What is a Pattern Database? Pattern Databases. Success Story #1. Success Story #2. Robert Holte Computing Science Dept. University of Alberta

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1 What is a Pattern Database Pattern Databases Robert Holte Computing Science Dept. University of Alberta November, 00 PDB = a heuristic stored as a lookup table Invented by Culberson and Schaeffer (99) created by abstracting the state space Key properties: guaranteed to be a lower bound guaranteed to be consistent the bigger the better (as a general rule) Success Story # Joe Culberson & Jonathan Schaeffer (99). 5-puzzle (0 states). hand-crafted patterns ( fringe (FR) and corner (CO)) Each PDB contains >500 million entries Used symmetries to compress and enhance the use of the PDBs Used in conjunction with Manhattan Distance (MD) Reduction in size of search tree: MD = * max(md,fr) MD = 7 * max(md,co) MD = 0 * max(md, dovetail(fr,co)) + tricks Success Story # Rich Korf (997) Rubik s Cube (0 9 states). hand-crafted patterns, all used together (max) Each PDB contains over million entries took hour to build all the PDBs Results: First time random instances had been solved optimally Hardest (solution length ) took 7 days Best known MD-like heuristic would have taken a century

2 Example: -puzzle Patterns created by domain abstraction ,0 states original state corresponding pattern Domain = blank 5 7 Domain = blank 5 7 Abstract = blank This abstraction produces 9 patterns Pattern Space Pattern Database goal pattern Pattern Distance to goal 0 Pattern Distance to goal

3 Calculating h(s) Domain Abstraction Given a state in the original problem 7 5 Compute the corresponding pattern Look up the abstract distance-to-goal Domain = blank 5 7 Abstract = blank 7 0,0 patterns Fundamental Questions -puzzle: A* vs. PDB size How to invent effective heuristics Create a simplified version of your problem. Use the exact distances in the simplified version as heuristic estimates in the original. How to use memory to speed up search Precompute all distances-to-goal in the simplified version of the problem and store them in a lookup table (pattern database). # nodes expanded (A*) pattern database size (# of abstract states)

4 Automatic Creation of Domain Abstractions Easy to enumerate all possible domain abstractions Domain = blank 5 7 Abstract = blank They form a lattice, e.g. Domain = blank 5 7 Abstract = blank is more abstract than the domain abstraction above 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. Making the Best Use of Memory Compression Results 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 Jonathan s new idea -disk -peg TOH, PDB based on disks No compression: 5Megs memory,. secs lossless compression: 5k memory,. secs Lossy compression: 9Megs, 5.9 secs 5-puzzle, additive PDB triple (7-7-) No compression: 57Megs memory, 0.09 secs Lossy compression, two PDB triples 57Megs memory, 0.0 secs

5 Max ing Multiple Heuristics Question Given heuristics h and h define h(s) = max ( h(s), h(s) ) Preserves key properties: lower bound consistency 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. large pattern database half-size pattern databases s ϕ s ϕ ϕ h(s) h (s) h (s) max 5

6 Many small pattern databases ϕ h (s) s ϕ n h n (s) Rubik s Cube PDB Size n,05,00 7,70,00,,00 Nodes Generated,5,9,9,99,09,99 5,,00 5,9,9 0,,00,5,5 max Summary Rubik s Cube CPU Time RATIO = State Space (x)-puzzle 9-pancake (,)-Topspin ( ops) (,)-Topspin ( ops) (x)-puzzle Best n Ratio Rubik s Cube. 5-puzzle (additive) 5. -puzzle (additive). to 5. #nodes generated using one PDB of size M #nodes generated using n PDBs of size M/n #PDBs Nodes Ratio Time Ratio time/node is.7x higher using six PDBs

7 Early Stopping Techniques for Reducing the Overhead of Multiple PDB lookup IDA* depth bound = 7 g(s) = Stop doing PDB lookups as soon as h > is found. Might result in extra IDA* iterations PDB (s) = 5 next bound is PDB (s) = 7 next bound is 0 Consistency-based Bounding Experimental Results A PDB (A) = PDB (A) = 7 5-puzzle, five additive PDBs (7-7-) Naïve: 0.5 secs Early Stopping: 0.0 secs B Because of consistency: PDB (B) PDB (B) No need to consult PDB Rubik s Cube, six non-additive PDBs Naïve: 7.5 secs Early Stopping:.955 secs Early Stopping and Bounding:. secs 7

8 Static Distribution of Heuristic Values Why Does Max ing Speed Up Search max of 5 small PDBs. large PDB..x nodes generated Runtime Distribution of Heuristic Values Saving Space If h and h are stored as pattern databases, max(h(s),h(s)) requires twice as much space as just one of them. How can we get the benefits of max without using any extra space dovetail two PDBs use smaller PDBs to define max

9 Dovetailing Dovetailing based on the blank 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. PDB PDB PDB PDB PDB PDB PDB PDB PDB PDB PDB PDB The hope: almost as good as max, but only half the memory. PDB PDB PDB PDB PDB PDB Any colouring is possible Dovetailing selection rule 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. 9

10 Dovetailing compared to Max ing Experimental Results Culberson & Schaeffer 99: Dovetailing two PDBs reduced #nodes generated by a factor of.5 compared to using either PDB alone max failures 7 dovetail failures Holte & Newton (unpublished): Dovetailing halved #nodes generated on average Example of Max Failing Depth Bound h h max(h,h) How to generalize Dovetailing to any abstractions of any space,05, 7,99 99,79,,05 9 5,0,5 0,97,9 0 TOTAL 5,5,, 0

11 Example Multiple Lookups in One Pattern Database 5 distance state goal Domain = blank 5 7 Abstract = blank Standard PDB lookup Second lookup, same PDB abstract state abstract goal abstract state abstract goal

12 Relevance Why is this lookup Two Key Properties () Distances are Symmetric relevant to the original state 5 7 () Distances are tile-independent Experimental Results -disk, -peg TOH, PDB of disks Normal: 7. secs Only the second lookup:. secs Both lookups:. secs Additive Pattern Databases 5-puzzle, additive PDB (-7) Normal: 0.0 secs Only the second lookup: 0.07 secs Both lookups: 0.0 secs

13 Adding instead of Max ing Manhattan Distance Heuristic 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. 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. This is advantageous because h (s) + h (s) max( h (s), h (s) ) MD(s) = + + = 5 goal state s M.D. as Additive PDBs () In General ϕ (x)= x if x = blank otherwise Partition the tiles in groups, G, G, G k PDB [ϕ (s) ] = ϕ i (x)= x blank if x G i otherwise ϕ (goal) ϕ (s) MD(s) = PDB [ϕ (s) ] + PDB [ϕ (s) ] + PDB [ϕ (s) ]

14 Korf & Felner s Method What s the Difference Partition the tiles in groups, G, G, G k ϕ i (x)= x if x G i blank if x = blank Moves of otherwise cost zero the blank cannot reach this position without disturbing tile or tile. On-demand distance calculation Hierarchical Search To build a PDB you must calculate all abstract distances-to-goal. Only a tiny fraction of them are needed to solve any individual problem. If you only intend to use the PDB to solve a few problems, calculate PDB entries only as you need them. Hierarchical Search

15 Calculate Distance by Searching at the Abstract Level Replace this line: by h(s) = PDB[ φ(s) ] h(s) = search(φ(s), φ(goal) ) (recursive) call to a search algorithm to compute abstract distance to goal for state s Hierarchical Search Abstract space, φ (φ (S)) φ Abstract space, φ (S) φ Original space, S 5-puzzle Results () 5-puzzle Results () Felner s 7-7- additive PDB: takes 0 minutes to build (,00 secs) Solves problems in 0.05 secs (on average) Felner s -7 additive PDB Takes 7 hours to build (5,00 secs) Solves problems in 0.0 secs Hierarchical IDA*, Gigabyte limit Using the same abstraction for all problems, solving takes secs (on average), or 07 secs if the cache is not cleared between problems Max ing over Corner & Fringe abstractions, solving takes 50 secs (on average) Using a customized abstraction for each problem, solving takes 7 secs (on average) 5

16 Thesis topics abound! General Dovetailing A Partial-Order on Domain Abstractions Lattice of domain abstractions Easy to enumerate all possible domain abstractions Domain = blank 5 7 Abstract = blank and to define a partial-order on them, e.g. Domain = blank 5 7 Abstract = blank is more abstract than the domain abstraction above.

17 The LCA of Abstractions General Dovetailing Domain = blank 5 7 Abstract = blank Domain = blank 5 7 Abstract = blank blank 5 7 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, lookup SELECT[ϕ(s)](s) 7