Best-Case and Worst-Case Behavior of Greedy Best-First Search - - PowerPoint PPT Presentation

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Best-Case and Worst-Case Behavior of Greedy Best-First Search - - PowerPoint PPT Presentation

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Best-Case and Worst-Case Behavior of Greedy Best-First Search Thomas Keller Malte Helmert Manuel Heusner University of Basel July 19th, 2018 Introduction Theoretical Results Experimental Results Conclusion Motivation A [Hart et

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Best-Case and Worst-Case Behavior of Greedy Best-First Search

Manuel Heusner Thomas Keller Malte Helmert

University of Basel

July 19th, 2018

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Introduction Theoretical Results Experimental Results Conclusion

Motivation

A∗ [Hart et al.,1968]

  • many potentially expanded states on last f-layer
  • tie-breaking is important
  • best case: shortest path
  • worst case: all potentially expanded states
  • polynomial-time computable in size of state space
  • M. Heusner, T. Keller, M. Helmert (Basel)

Best-Case and Worst-Case Behavior of Greedy Best-First Search 2/8

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SLIDE 3

Introduction Theoretical Results Experimental Results Conclusion

Motivation

A∗ [Hart et al.,1968]

  • many potentially expanded states on last f-layer
  • tie-breaking is important
  • best case: shortest path
  • worst case: all potentially expanded states
  • polynomial-time computable in size of state space

Greedy best-first search [Doran and Michie, 1966]

  • large heuristic plateaus
  • tie-breaking assumed to be important
  • best case: ?
  • worst case: ?
  • tractable?
  • M. Heusner, T. Keller, M. Helmert (Basel)

Best-Case and Worst-Case Behavior of Greedy Best-First Search 2/8

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SLIDE 4

Introduction Theoretical Results Experimental Results Conclusion

Complexity Results

Given a state space and a heuristic:

  • How many states does GBFS expand in its best case?
  • How many states does GBFS expand in its worst case?
  • M. Heusner, T. Keller, M. Helmert (Basel)

Best-Case and Worst-Case Behavior of Greedy Best-First Search 3/8

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SLIDE 5

Introduction Theoretical Results Experimental Results Conclusion

Complexity Results

Given a state space and a heuristic:

  • How many states does GBFS expand in its best case?
  • How many states does GBFS expand in its worst case?

NP-complete in general

  • overlapping benches and craters that can be reached on

different paths

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

Best-Case and Worst-Case Behavior of Greedy Best-First Search 3/8

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SLIDE 6

Introduction Theoretical Results Experimental Results Conclusion

Complexity Results

Given a state space and a heuristic:

  • How many states does GBFS expand in its best case?
  • How many states does GBFS expand in its worst case?

NP-complete in general

  • overlapping benches and craters that can be reached on

different paths

  • combinatorial problem

polynomial-time computable

  • in size of the state space
  • undirected edges
  • overlap-free craters and benches
  • M. Heusner, T. Keller, M. Helmert (Basel)

Best-Case and Worst-Case Behavior of Greedy Best-First Search 3/8

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Introduction Theoretical Results Experimental Results Conclusion

Background

  • locally characterized

progress states

  • based on high-water mark

4 3 2 1 A X B C D E F G H I Y J L K M N Z

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

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Introduction Theoretical Results Experimental Results Conclusion

Background

  • locally characterized

progress states

  • based on high-water mark
  • directed acyclic graph of

benches

3 2 1

A B C D E F G L B J K D G H I L E I I M K M

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

Best-Case and Worst-Case Behavior of Greedy Best-First Search 4/8

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Introduction Theoretical Results Experimental Results Conclusion

Background

  • locally characterized

progress states

  • based on high-water mark
  • directed acyclic graph of

benches

  • search run is sequence of

episodes

  • episode searches on single

bench along a bench path

3 2 1

A B C D E F G L B J K D G H I L E I I M K M

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

Best-Case and Worst-Case Behavior of Greedy Best-First Search 4/8

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Introduction Theoretical Results Experimental Results Conclusion

Background

  • locally characterized

progress states

  • based on high-water mark
  • directed acyclic graph of

benches

  • search run is sequence of

episodes

  • episode searches on single

bench along a bench path

  • crater relates to local

minimum

3 2 1

A B C D E F G L B J K D G H I L E I I M K M

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

Best-Case and Worst-Case Behavior of Greedy Best-First Search 4/8

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Introduction Theoretical Results Experimental Results Conclusion

Best-Case and Worst-Case Behavior

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A B C D E F G L B J K D G H I L E I I M K M

best case worst case expansions

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

Best-Case and Worst-Case Behavior of Greedy Best-First Search 5/8

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Introduction Theoretical Results Experimental Results Conclusion

Best-Case and Worst-Case Behavior

  • best case: minimize along

state path including all necessarily expanded crater states

3 2 1

A B C D E F G L B J K D G H I L E I I M K M

best case worst case expansions 6

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

Best-Case and Worst-Case Behavior of Greedy Best-First Search 5/8

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SLIDE 13

Introduction Theoretical Results Experimental Results Conclusion

Best-Case and Worst-Case Behavior

  • best case: minimize along

state path including all necessarily expanded crater states

  • worst case: maximize along

bench path including all potentially expanded bench states

3 2 1

A B C D E F G L B J K D G H I L E I I M K M

best case worst case expansions 6 9

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

Best-Case and Worst-Case Behavior of Greedy Best-First Search 5/8

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SLIDE 14

Introduction Theoretical Results Experimental Results Conclusion

Best-Case and Worst-Case Behavior

  • best case: minimize along

state path including all necessarily expanded crater states

  • worst case: maximize along

bench path including all potentially expanded bench states

  • beware of overlapping

benches and craters

3 2 1

A B C D E F G L B J K D G H I L E I I M K M

best case worst case expansions 6 9

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

Best-Case and Worst-Case Behavior of Greedy Best-First Search 5/8

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Introduction Theoretical Results Experimental Results Conclusion

Best-Case and Worst-Case Behavior

  • best case: minimize along

state path including all necessarily expanded crater states

  • worst case: maximize along

bench path including all potentially expanded bench states

  • beware of overlapping

benches and craters

3 2 1

A B C D E F G L B J K D G H I L E I I M K M

best case worst case expansions 6 9

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

Best-Case and Worst-Case Behavior of Greedy Best-First Search 5/8

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Introduction Theoretical Results Experimental Results Conclusion

Experimental Results

  • implemented algorithms for computing best and worst cases
  • state spaces of planning tasks from international planning

competitions

  • Fast Forward heuristic
  • M. Heusner, T. Keller, M. Helmert (Basel)

Best-Case and Worst-Case Behavior of Greedy Best-First Search 6/8

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SLIDE 17

Introduction Theoretical Results Experimental Results Conclusion

Experimental Results

  • implemented algorithms for computing best and worst cases
  • state spaces of planning tasks from international planning

competitions

  • Fast Forward heuristic
  • DAG of benches for 764 instances from 78 domains
  • M. Heusner, T. Keller, M. Helmert (Basel)

Best-Case and Worst-Case Behavior of Greedy Best-First Search 6/8

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Introduction Theoretical Results Experimental Results Conclusion

Experimental Results

  • implemented algorithms for computing best and worst cases
  • state spaces of planning tasks from international planning

competitions

  • Fast Forward heuristic
  • DAG of benches for 764 instances from 78 domains
  • best cases for 679 instances
  • M. Heusner, T. Keller, M. Helmert (Basel)

Best-Case and Worst-Case Behavior of Greedy Best-First Search 6/8

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SLIDE 19

Introduction Theoretical Results Experimental Results Conclusion

Experimental Results

  • implemented algorithms for computing best and worst cases
  • state spaces of planning tasks from international planning

competitions

  • Fast Forward heuristic
  • DAG of benches for 764 instances from 78 domains
  • best cases for 679 instances
  • worst cases for 739 instances
  • M. Heusner, T. Keller, M. Helmert (Basel)

Best-Case and Worst-Case Behavior of Greedy Best-First Search 6/8

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Introduction Theoretical Results Experimental Results Conclusion

Standard Tie-Breaking Strategies

101 102 103 104 105 106 50 100 150 200 250 300 expansions covered instances crater 100 101 102 103 104 50 100 150 200 250 300 350 400 expansions crater-free best fifo lifo rand worst

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

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Introduction Theoretical Results Experimental Results Conclusion

Conclusion

  • NP-complete in general
  • computing best and worst cases is often feasible
  • large impact of tie-breaking for less informed heuristics
  • room for improvement over standard tie-breaking strategies
  • M. Heusner, T. Keller, M. Helmert (Basel)

Best-Case and Worst-Case Behavior of Greedy Best-First Search 8/8

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Introduction Theoretical Results Experimental Results Conclusion

Conclusion

  • NP-complete in general
  • computing best and worst cases is often feasible
  • large impact of tie-breaking for less informed heuristics
  • room for improvement over standard tie-breaking strategies

Thank you for your attention!

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

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