Outline. Where Do Heuristics Come From? Part 3. Question. Max ing Multiple Heuristics

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1 Outline Where Do Heuristics Come From? Part Robert C. Holte Computing Science Department University of Alberta 00, Robert Holte Heuristics/Holte Part, Slide Part : Introduction Part : Details Part : Pattern Database Enhancements Taking the maximum of two or more PDBs Compression and Dovetailing of PDBs Additive PDBs Customized PDBs Multiple Lookups in One PDB Bonus! Related Algorithm (CFPD) Heuristics/Holte Part, Slide Max ing Multiple Heuristics Given heuristics h and h define h(s) = max ( h(s), h(s) ) Preserves key properties: lower bound consistency Question Given a fixed amount of memory, M, which gives the best heuristic? pattern database (PDB) of size M max ing PDBs of size M/ max ing PDBs of size M/ etc. Heuristics/Holte Part, Slide Heuristics/Holte Part, Slide

2 large pattern database s ϕ half-size pattern databases s ϕ ϕ h(s) h (s) h (s) max Heuristics/Holte Part, Slide Heuristics/Holte Part, Slide Many small pattern databases ϕ h (s) s ϕ n h n (s) Rubik s Cube* PDB Size n Nodes Generated,0,00 8,,89 7,70,800,9,99,,00,09,99,,00,9,89 0,,800,, max Heuristics/Holte Part, Slide 7 * easy problems Heuristics/Holte Part, Slide 8

3 Summary State Space Best n (x)-puzzle 0 9-pancake 0 (8,)-Topspin ( ops) 9 (8,)-Topspin (8 ops) 9 (x)-puzzle + Rubik s Cube -puzzle (additive) -puzzle (additive) 8 Ratio to. Rubik s Cube CPU Time #PDBs Nodes Ratio Time Ratio RATIO = #nodes generated using one PDB of size M #nodes generated using n PDBs of size M/n time/node is.7x higher using six PDBs Heuristics/Holte Part, Slide 9 Heuristics/Holte Part, Slide 0 Early Stopping Techniques for Reducing the Overhead of Multiple PDB lookups IDA* depth bound = 7 g(s) = Stop doing PDB lookups as soon as h > is found. Might result in extra IDA* iterations PDB (s) = next bound is 8 PDB (s) = 7 next bound is 0 Heuristics/Holte Part, Slide Heuristics/Holte Part, Slide

4 Consistency-based Bounding Experimental Results PDB (A) = PDB (A) = 7 -puzzle, five additive PDBs (7-7-) Naïve: 0. secs Early Stopping: 0.0 secs Because of consistency: PDB (B) PDB (B) No need to consult PDB Rubik s Cube, six non-additive PDBs Naïve: 7. secs Early Stopping: 8.9 secs Early Stopping and Bounding: 8.8 secs Heuristics/Holte Part, Slide Heuristics/Holte Part, Slide Static Distribution of Heuristic Values max of small PDBs. Why Does Max ing Speed Up Search? -puzzle, 00M states. large PDB..8x nodes generated Heuristics/Holte Part, Slide Heuristics/Holte Part, Slide

5 Runtime Distribution of Heuristic Values Example of Max Failing Depth Bound h h max(h,h) ,0,8 8 7, ,79,,0 9,80, 0,97,89 80 TOTAL,8, 8, Heuristics/Holte Part, Slide 7 Heuristics/Holte Part, Slide 8 Approaches Squeezing More into Memory Compress an individual Pattern Database Lossless compression Lossy compression must maintain admissibility Allows you to use a PDB bigger than will fit in memory use multiple PDBs instead of just one Merge two PDBs into one the same size Culberson & Schaeffer s dovetailing Heuristics/Holte Part, Slide 9 Heuristics/Holte Part, Slide 0

6 Compression Results -disk -peg TOH, PDB based on disks No compression: Megs memory,. secs lossless compression: k memory,.8 secs Lossy compression: 9Megs,.9 secs -puzzle, additive PDB triple (7-7-) No compression: 7Megs memory, 0.09 secs Lossy compression, two PDB triples 7Megs memory, 0.0 secs Dovetailing Given PDBs for a state space construct a hybrid containing some entries from each of them, so that the total number of entries is the same as in one of the originals. The hope: almost as good as max, but only half the memory. Heuristics/Holte Part, Slide Heuristics/Holte Part, Slide Dovetailing based on the blank PDB PDB PDB PDB PDB PDB PDB PDB PDB PDB PDB PDB Any colouring is possible???????????? PDB PDB PDB PDB???? PDB PDB?? Heuristics/Holte Part, Slide Heuristics/Holte Part, Slide

7 Dovetailing selection rule Dovetailing compared to Max ing Dovetailing requires a rule that maps each state, s, to one of the PDBs. Use that PDB to compute h(s). Any rule will work, but they won t all give the same performance. Intuitively, strict alternation between PDBs expected to be almost as good as max. max failures 7 dovetail failures Heuristics/Holte Part, Slide Heuristics/Holte Part, Slide Experimental Results Culberson & Schaeffer (99): Dovetailing two PDBs reduced #nodes generated by a factor of. compared to using either PDB alone How to generalize Dovetailing to any abstractions of any space? Holte & Newton (unpublished): Dovetailing halved #nodes generated on average Heuristics/Holte Part, Slide 7 Heuristics/Holte Part, Slide 8

8 A Partial-Order on Domain Abstractions Easy to enumerate all possible domain abstractions Domain = blank 7 8 Abstract = blank Lattice of domain abstractions and to define a partial-order on them, e.g. Domain = blank 7 8 Abstract = blank is more abstract than the domain abstraction above. Heuristics/Holte Part, Slide 9 Heuristics/Holte Part, Slide 0 The LCA of Abstractions General Dovetailing Domain = blank 7 8 Abstract = blank Domain = blank 7 8 Abstract = blank blank 7 8 blank LCA = least-abstract common abstraction Given PDB and PDB defined by ϕ and ϕ Find a common abstraction ϕ of ϕ and ϕ Because it is a common abstraction there exist ϕ and ϕ such that ϕ ϕ = ϕ ϕ = ϕ For every pattern, p, defined by ϕ, set SELECT[p] = ϕ or ϕ Keep every entry (p k,h) from PDB i for which SELECT[ϕ i (p k )]=i. Given state s. ϕ s = SELECT[ϕ(s)]. h(s) = PDB[ϕ s (s)] Heuristics/Holte Part, Slide Heuristics/Holte Part, Slide

9 Adding instead of Max ing Additive Pattern Databases Under some circumstances it is possible to add the values from two PDBs instead of just max ing them and still have an admissible heuristic. This is advantageous because* h (s) + h (s) max( h (s), h (s) ) * but see slide Compared to Max ing Heuristics/Holte Part, Slide Heuristics/Holte Part, Slide Manhattan Distance Heuristic M.D. as Additive PDBs () For a sliding-tile puzzle, Manhattan Distance looks at each tile individually, counts how many moves it is away from its goal position, and adds up these numbers. ϕ (x)= x if x = blank otherwise PDB [ϕ (s) ] = MD(s) = + + = ϕ (goal) ϕ (s) MD(s) = PDB [ϕ (s) ] + PDB [ϕ (s) ] goal state s + PDB [ϕ (s) ] Heuristics/Holte Part, Slide Heuristics/Holte Part, Slide

10 In General Korf & Felner s Method Partition the tiles in groups, G, G, G k Partition the tiles in groups, G, G, G k ϕ i (x)= x if x G i blank otherwise ϕ i (x)= x if x G i blank if x = blank otherwise Moves of cost zero Heuristics/Holte Part, Slide 7 Heuristics/Holte Part, Slide 8 What s the Difference? Compared to Max ing the blank cannot reach this position without disturbing tile or tile. If the PDBs were going to be max d instead of added, we would count all the moves in all the PDBs. Therefore the PDBs for adding have smaller entries than the corresponding PDBs for max ing. In initial experiments on the -puzzle, max ing returns a higher value than adding for about % of the states. Heuristics/Holte Part, Slide 9 Heuristics/Holte Part, Slide 0

11 8-7 Partition (7 million entries) Max ing After Adding * 7 8 tiles retain their identity 7 tiles retain their identity * movement of coloured tiles not counted Heuristics/Holte Part, Slide Heuristics/Holte Part, Slide 7-7- Partition ( million entries) Max ing after Adding For a given 7-7- partition, look up the values and add them. Do this for each of the five 7-7- partitions and take the maximum*. + Also compute Manhattan Distance, and use that if it is largest of all. This was also done for 8-7. Heuristics/Holte Part, Slide Heuristics/Holte Part, Slide

12 -Puzzle Results Partition 7-7- n Nodes Generated 7,9 Customized PDBs 8-7,8 8% reduction in #nodes generated, but only 0% reduction in CPU time. Heuristics/Holte Part, Slide Heuristics/Holte Part, Slide Space-Efficient PDBs Zhou & Hansen (AAAI, 00) Do not generate PDB entries that are provably not needed to solve the given problem. Prune abstract state A if f(a) > U, where U is an upper bound on the solution cost at the base level. To work well, needs a heuristic to guide the abstract search and a fairly tight U. Even then requires significantly more memory than Hierarchical IDA*. Silver, 00 Reverse Resumable A* Aims to minimize the number of PDB entries Backward search from abstract goal stops when abstract start is reached If h(x) is needed and has not been computed, resume the abstract search until you get it. Requires abstract Open and Closed lists. Heuristics/Holte Part, Slide 7 Heuristics/Holte Part, Slide 8

13 Super-Customization If customizing an abstraction for a given start state is a good idea, wouldn t it be even better to change abstractions in the middle of the search space to exploit local properties? This does pay off sometimes, even for PDBs: Felner, Korf & Hanan (00) Hernadvolgyi (00; also PhD thesis, chapter ) Multiple Lookups in One Pattern Database Heuristics/Holte Part, Slide 9 Heuristics/Holte Part, Slide 0 Use Symmetries Example c d Domain = blank 7 8 Abstract = blank c a e d e b mirror positions goal 7 distance(pos,c ) = distance(mirror(pos),c) state normal PDB lookup mirror lookup Heuristics/Holte Part, Slide Heuristics/Holte Part, Slide

14 Dual PDB Lookups Standard PDB lookup 8 distance? state goal abstract state abstract goal Domain = blank 7 8 Abstract = blank Heuristics/Holte Part, Slide Heuristics/Holte Part, Slide Dual lookup, same PDB Relevance? Why is this lookup abstract state Heuristics/Holte Part, Slide abstract goal relevant to the original state? 8 7 Heuristics/Holte Part, Slide

15 Tile In a PDB for tile, this lookup Two Key Properties () Distances are Symmetric is relevant to the original state. 8 7 Heuristics/Holte Part, Slide 7 () Distances are tile-independent Heuristics/Holte Part, Slide 8 Third Key Property () Can determine which tiles in the given state correspond to the key tiles in the goal state. Domain = blank 7 8 Abstract = blank Fourth Key Property () The tiles that correspond to the key tiles in the goal state occur in the goal state.??? Domain = blank 7 8 Abstract = blank state 7 8 goal 7 state 7 8 goal Heuristics/Holte Part, Slide 9 Heuristics/Holte Part, Slide 0

16 Experimental Results -disk, -peg TOH, PDB of disks Normal: 7. secs Only the dual lookup:. secs Both lookups:. secs -puzzle, additive PDB (8-7) Normal: 0.0 secs Only the dual lookup: 0.07 secs Both lookups: 0.0 secs Heuristics/Holte Part, Slide Dual Not Always Consistent 7 Domain = 7 Abstract = 7 Heuristics/Holte Part, Slide goal (7-pancake puzzle) abstract distance = 0 7 abstract distance = Bidirectional Pathmax h(a) = X Related Algorithm CFPD h(b) = 7 h(c) = X Heuristics/Holte Part, Slide Heuristics/Holte Part, Slide

17 CFDP CFDP - Example Coarse-to-Fine Dynamic Programming Works on continuous or discrete spaces. Most easily explained if space is a trellis (level structure). Abstraction = grouping states on the same level. Multiple levels of abstraction. Resembles refinement, but guaranteed to find optimal solution. Application: finding optimal convex region boundaries in an image. S More A C E H B D F I G Heuristics/Holte Part, Slide Heuristics/Holte Part, Slide CFDP Coarsest States CFDP Abstract Edges S 0 0 A C 0 B D 0 0 G Optimal solution S G 0 E 0 F 0 0 H 0 I More, all > 0 at coarsest level Heuristics/Holte Part, Slide 7 More, all > 0 at coarsest level Heuristics/Holte Part, Slide 8

18 CFDP Refine Optimal Path CFDP Refine Optimal Path Optimal solution 0 S A C G Optimal solution 0 S A C 0 B D 0 G More, all > 0 at coarsest level Heuristics/Holte Part, Slide 9 More, all > 0 at coarsest level Heuristics/Holte Part, Slide 70 CFDP Refine Again CFDP Final Iteration Optimal solution 0 S A C 0 B D 0 0 G Final solution 0 S 0 A C 0 B D 0 0 G 0 0 F 0 0 E 0 F 0 0 I H 0 I More, all > 0 at coarsest level Heuristics/Holte Part, Slide 7 More, all > 0 at Still at the coarsest level coarsest level Heuristics/Holte Part, Slide 7