2. UNINFORMED SEARCH

3. Problem - solving agents

4. Example : Romania  On holiday in Romania ; currently in Arad.  Flight leaves tomorrow from Bucharest

5. What do we need to define ?

6. Problem Formulation  The process of defining actions, states and goal.  States :  Cities ( e. g. Arad, Sibiu, Bucharest, etc )  Actions :  GoTo ( adjacent city )  Goal :  Bucharest Why not “turn left 5 degrees” or “walk 100 meters forward…”?

7. Abstraction  The process of removing details from a representation.  Simplifies the problem  Makes problems tractable ( possible to solve )  Humans are great at this !  Imagine a hierarchy in which another agent takes care of the lower level details, such as navigating from the city center to the highway.

8. Back to Arad…  We are in Arad and need to find our way to Bucharest.

9. Step 1 – Check Goal Condition  Check, are we at the goal ?  ( obviously not in this case, but we need to check in case we were )

10. Step 2 – Expand Current Node  Enumerate all the possible actions you could take from the current state  Formally : apply each legal action to the current state, thereby generating a new set of states.  From Arad can go to :  Sibiu  Timisoara  Zerind

11. Step 3 – Select which action to perform  Perform one of the possible actions ( e. g. GoTo ( Sibiu ))  Then go back to Step 1 and repeat.

12. This is an example of Tree Search  Exploration of state space by generating successors of already - explored states ( a. k. a. expanding states )  Usually performed offline, as a simulation  Returns the sequence of actions that should be performed to reach the goal, or that the goal is unreachable.

13. Example : Romania

14. Tree Search

15. Tree search example

16. Tree search example : start with Sibiu

17. Tree search example : need to process the descendants of Sibiu Note that we can loop back to Arad. Have to make sure we don’t go in circles forever!

18. Tree search algorithms

19. Implementation : general tree search a.k.a. frontier

20. This is the part that distinguishes different search strategies

21. Search strategies  A search strategy is defined by picking the order of node expansion

22. Uninformed search strategies  Uninformed search strategies use only the information available in the problem definition  What does it mean to be uninformed ?  You only know the topology of which states are connected by which actions. No additional information.  Later we ’ ll talk about informed search, in which you can estimate which actions are likely to be better than others.

23. Breadth - first search  Expand shallowest unexpanded node  Implementation :  Fringe is a FIFO queue, i. e., new successors go at end

24. Breadth - first search  Expand shallowest unexpanded node  Implementation :  Fringe is a FIFO queue, i. e., new successors go at end

25. Breadth - first search  Expand shallowest unexpanded node  Implementation :  Fringe is a FIFO queue, i. e., new successors go at end

26. Breadth - first search  Expand shallowest unexpanded node  Implementation :  Fringe is a FIFO queue, i. e., new successors go at end

27. BFS on a Graph

28. Search Strategy Evaluation : finding solutions  Strategies are evaluated along the following dimensions :  completeness : does it always find a solution if one exists ?  optimality : does it always find a least - cost solution ?

29. Search Strategy Evaluation : complexity ( cost )  Two types of complexity  time complexity : number of nodes visited  space complexity : maximum number of nodes in memory  Time and space complexity are measured in terms of  b : maximum branching factor of the search tree ( may  ∞ )  d : depth of the least - cost solution  m : maximum depth of the state space ( may be ∞ )

30. Properties of breadth - first search  Complete ?  Yes ( if b is finite )  Optimal ?  Yes ( if cost = 1 per step )  Time ?  1+ b + b 2 + b 3 + … + b d = O ( b d )  Space ?  O ( b d ) ( keeps every node in memory )

31. Problems of breadth first search  Space is the biggest problem ( more than time )  Example from book, BFS b =10 to depth of 10  3 hours ( not so bad )  10 terabytes of memory ( really bad )  Only reason speed is not a problem is you run out of memory first

32. Problems of breadth first search  BFS is not optimal if the cost of some actions is greater than others…

33. Uniform - cost search  For graphs with actions of different cost  Equivalent to breadth - first if step costs all equal  Expand least - cost unexpanded node  Implementation :  fringe = queue sorted by path cost g ( n ), from smallest to largest ( i. e. a priority queue )

34. Uniform - cost search

35.  Complete ?  Yes, if step cost ≥ ε  Time ?  O ( b ceiling ( C */ ε ) ) where C * is the cost of the optimal solution  Space ?  # of nodes with g ≤ cost of optimal solution, O ( b ceiling ( C */ ε ) )  Optimal ?  Yes – nodes expanded in increasing order of g ( n ) See book for detailed analysis.

36. Depth - first search  Expand deepest unexpanded node  Implementation :  fringe = LIFO queue, i. e., put successors at front ( i. e. a stack )

37. Depth - first search  Expand deepest unexpanded node  Implementation :  fringe = LIFO queue, i. e., put successors at front

38. Depth - first search  Expand deepest unexpanded node  Implementation :  fringe = LIFO queue, i. e., put successors at front

39. Depth - first search  Expand deepest unexpanded node  Implementation :  fringe = LIFO queue, i. e., put successors at front

40. Depth - first search  Expand deepest unexpanded node  Implementation :  fringe = LIFO queue, i. e., put successors at front

41. Depth - first search  Expand deepest unexpanded node  Implementation :  fringe = LIFO queue, i. e., put successors at front

42. This is the part that distinguishes different search algorithms

43. Search Solution  Each node needs to keep track of its parent  Once the goal is found, traverse up the tree to the root to find the solution

44. Properties of depth - first search  Complete ?  No : fails in infinite - depth spaces  Yes : in finite spaces  Optimal ?  No  Time ?  O ( b m ): ( m is max depth of state space )  terrible if m is much larger than d  but if solutions are plentiful, may be much faster than breadth - first  Space ?  O ( bm ), i. e., linear space !

45. Depth - limited search  depth - first search with depth limit l ( i. e., don ’ t expand nodes past depth l )  … will fail if the goal is below the depth limit

46. Iterative deepening search

47. Iterative deepening search l =0

48. Iterative deepening search l =1

49. Iterative deepening search l =2

50. Iterative deepening search l =3

51. Properties of iterative deepening search  Complete ?  Yes  Time ?  ( d +1) b 0 + d b 1 + ( d -1) b 2 + … + b d = O ( b d )  Space ?  O ( bd )  Optimal ?  Yes, if step cost = 1

52. Bidirectional Search  Run two simultaneous searches  One forward from the start  One backward from the goal  Hope that the searches meet in the middle  b d /2 + b d /2 << b d

53. Summary of algorithms

54. Graph search The closed set keeps track of loops in the graph so that the search terminates.

55. Questions ? Waitlisted ? Talk to me after class.