On Variable Dependencies and Compressed Pattern Databases

Transcription

1 On Variable Dependencies and Compressed Pattern Databases Malte Helmert 1 Nathan Sturtevant Ariel elner 1 University of Basel, Switzerland University of Denver, USA Ben Gurion University, Israel SoCS 017

2 Introduction

3 Quotation previous work on compressed pattern databases: Sturtevant, elner and Helmert (SoCS 014) 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 (elner et al., 007). this paper: try to understand why

4 Compressed PDBs E G H

5 Compressed PDBs E G H h (A) = 6

6 Compressed PDBs E G H E G H h (A) = 6

7 Compressed PDBs E G H E G H h (A) = 6 AB CD E GH IJ KL 4 1 0

8 Compressed PDBs E G H E G H h (A) = 6 h PDB (A) = 4 AB CD E GH IJ KL 4 1 0

9 Compressed PDBs E G H E G H h (A) = 6 h PDB (A) = 4 PDB (A) = AB CD E GH IJ KL

10 Compressed PDBs E G H E G H h (A) = 6 h PDB (A) = 4 PDB (A) = AB CD E GH IJ KL

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

12 Experimental Results State Space M/N h MOD DIV random h C Hanoi Sliding Tiles A Sliding Tiles B TopSpin Hanoi: 4 pegs and 16 disks; pattern with 15 disks Sliding Tiles A: 4 4 puzzle; pattern blank, 1,,, 4, 5, 6 Sliding Tiles B: 4 4 puzzle; pattern 6, 5, 4,,, 1, blank TopSpin: 18 tokens and turnstile size 4; pattern with 7 tokens all use lexicographic ranking

13 Experimental Results State Space M/N h MOD DIV random h C Hanoi Sliding Tiles A Sliding Tiles B TopSpin better than h C on average Hanoi: 4 pegs and 16 disks; pattern with 15 disks Sliding Tiles A: 4 4 puzzle; pattern blank, 1,,, 4, 5, 6 Sliding Tiles B: 4 4 puzzle; pattern 6, 5, 4,,, 1, blank TopSpin: 18 tokens and turnstile size 4; pattern with 7 tokens all use lexicographic ranking

14 Experimental Results State Space M/N h MOD DIV random h C Hanoi Sliding Tiles A Sliding Tiles B TopSpin worse than h C on average Hanoi: 4 pegs and 16 disks; pattern with 15 disks Sliding Tiles A: 4 4 puzzle; pattern blank, 1,,, 4, 5, 6 Sliding Tiles B: 4 4 puzzle; pattern 6, 5, 4,,, 1, blank TopSpin: 18 tokens and turnstile size 4; pattern with 7 tokens all use lexicographic ranking

15 Experimental Results State Space M/N h MOD DIV random h C Hanoi Sliding Tiles A Sliding Tiles B TopSpin equal to h C on average Hanoi: 4 pegs and 16 disks; pattern with 15 disks Sliding Tiles A: 4 4 puzzle; pattern blank, 1,,, 4, 5, 6 Sliding Tiles B: 4 4 puzzle; pattern 6, 5, 4,,, 1, blank TopSpin: 18 tokens and turnstile size 4; pattern with 7 tokens all use lexicographic ranking

16 Good News

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

18 Dominance of Compressed PDBs: Proof Idea E G H E G H h (A) = 6 h (A) = 4 (A) = AB CD E GH IJ KL

19 Dominance of Compressed PDBs: Proof Idea E G H E G H E G H h (A) = 6 h (A) = 4 (A) = AB CD E GH IJ KL AB CD E GH IJ KL 1 0

20 Dominance of Compressed PDBs: Proof Idea E G H E G H E G H h (A) = 6 h (A) = 4 (A) = h C (A) = AB CD E GH IJ KL AB CD E GH IJ KL 1 0

21 Dominance of Compressed PDBs: Experimental Results State Space M/N h MOD DIV random h C Hanoi Sliding Tiles A Sliding Tiles B TopSpin Hanoi: 4 pegs and 16 disks; pattern with 15 disks Sliding Tiles A: 4 4 puzzle; pattern blank, 1,,, 4, 5, 6 Sliding Tiles B: 4 4 puzzle; pattern 6, 5, 4,,, 1, blank TopSpin: 18 tokens and turnstile size 4; pattern with 7 tokens all use lexicographic ranking

22 Dominance of Compressed PDBs: Experimental Results State Space M/N h MOD DIV random h C Hanoi Sliding Tiles A Sliding Tiles B TopSpin (s) h C (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,,, 4, 5, 6 Sliding Tiles B: 4 4 puzzle; pattern 6, 5, 4,,, 1, blank TopSpin: 18 tokens and turnstile size 4; pattern with 7 tokens all use lexicographic ranking

23 Bad News

24 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 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 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 Improvements vs. Dependencies Theorem (no improvements without dependencies) Consider the patterns C in an undirected state space. Let be a compressed PDB heuristic with a compression regime compatible with the refinement relation between and C. If no variable in C depends on any variable in \ C, then for all states s. (s) = h C (s)

28 Improvements vs. Dependencies: Proof Idea E G H E G H E G H h (A) = 4 h (A) = (A) = h C (A) = AB CD E GH IJ KL AB CD E GH IJ KL 1 0

29 Improvements vs. Dependencies: Experimental Results State Space M/N h MOD DIV random h C Hanoi Sliding Tiles A Sliding Tiles B TopSpin Hanoi: 4 pegs and 16 disks; pattern with 15 disks Sliding Tiles A: 4 4 puzzle; pattern blank, 1,,, 4, 5, 6 Sliding Tiles B: 4 4 puzzle; pattern 6, 5, 4,,, 1, blank TopSpin: 18 tokens and turnstile size 4; pattern with 7 tokens all use lexicographic ranking

30 Improvements vs. Dependencies: Experimental Results State Space M/N h MOD DIV random h C Hanoi Sliding Tiles A Sliding Tiles B TopSpin (s) = h C (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,,, 4, 5, 6 Sliding Tiles B: 4 4 puzzle; pattern 6, 5, 4,,, 1, blank TopSpin: 18 tokens and turnstile size 4; pattern with 7 tokens all use lexicographic ranking

31 Related Work in Classical Planning our result: = h C for undirected state spaces under certain dependency conditions literature (Haslum et al. 007; Pommerening et al. 01): h = h C for arbitrary state spaces under certain (different) dependency conditions neither result entails the other many more details in paper

32 Related Work in Classical Planning our result: = h C for undirected state spaces under certain dependency conditions literature (Haslum et al. 007; Pommerening et al. 01): h = h C for arbitrary state spaces under certain (different) dependency conditions neither result entails the other many more details in paper

33 Conclusion

34 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)

35 Thank You Thank you for your attention!