Using Abstraction to Speed Up Search Robert Holte University of Ottawa

Textbook Computer Science Problem: in a given graph find a shortest path between two nodes (start, goal) (explicit) Graph Representations adjacency matrix – O(N 2 ) adjacency list – O(N+E) Algorithms Dijkstra (if all edge weights are non-negative) Bellman-Ford (all shortest paths from start) time and space – low-order polynomials of N and E

Portion of the London Underground Graph

“Implicit” Graphs Instead of an adjacency list/matrix, Define a graph by a successor function, succ, and one or more “seed” nodes. Nodes of the graph = transitive closure of succ applied to the seed nodes An edge exists from n 1 to n 2 iff n 2  succ(n 1 ) A graph can be exponentially larger than its description.

Towers of Hanoi puzzle for D disks: –graph has 3 D nodes –description is O(D 2 )

3-disk State Space

Speed up Search by Using Extra Information Solution “skeleton” = chain of subgraphs leading from start to goal –Refinement: search in the first subgraph until you reach the second, then search in the second until you reach the third, etc. –very fast but path found not guaranteed to be optimal Heuristic = h(n) = estimate of the distance from node n to the goal node. –if it never overestimates, it can eliminate (prune) parts of the graph and still guarantee finding an optimal path

Transform the edge weights to account for h(n): becomes Use Dijkstra (if h(n 1 )  e + h(n 2 )) or Bellman-Ford (all shortest paths) Distance to n increases by h(n), so a more accurate heuristic prunes more nodes. How to Search with a Heuristic e n1n1 n2n2 e – h(n 1 )+h(n 2 ) n1n1 n2n2

My Research Aim: to generate heuristics or skeletal solutions automatically (from an explicit or implicit graph) Method: 1.Create an abstraction of the given graph 2.Find a solution path in the abstract graph 3.Use the abstract solution as a skeletal solution, or use its length as a heuristic

Example: Towers of Hanoi start goal

Abstraction: “Ignore the smallest disk” start goal

Abstract Solution start goal

My Research Contributions (1) Methods of Abstraction –explicit graphs: STAR abstraction –implicit graphs: domain abstraction –select among abstractions using Korf & Reid’s method for predicting search time Search Methods –refinement: AltO –heuristic: Hierarchical A*

My Research Contributions (2) Conceptual Advances –Unified view of refinement and heuristic search, enabling a continuous tradeoff between solution quality and search time. –Demonstrate, and analyze, failures of abstraction techniques on implicit graphs. –Demonstrate viability of hierarchical heuristic search. –Investigate the relation between a heuristic’s search time and its size as a lookup table. –Transfer techniques between AI and CS.

STAR abstraction Abstract state = connected subgraph (all nodes within distance R of a given node) Fully automatic (but need to amortize cost) Fine control over granularity (R) Can be applied recursively (abstraction hierarchy) Guarantees refinement will succeed Produces a consistent heuristic

AltO Refinement Technique Conduct abstract search in the opposite direction to the base-level search; Use the whole abstract search graph for refinement. 12% faster than standard refinement and produces better solutions (20% shorter) Robust – not sensitive to the abstract solution found, and not very sensitive to the radius of abstraction times faster than A* (same abstraction) and solutions usually within 30% of optimal

Hierarchical A* Heuristic search: h(n) is needed for every node reached during search. Hierarchical A*: Compute h(n) on demand. If h(n) is not already known compute it by searching at the abstract level. The search at an abstract level can be informed by a heuristic defined by a higher abstract level.  many abstract searches for one base-level search  naïve implementation is 10 times slower than blind base-level search.

Making Hierarchical A* Efficient Key Observation: For one base-level search all the abstract searches have the same goal so distance-to-goal information from one abstract search can used to speedup subsequent abstract searches. Example: “P-g caching” Suppose an abstract search finds an optimal path of length P, and during search it reaches node n at distance g(n) from the abstract start node. Then P-g(n) is a consistent heuristic (at the abstract level) and is often more accurate than the h(n) value used during this search.

Pattern Databases Instead of computing h(n) on demand, precompute distance-to-goal for all abstract states, and store them in a lookup table. Joe Culberson & Jonathan Schaeffer (1994) 1000-times speedup over best hand-crafted heuristics for the 15-puzzle (10 13 states). Rich Korf (1997): first optimal solutions of random instances of Rubik’s cube (10 19 states).

My Pattern Database Research Pattern databases generated semi-automatically, not hand-crafted Identify obstacles to fully automatic generation Large-scale studies involving thousands of pattern databases Study pattern databases in isolation, not in conjunction with a hand-crafted heuristic Evaluate Korf & Reid’s method for predicting search time

Domain Abstraction puzzle 181,440 nodes

Domain Abstraction puzzle 181,440 nodes 678 abstraction 3024 nodes

Domain Abstraction puzzle 181,440 nodes abstraction 5040 nodes

Performance versus Size pattern database size (# abstract states) # nodes expanded

Performance of Pattern Databases # nodes expanded pattern database size (# of abstract states)

Prediction of Search Time actual # nodes expanded Korf & Reid’s prediction

Future Research weighted, directed graphs multiple abstractions, blended abstractions compression of pattern databases understand & avoid failure modes alternative methods of abstraction how best to use limited memory