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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,

  1. 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, Israel SoCS 2017

  2. Introduction Good News Bad News Conclusion Introduction

  3. 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

  4. Introduction Good News Bad News Conclusion Compressed PDBs A B C D E F G H I J K L

  5. Introduction Good News Bad News Conclusion Compressed PDBs A B C D E F G H I J K L h ∗ (A) = 6

  6. Introduction Good News Bad News Conclusion Compressed PDBs A B C D A B C D ⇒ E F G H E F G H I J K L I J K L h ∗ (A) = 6

  7. Introduction Good News Bad News Conclusion Compressed PDBs A B C D A B C D ⇒ E F G H E F G H I J K L I J K L ⇓ h ∗ (A) = 6 AB 4 CD 3 2 EF GH 3 IJ 1 KL 0

  8. Introduction Good News Bad News Conclusion Compressed PDBs A B C D A B C D ⇒ E F G H E F G H I J K L I J K L ⇓ h ∗ (A) = 6 AB 4 h PDB (A) = 4 CD 3 2 EF GH 3 IJ 1 KL 0

  9. Introduction Good News Bad News Conclusion Compressed PDBs A B C D A B C D ⇒ E F G H E F G H I J K L I J K L ⇓ h ∗ (A) = 6 AB 4 h PDB (A) = 4 3 CD 3 h comp PDB (A) = 3 2 EF 2 GH 3 IJ 1 0 KL 0

  10. Introduction Good News Bad News Conclusion Compressed PDBs A B C D A B C D ⇒ E F G H E F G H I J K L I J K L ⇓ h ∗ (A) = 6 AB 4 h PDB (A) = 4 3 CD 3 h comp PDB (A) = 3 2 EF 2 GH 3 IJ 1 0 KL 0

  11. Introduction Good News Bad News Conclusion Comparing PDBs to Compressed PDBs Assume we have N units of memory. Consider three heuristics: h F : fine-grained PDB ( M ≫ N entries) h comp : compressed fine-grained PDB ( N entries) F h C : coarse-grained PDB ( N entries) Which one should we use, h comp or h C ? F

  12. Introduction Good News Bad News Conclusion Experimental Results h comp F State Space M / N h F MOD DIV random h C 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

  13. Introduction Good News Bad News Conclusion Experimental Results h comp F State Space M / N h F MOD DIV random h C 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 h comp better than h C on average F 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

  14. Introduction Good News Bad News Conclusion Experimental Results h comp F State Space M / N h F MOD DIV random h C 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 h comp worse than h C on average F 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

  15. Introduction Good News Bad News Conclusion Experimental Results h comp F State Space M / N h F MOD DIV random h C 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 h comp equal to h C on average F 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

  16. Introduction Good News Bad News Conclusion Good News

  17. Introduction Good News Bad News Conclusion Dominance of Compressed PDBs Theorem (dominance of compressed PDBs) Let h F and h C be heuristics such that h F is a refinement of h C . Consider compressed heuristics with a compression regime that is compatible with h F and h C . Then h comp ( s ) ≥ h C ( s ) F for all states s. informally: compression step applies further abstraction on top of the abstraction h F

  18. Introduction Good News Bad News Conclusion Dominance of Compressed PDBs: Proof Idea A B C D A B C D ⇒ E F G H E F G H I J K L I J K L ⇓ h ∗ (A) = 6 AB 4 h F (A) = 4 3 CD 3 h comp (A) = 3 F 2 EF 2 GH 3 IJ 1 0 KL 0

  19. Introduction Good News Bad News Conclusion Dominance of Compressed PDBs: Proof Idea A B C D A B C D A B C D ⇒ ⇒ E F G H E F G H E F G H I J K L I J K L I J K L ⇓ ⇓ h ∗ (A) = 6 AB 4 AB h F (A) = 4 3 2 CD CD 3 h comp (A) = 3 F 2 EF EF 2 1 GH GH 3 IJ 1 IJ 0 0 KL KL 0

  20. Introduction Good News Bad News Conclusion Dominance of Compressed PDBs: Proof Idea A B C D A B C D A B C D ⇒ ⇒ E F G H E F G H E F G H I J K L I J K L I J K L ⇓ ⇓ h ∗ (A) = 6 AB 4 AB h F (A) = 4 3 2 CD CD 3 h comp (A) = 3 F 2 EF EF 2 1 h C (A) = 2 GH GH 3 IJ 1 IJ 0 0 KL KL 0

  21. Introduction Good News Bad News Conclusion Dominance of Compressed PDBs: Experimental Results h comp F State Space M / N h F MOD DIV random h C 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

  22. Introduction Good News Bad News Conclusion Dominance of Compressed PDBs: Experimental Results h comp F State Space M / N h F MOD DIV random h C 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 h comp ( s ) ≥ h C ( s ) for all states according to the theorem F 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

  23. Introduction Good News Bad News Conclusion Bad News

  24. 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”).

  25. 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

  26. 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

  27. 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 h comp be a compressed PDB heuristic with a compression F regime compatible with the refinement relation between F and C. If no variable in C depends on any variable in F \ C, then h comp ( s ) = h C ( s ) F for all states s.

  28. Introduction Good News Bad News Conclusion Improvements vs. Dependencies: Proof Idea A B C D A B C D A B C D ⇒ ⇒ E F G H E F G H E F G H I J K L I J K L I J K L ⇓ ⇓ h ∗ (A) = 4 AB 3 AB h F (A) = 3 2 2 CD CD 2 h comp (A) = 2 F 2 EF EF 1 1 h C (A) = 2 GH GH 1 IJ 1 IJ 0 0 KL KL 0

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