of the abstraction hF
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Introduction Good News Bad News Conclusion

Dominance of Compressed PDBs: Proof Idea

A B C D E F G H I J K L

h∗(A) = 6 hF(A) = 4 hcomp

F

(A) = 3 ⇒

A B C D E F G H I J K L

⇓

AB CD EF GH IJ KL 4 3 2 3 1 3 2

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SLIDE 19

Introduction Good News Bad News Conclusion

Dominance of Compressed PDBs: Proof Idea

A B C D E F G H I J K L

h∗(A) = 6 hF(A) = 4 hcomp

F

(A) = 3 ⇒

A B C D E F G H I J K L

⇓

AB CD EF GH IJ KL 4 3 2 3 1 3 2

⇒

A B C D E F G H I J K L

⇓

AB CD EF GH IJ KL 2 1

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Introduction Good News Bad News Conclusion

Dominance of Compressed PDBs: Proof Idea

A B C D E F G H I J K L

h∗(A) = 6 hF(A) = 4 hcomp

F

(A) = 3 hC(A) = 2 ⇒

A B C D E F G H I J K L

⇓

AB CD EF GH IJ KL 4 3 2 3 1 3 2

⇒

A B C D E F G H I J K L

⇓

AB CD EF GH IJ KL 2 1

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Introduction Good News Bad News Conclusion

Dominance of Compressed PDBs: Experimental Results

hcomp

F

State Space M/N hF MOD DIV random hC Hanoi 4 104.32 87.04 103.76 90.08 87.04 Sliding Tiles A 10 34.99 29.89 32.08 26.38 32.08 Sliding Tiles B 10 34.99 30.50 32.84 26.38 15.29 TopSpin 12 10.78 9.29 9.59 8.73 9.59 Hanoi: 4 pegs and 16 disks; pattern with 15 disks Sliding Tiles A: 4 × 4 puzzle; pattern blank, 1, 2, 3, 4, 5, 6 Sliding Tiles B: 4 × 4 puzzle; pattern 6, 5, 4, 3, 2, 1, blank TopSpin: 18 tokens and turnstile size 4; pattern with 7 tokens all use lexicographic ranking

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Introduction Good News Bad News Conclusion

Dominance of Compressed PDBs: Experimental Results

hcomp

F

State Space M/N hF MOD DIV random hC Hanoi 4 104.32 87.04 103.76 90.08 87.04 Sliding Tiles A 10 34.99 29.89 32.08 26.38 32.08 Sliding Tiles B 10 34.99 30.50 32.84 26.38 15.29 TopSpin 12 10.78 9.29 9.59 8.73 9.59 hcomp

F

(s) ≥ hC(s) for all states according to the theorem Hanoi: 4 pegs and 16 disks; pattern with 15 disks Sliding Tiles A: 4 × 4 puzzle; pattern blank, 1, 2, 3, 4, 5, 6 Sliding Tiles B: 4 × 4 puzzle; pattern 6, 5, 4, 3, 2, 1, blank TopSpin: 18 tokens and turnstile size 4; pattern with 7 tokens all use lexicographic ranking

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Introduction Good News Bad News Conclusion

Bad News

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Introduction Good News Bad News Conclusion

State Variables

States are described in terms of state variables. Examples: Towers of Hanoi: position of one disk sliding tiles: position of a tile (or blank) TopSpin: position of a token PDBs project to a subset of variables (the “pattern”).

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Introduction Good News Bad News Conclusion

Variable Dependencies

Variable u depends on variable v if changing u is conditioned in any way on v. Towers of Hanoi sliding tiles TopSpin

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Introduction Good News Bad News Conclusion

Variable Dependencies

Variable u depends on variable v if changing u is conditioned in any way on v. Towers of Hanoi sliding tiles TopSpin

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Introduction Good News Bad News Conclusion

Improvements vs. Dependencies

Theorem (no improvements without dependencies) Consider the patterns F ⊇ C in an undirected state space. Let hcomp

F

be a compressed PDB heuristic with a compression regime compatible with the refinement relation between F and C. If no variable in C depends on any variable in F \ C, then hcomp

F

(s) = hC(s) for all states s.

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Introduction Good News Bad News Conclusion

Improvements vs. Dependencies: Proof Idea

A B C D E F G H I J K L

h∗(A) = 4 hF(A) = 3 hcomp

F

(A) = 2 hC(A) = 2 ⇒

A B C D E F G H I J K L

⇓

AB CD EF GH IJ KL 3 2 2 1 1 2 1

⇒

A B C D E F G H I J K L

⇓

AB CD EF GH IJ KL 2 1

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Introduction Good News Bad News Conclusion

Improvements vs. Dependencies: Experimental Results

hcomp

F

State Space M/N hF MOD DIV random hC Hanoi 4 104.32 87.04 103.76 90.08 87.04 Sliding Tiles A 10 34.99 29.89 32.08 26.38 32.08 Sliding Tiles B 10 34.99 30.50 32.84 26.38 15.29 TopSpin 12 10.78 9.29 9.59 8.73 9.59 Hanoi: 4 pegs and 16 disks; pattern with 15 disks Sliding Tiles A: 4 × 4 puzzle; pattern blank, 1, 2, 3, 4, 5, 6 Sliding Tiles B: 4 × 4 puzzle; pattern 6, 5, 4, 3, 2, 1, blank TopSpin: 18 tokens and turnstile size 4; pattern with 7 tokens all use lexicographic ranking

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Introduction Good News Bad News Conclusion

Improvements vs. Dependencies: Experimental Results

hcomp

F

State Space M/N hF MOD DIV random hC Hanoi 4 104.32 87.04 103.76 90.08 87.04 Sliding Tiles A 10 34.99 29.89 32.08 26.38 32.08 Sliding Tiles B 10 34.99 30.50 32.84 26.38 15.29 TopSpin 12 10.78 9.29 9.59 8.73 9.59 hcomp

F

(s) = hC(s) for all states according to the theorem Hanoi: 4 pegs and 16 disks; pattern with 15 disks Sliding Tiles A: 4 × 4 puzzle; pattern blank, 1, 2, 3, 4, 5, 6 Sliding Tiles B: 4 × 4 puzzle; pattern 6, 5, 4, 3, 2, 1, blank TopSpin: 18 tokens and turnstile size 4; pattern with 7 tokens all use lexicographic ranking

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Introduction Good News Bad News Conclusion

Related Work in Classical Planning

  • ur result:

hcomp

F

= hC for undirected state spaces under certain dependency conditions literature (Haslum et al. 2007; Pommerening et al. 2013): hF = hC for arbitrary state spaces under certain (different) dependency conditions neither result entails the other many more details in paper

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Introduction Good News Bad News Conclusion

Related Work in Classical Planning

  • ur result:

hcomp

F

= hC for undirected state spaces under certain dependency conditions literature (Haslum et al. 2007; Pommerening et al. 2013): hF = hC for arbitrary state spaces under certain (different) dependency conditions neither result entails the other many more details in paper

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Introduction Good News Bad News Conclusion

Conclusion

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Introduction Good News Bad News Conclusion

Conclusion

When is entry compression a good idea? never bad when compatible with refinement never good when refinement does not capture a dependency What does this mean for the benchmarks? Towers of Hanoi: must compress smaller disks away sliding tile: compressing blank the only useful refinement TopSpin: no dependencies, hence no gain (ditto: Pancakes, Rubik’s Cube)

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Introduction Good News Bad News Conclusion

Thank You

Thank you for your attention!