CS 188: Artificial Intelligence

1. CS 188: Artificial Intelligence Search Continued Instructors: Anca Dragan University of California, Berkeley [These slides adapted from Dan Klein and Pieter Abbeel; ai.berkeley.edu]

2. Recap: Search

3. Search • Search problem: • States (abstraction of the world) • Actions (and costs) • Successor function (world dynamics): • {s’|s,a->s’} • Start state and goal test

4. Depth-First Search

5. Depth-First Search G a c b a c b e d f e d f S h S h e e p d p r p q q q h h r r b c p p q q f f a a q q c c G G a a Strategy: expand a deepest node first Implementation: Fringe is a LIFO stack r

6. Search Algorithm Properties

7. Search Algorithm Properties • Complete: Guaranteed to find a solution if one exists? • Return in finite time if not? • Optimal: Guaranteed to find the least cost path? • Time complexity? • Space complexity? • Cartoon of search tree: • b is the branching factor • m is the maximum depth • solutions at various depths • Number of nodes in entire tree? • 1 + b + b2 + …. bm = O(bm) 1 node b b nodes … b2 nodes m tiers bm nodes

8. Depth-First Search (DFS) Properties • What nodes DFS expand? • Some left prefix of the tree. • Could process the whole tree! • If m is finite, takes time O(bm) • How much space does the fringe take? • Only has siblings on path to root, so O(bm) • Is it complete? • m could be infinite, so only if we prevent cycles (more later) • Is it optimal? • No, it finds the “leftmost” solution, regardless of depth or cost 1 node b b nodes … b2 nodes m tiers bm nodes

9. Breadth-First Search

10. Breadth-First Search G a c b e d f S S e e p h d Search Tiers q h h r r b c p r q p p q q f f a a q q c c G G a a Strategy: expand a shallowest node first Implementation: Fringe is a FIFO queue

11. Breadth-First Search (BFS) Properties • What nodes does BFS expand? • Processes all nodes above shallowest solution • Let depth of shallowest solution be s • Search takes time O(bs) • How much space does the fringe take? • Has roughly the last tier, so O(bs) • Is it complete? • s must be finite if a solution exists, so yes! (if no solution, still need depth != ∞) • Is it optimal? • Only if costs are all 1 (more on costs later) 1 node b b nodes … s tiers b2 nodes bs nodes bm nodes

12. Video of Demo Maze Water DFS/BFS (part 1)

13. Video of Demo Maze Water DFS/BFS (part 2)

14. Iterative Deepening • Idea: get DFS’s space advantage with BFS’s time / shallow-solution advantages • Run a DFS with depth limit 1. If no solution… • Run a DFS with depth limit 2. If no solution… • Run a DFS with depth limit 3. ….. • Isn’t that wastefully redundant? • Generally most work happens in the lowest level searched, so not so bad! b …

15. Cost-Sensitive Search GOAL a c b e d f START h p r q

16. Cost-Sensitive Search BFS finds the shortest path in terms of number of actions. It does not find the least-cost path. We will now cover a similar algorithm which does find the least-cost path. GOAL a 2 2 c b 3 2 1 8 e 2 d 3 f 9 8 2 START h 4 2 1 4 15 p r q How?

17. Uniform Cost Search

18. Uniform Cost Search G a c b e d f S h S e e p d p r q q h h r r b c p p q q f f a a q q c c G G a a 2 Strategy: expand a cheapest node first: Fringe is a priority queue (priority: cumulative cost) 8 1 2 2 3 2 9 8 1 1 15 0 1 9 3 16 11 5 17 4 11 Cost contours 7 6 13 8 11 10

19. Uniform Cost Search (UCS) Properties • What nodes does UCS expand? • Processes all nodes with cost less than cheapest solution! • If that solution costs C* and arcs cost at least  ,then the “effective depth” is roughly C*/ • Takes time O(bC*/) (exponential in effective depth) • How much space does the fringe take? • Has roughly the last tier, so O(bC*/) • Is it complete? • Assuming best solution has a finite cost and minimum arc cost is positive, yes! (if no solution, still need depth != ∞) • Is it optimal? • Yes! (Proof via A*) b c  1 … c  2 C*/ “tiers” c  3

20. Uniform Cost Issues • Remember: UCS explores increasing cost contours • The good: UCS is complete and optimal! • The bad: • Explores options in every “direction” • No information about goal location • We’ll fix that soon! c  1 … c  2 c  3 Start Goal [Demo: empty grid UCS (L2D5)] [Demo: maze with deep/shallow water DFS/BFS/UCS (L2D7)]

21. Video of Demo Empty UCS

22. Video of Demo Maze with Deep/Shallow Water --- DFS, BFS, or UCS? (part 1)

23. Video of Demo Maze with Deep/Shallow Water --- DFS, BFS, or UCS? (part 2)

24. Video of Demo Maze with Deep/Shallow Water --- DFS, BFS, or UCS? (part 3)

25. The One Queue • All these search algorithms are the same except for fringe strategies • Conceptually, all fringes are priority queues (i.e. collections of nodes with attached priorities) • Practically, for DFS and BFS, you can avoid the log(n) overhead from an actual priority queue, by using stacks and queues • Can even code one implementation that takes a variable queuing object

26. Up next: Informed Search • Uninformed Search • DFS • BFS • UCS • Informed Search • Heuristics • Greedy Search • A* Search • Graph Search

27. Search Heuristics • A heuristic is: • A function that estimates how close a state is to a goal • Designed for a particular search problem • Pathing? • Examples: Manhattan distance, Euclidean distance for pathing 10 5 11.2

28. Example: Heuristic Function h(x)

29. Greedy Search

30. Greedy Search • Expand the node that seems closest… • Is it optimal? • No. Resulting path to Bucharest is not the shortest!

31. Greedy Search b • Strategy: expand a node that you think is closest to a goal state • Heuristic: estimate of distance to nearest goal for each state • A common case: • Best-first takes you straight to the (wrong) goal • Worst-case: like a badly-guided DFS … b … [Demo: contours greedy empty (L3D1)] [Demo: contours greedy pacman small maze (L3D4)]

32. Video of Demo Contours Greedy (Empty)

33. Video of Demo Contours Greedy (Pacman Small Maze)

34. A* Search

35. A* Search UCS Greedy A*

36. Combining UCS and Greedy • Uniform-costorders by path cost, or backward cost g(n) • Greedyorders by goal proximity, or forward cost h(n) • A* Search orders by the sum: f(n) = g(n) + h(n) g = 0 h=6 8 S g = 1 h=5 e h=1 a 1 1 3 2 g = 9 h=1 g = 2 h=6 S a d G g = 4 h=2 b d e h=6 h=5 1 h=2 h=0 1 g = 3 h=7 g = 6 h=0 g = 10 h=2 c b c G d h=7 h=6 g = 12 h=0 G Example: TegGrenager

37. When should A* terminate? • Should we stop when we enqueue a goal? • No: only stop when we dequeue a goal h = 2 g h + A 2 2 S 0 3 3 S G S->A 2 2 4 h = 3 h = 0 S->B 2 1 3 2 B 3 S->B->G 5 0 5 h = 1 S->A->G 4 0 4

38. Is A* Optimal? h = 6 • What went wrong? • Actual bad goal cost < estimated good goal cost • We need estimates to be less than actual costs! A 1 3 g h + S 0 7 7 S h = 7 G h = 0 S->A 1 6 7 S->G 5 0 5 5

39. Admissible Heuristics

40. Idea: Admissibility Inadmissible (pessimistic) heuristics break optimality by trapping good plans on the fringe Admissible (optimistic) heuristics slow down bad plans but never outweigh true costs

41. Admissible Heuristics • A heuristic h is admissible(optimistic) if: where is the true cost to a nearest goal • Examples: • Coming up with admissible heuristics is most of what’s involved in using A* in practice. 0.0 11.5 15

42. Optimality of A* Tree Search

43. Optimality of A* Tree Search Assume: • A is an optimal goal node • B is a suboptimal goal node • h is admissible Claim: • A will exit the fringe before B …

44. Optimality of A* Tree Search: Blocking Proof: • Imagine B is on the fringe • Some ancestor n of A is on the fringe, too (maybe A!) • Claim: n will be expanded before B • f(n) is less or equal to f(A) … Definition of f-cost Admissibility of h h = 0 at a goal

45. Optimality of A* Tree Search: Blocking Proof: • Imagine B is on the fringe • Some ancestor n of A is on the fringe, too (maybe A!) • Claim: n will be expanded before B • f(n) is less or equal to f(A) • f(A) is less than f(B) … B is suboptimal h = 0 at a goal

46. Optimality of A* Tree Search: Blocking Proof: • Imagine B is on the fringe • Some ancestor n of A is on the fringe, too (maybe A!) • Claim: n will be expanded before B • f(n) is less or equal to f(A) • f(A) is less than f(B) • n expands before B • All ancestors of A expand before B • A expands before B • A* search is optimal …

47. Properties of A* Uniform-Cost A* b b … …

48. UCS vs A* Contours • Uniform-cost expands equally in all “directions” • A* expands mainly toward the goal, but does hedge its bets to ensure optimality Start Goal Goal Start [Demo: contours UCS / greedy / A* empty (L3D1)] [Demo: contours A* pacman small maze (L3D5)]

49. Video of Demo Contours (Empty) -- UCS

50. Video of Demo Contours (Empty) -- Greedy