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Search Behavior of Greedy Best-First Search Manuel Heusner May 10th, 2019 University of Basel State Spaces 1/27 State Space Search 1 3 1 3 2 1 3 1 2 2 3 2 1 1 3 3 3 3 1 1 1 3 3 1 2 2 1 3 3 1 2/27 State Space Search

  1. Search Behavior of Greedy Best-First Search Manuel Heusner May 10th, 2019 University of Basel

  2. State Spaces 1/27

  3. State Space Search 1 3 1 3 2 1 3 1 2 2 3 2 1 1 3 3 3 3 1 1 1 3 3 1 2 2 1 3 3 1 2/27

  4. State Space Search input: • initial state 1 3 1 3 2 1 3 1 2 2 3 2 1 1 3 3 3 3 1 1 1 3 3 1 2 2 1 3 3 1 2/27

  5. State Space Search input: • initial state • goal test function 1 3 1 3 2 1 3 1 2 2 3 2 1 1 3 3 3 3 1 1 1 3 3 1 2 2 1 3 3 1 2/27

  6. State Space Search input: • initial state • goal test function 1 3 1 3 3 • successor generator 2 1 1 3 1 2 2 2 3 2 1 1 3 3 3 3 1 1 1 3 3 1 2 2 1 3 3 1 2/27

  7. State Space Search input: • initial state • goal test function 1 1 3 1 3 1 3 3 • successor generator • transition cost function 2 2 1 1 3 1 3 1 2 2 2 2 3 3 2 2 1 1 1 1 3 3 3 3 3 3 3 3 1 1 1 1 1 3 1 3 3 3 1 1 2 2 2 2 1 3 1 3 3 3 1 1 2/27

  8. State Space Search input: • initial state • goal test function 3 1 3 1 3 • successor generator • transition cost function 2 1 1 3 1 2 2 3 2 1 1 3 3 3 3 1 1 output: 1 3 2 3 1 • solution path 2 2 1 3 3 2 1 2/27

  9. State Space Search input: • initial state • goal test function 1 3 1 3 • successor generator • transition cost function 2 1 3 1 2 2 3 2 1 1 3 3 3 3 1 1 output: 1 3 3 1 • solution path 2 2 1 3 3 1 additional information: • heuristic � heuristic best-first search 2/27

  10. Motivation information of A ∗ • c ∗ : optimal solution path cost • f ( s ): estimate of optimal solution path cost 3/27

  11. Motivation information of A ∗ • c ∗ : optimal solution path cost • f ( s ): estimate of optimal solution path cost behavior of A*: • necessary: f ( s ) < c ∗ • never: f ( s ) > c ∗ • potential: f ( s ) = c ∗ • worst case: necessary & potential • best case: necessary & shortest path of potential states • progress: increase of f -value 3/27

  12. Motivation information of A ∗ • c ∗ : optimal solution path cost • f ( s ): estimate of optimal solution path cost behavior of A*: • necessary: f ( s ) < c ∗ • never: f ( s ) > c ∗ • potential: f ( s ) = c ∗ • worst case: necessary & potential • best case: necessary & shortest path of potential states • progress: increase of f -value Can we get similar results for greedy best-first search? 3/27

  13. Guiding Questions Given a state space and a heuristic: • When does GBFS make search progress? • Which states does GBFS potentially, never or necessarily expand? • Which are the best-case and worst-case search runs of GBFS? 4/27

  14. Greedy Best-First Search 6 A 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 5/27

  15. Greedy Best-First Search 6 A 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 5/27

  16. Greedy Best-First Search 6 A 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 5/27

  17. Greedy Best-First Search 6 A 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 5/27

  18. Greedy Best-First Search 6 A 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 5/27

  19. Greedy Best-First Search 6 A 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 5/27

  20. Greedy Best-First Search 6 A 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 5/27

  21. Greedy Best-First Search 6 A 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 5/27

  22. Greedy Best-First Search 6 A 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 5/27

  23. Greedy Best-First Search 6 A 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 5/27

  24. Greedy Best-First Search 6 A 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 5/27

  25. Greedy Best-First Search 6 A 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 5/27

  26. Greedy Best-First Search 6 A 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 5/27

  27. Greedy Best-First Search 6 A 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 5/27

  28. When does GBFS make search progress? 6/27

  29. High-Water Mark of State [Wilt & Ruml,2014] 7/27

  30. High-Water Mark of State [Wilt & Ruml,2014] The highest h-value that GBFS reaches during a search run starting in a state. 7/27

  31. High-Water Mark of State [Wilt & Ruml,2014] The highest h-value that GBFS reaches during a search run starting in a state. 6 A 5 B 4 C D E F G 3 H I J K L M 2 N O P Q R 1 S T U V W X 0 Y Z 7/27

  32. High-Water Mark of State [Wilt & Ruml,2014] The highest h-value that GBFS reaches during a search run starting in a state. 6 A 5 B 4 C D E F G 3 H I J K L M 2 N O P Q R 1 S T U V W X 0 Y Z  min ρ ∈ P ( s ) (max s ∈ ρ h ( s )) if P ( s ) � = ∅  hwm ( s ) := ∞ otherwise  7/27

  33. High-Water Mark Pruning [Wilt & Ruml,2014] GBFS never expands a state s with h ( s ) > hwm ( s init ). 8/27

  34. High-Water Mark Pruning [Wilt & Ruml,2014] GBFS never expands a state s with h ( s ) > hwm ( s init ). 6 A 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 8/27

  35. Search Progress 6 A 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 9/27

  36. Search Progress 6 A 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 9/27

  37. Search Progress 6 A 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 9/27

  38. Search Progress high-water mark of set of states: 6 A hwm ( S ) := min s ∈ S ( hwm ( s )) 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 9/27

  39. Search Progress high-water mark of set of states: 6 A hwm ( S ) := min s ∈ S ( hwm ( s )) 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 9/27

  40. Search Progress high-water mark of set of states: 6 A hwm ( S ) := min s ∈ S ( hwm ( s )) 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 9/27

  41. Search Progress high-water mark of set of states: 6 A hwm ( S ) := min s ∈ S ( hwm ( s )) 5 B 4 C D E F G progress state: 3 H I J K L M hwm ( s ) > hwm ( succ ( s )) 2 Q N O P R 1 S T U V W X 0 Y Z 9/27

  42. Search Progress high-water mark of set of states: 6 A hwm ( S ) := min s ∈ S ( hwm ( s )) 5 B 4 C D E F G progress state: 3 H I J K L M hwm ( s ) > hwm ( succ ( s )) 2 Q N O P R 1 S T U V W X episodes of local searches! 0 Y Z 9/27

  43. Search Progress high-water mark of set of states: 6 A hwm ( S ) := min s ∈ S ( hwm ( s )) 5 B 4 C D E F G progress state: 3 H I J K L M hwm ( s ) > hwm ( succ ( s )) 2 Q N O P R 1 S T U V W X episodes of local searches! 0 Y Z 9/27

  44. Search Progress high-water mark of set of states: 6 A hwm ( S ) := min s ∈ S ( hwm ( s )) 5 B 4 C D E F G progress state: 3 H I J K L M hwm ( s ) > hwm ( succ ( s )) 2 Q N O P R 1 S T U V W X episodes of local searches! 0 Y Z 9/27

  45. Search Progress high-water mark of set of states: 6 A hwm ( S ) := min s ∈ S ( hwm ( s )) 5 B 4 C D E F G progress state: 3 H I J K L M hwm ( s ) > hwm ( succ ( s )) 2 Q N O P R 1 S T U V W X episodes of local searches! 0 Y Z 9/27

  46. Search Progress high-water mark of set of states: 6 A hwm ( S ) := min s ∈ S ( hwm ( s )) 5 B 4 C D E F G progress state: 3 H I J K L M hwm ( s ) > hwm ( succ ( s )) 2 Q N O P R 1 S T U V W X episodes of local searches! Search Progress 0 Y Z GBFS makes progress when expanding a progress state. 9/27

  47. Which states does GBFS potentially or never expand? 10/27

  48. Progress States 6 A 5 B 4 C D E F G 3 H I J K L M 2 Q N O P R 1 S T U V W X 0 Y Z 11/27

  49. Benches 6 A A 5 B B 4 C C D D E E F F G G 3 H H I I J J K K L L M M • progress state s induces bench B ( s ) 2 Q Q N N O O P P R R 1 S S T T U U V V W W X X 0 Y Y Z Z 12/27

  50. Benches 6 A A 5 B B 4 C C D D E E F F G G 3 H H I I J J K K L L M M • progress state s induces bench B ( s ) 2 Q Q N N O O P P R R 1 S S T T U U V V W W X X 0 Y Y Z Z 12/27

  51. Bench Space B ( B ) 4 B ( D ) B ( G ) 3 B ( C ) 2 • connects the benches via 1 B ( N ) B ( K ) progress states 0 B ( U ) −∞ B ( Z ) 13/27

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