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Understandig the Search Behaviour of Greedy Best-First Search - - PowerPoint PPT Presentation

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Understandig the Search Behaviour of Greedy Best-First Search Manuel Heusner Thomas Keller Malte Helmert University of Basel June 16th, 2017 Introduction High-Water Marks Benches Craters Conclusion Introduction M. Heusner , T. Keller, M.

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Understandig the Search Behaviour of Greedy Best-First Search

Manuel Heusner Thomas Keller Malte Helmert

University of Basel

June 16th, 2017

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Introduction High-Water Marks Benches Craters Conclusion

Introduction

  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

Open Questions

  • Which states is GBFS guaranteed to expand?
  • Which states is GBFS guaranteed not to expand?
  • Which states may GBFS potentially expand?

Note: Partly answered for A∗ (based on f-value) and for GBFS (based on high-water mark).

  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

State Space Topology

  • state space: generative model with initial state, goal states

and successor function

  • heuristic: assigns non-negative values to states
  • state space topology: state space + heuristic
  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

State Space Topology

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X Y Z A B E C D F G H I J K L M U N P Q R S T

  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

Greedy Best-First Search

  • expansion : generates successors of a state
  • greedy best-first search: iteratively expands states with lowest

heuristic value

  • tie-breaking: selects a state among states with equal heuristic

values

  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

Greedy Best-First Search

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 A

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Introduction High-Water Marks Benches Craters Conclusion

Greedy Best-First Search

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X A B C D

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Introduction High-Water Marks Benches Craters Conclusion

Greedy Best-First Search

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X A B E C D

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Introduction High-Water Marks Benches Craters Conclusion

Greedy Best-First Search

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X A B E C D F G

  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

Greedy Best-First Search

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X A B E C D F G J P

  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

Greedy Best-First Search

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X A B E C D F G J K P

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Introduction High-Water Marks Benches Craters Conclusion

Greedy Best-First Search

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X A B E C D F G I J K P

  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

Greedy Best-First Search

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X A B E C D F G I J K P

  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

Greedy Best-First Search

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X A B E C D F G I J K U N P

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Introduction High-Water Marks Benches Craters Conclusion

Greedy Best-First Search

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X A B E C D F G I J K U N P T

  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

High-Water Marks

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Introduction High-Water Marks Benches Craters Conclusion

High-Water Marks

Definition (high-water mark) The high-water mark is the largest heuristic value of a state that GBFS starting from a state (or a set of states) must expand before reaching a goal state.

  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

High-Water Mark of State

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X Y Z A B E C D F G H I J K L M U N P Q R S T

high-water mark of state P: 4

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Introduction High-Water Marks Benches Craters Conclusion

High-Water Mark of State Set

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X Y Z A B E C D F G H I J K L M U N P Q R S T

high-water mark of state set {J, P}: 3

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Introduction High-Water Marks Benches Craters Conclusion

Earlier Result

Theorem (Wilt & Ruml, SoCS 2014) GBFS is guaranteed to not expand a state whose heuristic value is larger than high-water mark of initial state.

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Introduction High-Water Marks Benches Craters Conclusion

Earlier Result

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X Y Z A B E C D F G H I J K L M U N P Q R S T

never expanded states: {X}

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Introduction High-Water Marks Benches Craters Conclusion

Benches

  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

Bench Exit States

Definition (bench exit state) Bench exit state is a state which has a successor that has lower high-water mark or that is a goal state.

  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

Bench Exit States

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X Y Z A B E C D F G H I J K L M U N P Q R S T

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Introduction High-Water Marks Benches Craters Conclusion

Bench Exit States

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X Y Z A B E C D F G H I J K L M U N P Q R S T

  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

Bench Exit Property

Theorem (bench exit property) Whenever GBFS expands a bench exit state, all previously generated states will never be expanded for the rest of the algorithm run. Note: GBFS makes progress when bench exit state is expanded.

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Introduction High-Water Marks Benches Craters Conclusion

Bench Exit Property

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 A

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Introduction High-Water Marks Benches Craters Conclusion

Bench Exit Property

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X A B C D

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Introduction High-Water Marks Benches Craters Conclusion

Bench Exit Property

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X B C D

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Introduction High-Water Marks Benches Craters Conclusion

Bench Exit Property

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X B E C D

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Introduction High-Water Marks Benches Craters Conclusion

Bench Exit Property

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X B E C D F G

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Introduction High-Water Marks Benches Craters Conclusion

Bench Exit Property

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X B E C D F G J P

  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

Bench Exit Property

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 J P

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Introduction High-Water Marks Benches Craters Conclusion

Bench Exit Property

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 J K P

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Introduction High-Water Marks Benches Craters Conclusion

Bench Exit Property

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 I J K P

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Introduction High-Water Marks Benches Craters Conclusion

Bench Exit Property

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 G I J K P

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Introduction High-Water Marks Benches Craters Conclusion

Bench Exit Property

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 G I J K U N P

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Introduction High-Water Marks Benches Craters Conclusion

Bench Exit Property

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 U N

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Introduction High-Water Marks Benches Craters Conclusion

Bench Exit Property

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 U N T

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Introduction High-Water Marks Benches Craters Conclusion

Bench Exit Property

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 T

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Introduction High-Water Marks Benches Craters Conclusion

Benches

Definition (bench) A bench contains all states that GBFS starting with a given set of states can expand until expansion of a bench exit state. It is empty if the given set of states contains a goal state. It is associated with high-water mark of the given set of states.

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Introduction High-Water Marks Benches Craters Conclusion

Benches

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X Y Z A B E C D F G H I J K L M U N P Q R S T

states on bench defined by {J, P}: {I, J, P, K}

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Introduction High-Water Marks Benches Craters Conclusion

Bench Transition Systems

Definition (bench transition system) A bench transition system contains all benches which are reachable from the bench that starts with the initial state. A successor bench is defined by the successor states of a bench exit state.

  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

Bench Transition Systems

Example

h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 A C D F G I J K I J L Q R S M N P T

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Introduction High-Water Marks Benches Craters Conclusion

Results

Theorem GBFS potentially expands a state if it is on at least one bench from bench transition system. Theorem GBFS is guaranteed to not expand a state that is not on a bench

  • f the bench transition system.
  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

Results

Example

h = 6 h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 X Y Z A B E C D F G H I J K L M U N P Q R S T

never expanded states: {B, E, H, T, U, X, Y, Z}

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Introduction High-Water Marks Benches Craters Conclusion

Craters

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Introduction High-Water Marks Benches Craters Conclusion

Surfaces

Definition (surface) A state is on the surface of a bench if its heuristic value is the high-water mark of the bench. Note: Is often called heuristic plateau or uninformed heuristic region.

  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

Surfaces

Example

h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 A C D F G I J K I J L Q R S M N P T

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Introduction High-Water Marks Benches Craters Conclusion

Crater Entry States

Definition (crater entry state) A crater entry state is a state that is on the surface of a bench and that has a successor which is on a bench but not on a surface.

  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

Crater Entry States

Example

h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 A C D F G I J K I J L Q R S M N P T

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Introduction High-Water Marks Benches Craters Conclusion

Craters

Definition (crater) A crater contains all states that GBFS starting with a given crater entry state expands until expansion of a state from the surface. Note: Is often called local minimum or uninformed heuristic region.

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Introduction High-Water Marks Benches Craters Conclusion

Craters

Example

h = 5 h = 4 h = 3 h = 2 h = 1 h = 0 A C D F G I J K I J L Q R S M N P T

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Introduction High-Water Marks Benches Craters Conclusion

Result

Theorem Whenever GBFS expands a crater entry state s, then GBFS is guaranteed to expand all states in the crater defined by s.

  • M. Heusner, T. Keller, M. Helmert (Basel)

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Introduction High-Water Marks Benches Craters Conclusion

Conclusion

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Introduction High-Water Marks Benches Craters Conclusion

Conclusion

  • exact characterization of potentially expanded and never

expanded states

  • characterization of surely expanded states given some

conditions

  • better understanding of search behaviour and search progress
  • M. Heusner, T. Keller, M. Helmert (Basel)

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