Using Abstraction to Speed Up Search

Robert Holte

University of Ottawa

Textbook Computer ScienceProblem: in a given graph find a shortest path between

two nodes (start, goal)

(explicit) Graph Representationsadjacency matrix – O(N2)adjacency list – O(N+E)

AlgorithmsDijkstra (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 n1 to n2 iff n2 succ(n1)

A graph can be exponentially larger than its description.

Towers of Hanoi puzzlefor D disks:

– graph has 3D nodes– description is O(D2)

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(n1) e + h(n2)) 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

en1 n2

e – h(n1)+h(n2)n1 n2

My Research

Aim:to generate heuristics or skeletal solutions automatically(from an explicit or implicit graph)

Method:1. Create an abstraction of the given graph2. Find a solution path in the abstract graph3. Use the abstract solution as a skeletal solution, or use

its length as a heuristic

Example: Towers of Hanoistart

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

• 10-40 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 (1013 states).

Rich Korf (1997): first optimal solutions of random instances of Rubik’s cube (1019 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

1 2

3 4 5

6 7 8

8-puzzle181,440 nodes

Domain Abstraction

1 2

3 4 5

6 7 8

8-puzzle181,440 nodes

6 7 8

<5> abstraction3024 nodes

Domain Abstraction

1 2

3 4 5

6 7 8

8-puzzle181,440 nodes

<3,3,2> abstraction5040 nodes

Performance versus Size

pattern database size (# abstract states)

# no

des

expa

nded

Performance of Pattern Databases

# no

des

expa

nded

pattern database size (# of abstract states)

Prediction of Search Time

actual # nodes expanded

Kor

f &

Rei

d’s

pred

icti

on

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