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

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 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

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

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

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

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

Introduction Theoretical Results Experimental Results Conclusion Background • locally characterized progress states • based on high-water mark 4 A X 3 B C D E 2 Y F G H I 1 J K L M 0 N Z M. Heusner , T. Keller, M. Helmert (Basel) Best-Case and Worst-Case Behavior of Greedy Best-First Search 4/8

Introduction Theoretical Results Experimental Results Conclusion Background • locally characterized A B C D E progress states 3 F G • based on high-water mark L • directed acyclic graph of D E benches 2 G H I I L B 1 I J K M 0 K M M. Heusner , T. Keller, M. Helmert (Basel) Best-Case and Worst-Case Behavior of Greedy Best-First Search 4/8

Introduction Theoretical Results Experimental Results Conclusion Background • locally characterized A B C D E progress states 3 F G • based on high-water mark L • directed acyclic graph of D E benches 2 G H I I L • search run is sequence of episodes B • episode searches on single 1 I J K M bench along a bench path 0 K M M. Heusner , T. Keller, M. Helmert (Basel) Best-Case and Worst-Case Behavior of Greedy Best-First Search 4/8

Introduction Theoretical Results Experimental Results Conclusion Background • locally characterized A B C D E progress states 3 F G • based on high-water mark L • directed acyclic graph of D E benches 2 G H I I L • search run is sequence of episodes B • episode searches on single 1 I J K M bench along a bench path 0 • crater relates to local K M minimum M. Heusner , T. Keller, M. Helmert (Basel) Best-Case and Worst-Case Behavior of Greedy Best-First Search 4/8

Introduction Theoretical Results Experimental Results Conclusion Best-Case and Worst-Case Behavior A B C D E 3 F G L D E 2 G H I I L B 1 I J K M 0 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

Introduction Theoretical Results Experimental Results Conclusion Best-Case and Worst-Case Behavior A B C D E 3 • best case: minimize along F G L state path including all necessarily expanded crater D E states 2 G H I I L B 1 I J K M 0 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

Introduction Theoretical Results Experimental Results Conclusion Best-Case and Worst-Case Behavior A B C D E 3 • best case: minimize along F G L state path including all necessarily expanded crater D E states 2 G H I I L • worst case: maximize along bench path including all B potentially expanded bench 1 I J K M states 0 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

Introduction Theoretical Results Experimental Results Conclusion Best-Case and Worst-Case Behavior A B C D E 3 • best case: minimize along F G L state path including all necessarily expanded crater D E states 2 G H I I L • worst case: maximize along bench path including all B potentially expanded bench 1 I J K M states 0 • beware of overlapping K M best case worst case benches and craters expansions 6 9 M. Heusner , T. Keller, M. Helmert (Basel) Best-Case and Worst-Case Behavior of Greedy Best-First Search 5/8

Introduction Theoretical Results Experimental Results Conclusion Best-Case and Worst-Case Behavior A B C D E 3 • best case: minimize along F G L state path including all necessarily expanded crater D E states 2 G H I I L • worst case: maximize along bench path including all B potentially expanded bench 1 I J K M states 0 • beware of overlapping K M best case worst case benches and craters expansions 6 9 M. Heusner , T. Keller, M. Helmert (Basel) Best-Case and Worst-Case Behavior of Greedy Best-First Search 5/8

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

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

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

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

Introduction Theoretical Results Experimental Results Conclusion Standard Tie-Breaking Strategies crater crater-free 300 400 covered instances 250 350 300 200 best 250 150 200 fifo 150 100 lifo 100 rand 50 50 worst 0 0 10 1 10 2 10 3 10 4 10 5 10 6 10 0 10 1 10 2 10 3 10 4 expansions expansions M. Heusner , T. Keller, M. Helmert (Basel) Best-Case and Worst-Case Behavior of Greedy Best-First Search 7/8

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

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