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Dr. Ali Abdalla Lectures

Dr. Ali Abdalla Lectures

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  • 1. What is Search? Dr. Ali Abdallah(dr_ali_abdallah@hotmail.com)
  • 2. What is Search?• Search is something we do all the time without realizing it. – Finding this room required search. – So does finding the best insurance deal. – Or finding your keys.• Computers are not, naturally good at search.• But if they could do it, they would be very fast!
  • 3. Computers Bad At Search?• Yes, they can’t decide for themselves how to search.• We have to tell them how to search. – Even when we do tell them how, they will still not understand what they are doing!• If computers are to be efficient at search, we must very carefully tell them how to do it.
  • 4. What is Search For?• All AI is search! – Not totally true (obviously) but more true than you might think. – Finding a good/best solution to a problem amongst many possible solutions.• Ways of methodically deciding which solutions are better than others.
  • 5. Three Types of Search• Blind Search: – Searching the possibilities, in some predefined order. The types of search we do in a maze.• Heuristic Search: – Searching the possibilities in order of their perceived value.• Stochastic Search: – Randomly searching in a cleaver way.
  • 6. Generalized Search algorithm function GeneralSearch(QueuingFunction) queue = MakeInitialQueue() loop if queue is empty then return failure node = RemoveFromFront( queue ) if GoalTest( node ) succeeds then return node queue = QueuingFunction( queue, Expand(node) ) end loop The choice of queuing function is the main feature
  • 7. Example: Arad to Buchareststart end
  • 8. Representing Search Arad Zerind Sibiu Timisoara Arad Oradea Fagaras Rimnicu Vilcea Sibiu Bucharest
  • 9. Search Terminology• Completeness – solution will be found, if it exists• Time complexity – number of nodes expanded• Space complexity – number of nodes in memory• Optimality – least cost solution will be found
  • 10. Blind (Uninformed) Search(Where we systematically explore alternatives) R&N: Chap. 3, Sect. 3.3–5
  • 11. Simple Problem-Solving-AgentAgent Algorithm  s0  sense/read state  GOAL?  select/read goal test  SUCCESSORS  read successor function  solution  search(s0, G, Succ)  perform(solution)
  • 12. Searching the State Space Search tree Note that some states are visited multiple times
  • 13. Basic Search Concepts Search tree Search node Node expansion Fringe of search tree Search strategy: At each stage it determines which node to expand
  • 14. Search Nodes ≠ States 8 2 3 4 7 5 1 6 8 2 7 3 4 5 1 6 8 2 8 2 8 4 2 8 2 3 4 7 3 4 7 3 7 3 4 7 5 1 6 5 1 6 5 1 6 5 1 6
  • 15. Search Nodes ≠ States 8 2 3 4 7 5 1 6 8 2 7 If states are allowed to be revisited, 3 4 the search tree may be infinite even 5 1 6 when the state space is finite 8 2 8 2 8 4 2 8 2 3 4 7 3 4 7 3 7 3 4 7 5 1 6 5 1 6 5 1 6 5 1 6
  • 16. Data Structure of a Node 8 2 STATE 3 4 7 PARENT-NODE 5 1 6 BOOKKEEPING CHILDREN Action Right Depth 5 ... Path-Cost 5 yes Expanded Depth of a node N = length of path from root to N (Depth of the root = 0)
  • 17. Node expansionThe expansion of a node N of the searchtree consists of: – Evaluating the successor function on STATE(N) – Generating a child of N for each state returned by the function
  • 18. Fringe and Search Strategy The fringe is the set of all search nodes that haven’t been expanded yet8 2 Is it identical3 4 7 to the set of5 1 6 leaves?8 2 73 45 1 6 8 2 8 2 8 4 2 8 2 3 4 7 3 4 7 3 7 3 4 7 5 1 6 5 1 6 5 1 6 5 1 6
  • 19. Fringe and Search Strategy The fringe is the set of all search nodes that haven’t been expanded yet It is implemented as a priority queue FRINGE • INSERT(node,FRINGE) • REMOVE(FRINGE) The ordering of the nodes in FRINGE defines the search strategy
  • 20. Search Algorithm If GOAL?(initial-state) then return initial-state INSERT(initial-node,FRINGE) Repeat:  If empty(FRINGE) then return failure  n  REMOVE(FRINGE)  s  STATE(n)  For every state s’ in SUCCESSORS(s)  Create a new node n’ as a child of n  If GOAL?(s’) then return path or goal state  INSERT(n’,FRINGE)
  • 21. Performance Measures Completeness A search algorithm is complete if it finds a solution whenever one exists [What about the case when no solution exists?] Optimality A search algorithm is optimal if it returns an optimal solution whenever a solution exists Complexity It measures the time and amount of memory required by the algorithm
  • 22. Important Parameters• Maximum number of successors of any state  branching factor b of the search tree• Minimal length of a path between the initial and a goal state  depth d of the shallowest goal node in the search tree
  • 23. Blind vs. Heuristic Strategies Blind (or un-informed) strategies do not exploit state descriptions to select which node to expand next Heuristic (or informed) strategies exploits state descriptions to select the “most promising” node to expand
  • 24. Example For a blind strategy, N1 and N2 8 2 are just two nodes (at some depth in the search tree) STATE 3 4 7 N1 5 1 6 1 2 3 1 2 3 4 5 6 STATE 4 5 7 8 N2 7 8 6 Goal state
  • 25. Example For a heuristic strategy counting the number of misplaced tiles, 8 2 N2 is more promising than N1 STATE 3 4 7 N1 5 1 6 1 2 3 1 2 3 4 5 6 STATE 4 5 7 8 N2 7 8 6 Goal state
  • 26. Important Remark Some search problems, such as the (n2-1)- puzzle, are NP-hard One can’t expect to solve all instances of such problems in less than exponential time One may still strive to solve each instance as efficiently as possible
  • 27. Blind Strategies Breadth-first • Bidirectional Depth-first • Depth-limited • Iterative deepening Arc cost Uniform-Cost = c(action) ≥ ε > 0 (variant of breadth-first)
  • 28. Breadth-First StrategyNew nodes are inserted at the end of FRINGE 1 2 3 FRINGE = (1) 4 5 6 7
  • 29. Breadth-First StrategyNew nodes are inserted at the end of FRINGE 1 2 3 FRINGE = (2, 3) 4 5 6 7
  • 30. Breadth-First StrategyNew nodes are inserted at the end of FRINGE 1 2 3 FRINGE = (3, 4, 5) 4 5 6 7
  • 31. Breadth-First StrategyNew nodes are inserted at the end of FRINGE 1 2 3 FRINGE = (4, 5, 6, 7) 4 5 6 7
  • 32. Evaluation b: branching factor d: depth of shallowest goal node Breadth-first search is: • Complete • Optimal if step cost is 1 Number of nodes generated: 1 + b + b2 + … + bd = (bd+1-1)/(b-1) = O(bd)  Time and space complexity is O(bd)
  • 33. Time and Memory Requirements d # Nodes Time Memory 2 111 .01 msec 11 Kbytes 4 11,111 1 msec 1 Mbyte 6 ~106 1 sec 100 Mb 8 ~108 100 sec 10 Gbytes 10 ~1010 2.8 hours 1 Tbyte 12 ~1012 11.6 days 100 Tbytes 14 ~1014 3.2 years 10,000 TbytesMemory requirements vs Time requirementsAssumptions: b = 10; 1,000,000 nodes/sec; 100bytes/node
  • 34. RemarkIf a problem has no solution, breadth-first mayrun for ever (if the state space is infinite orstates can be revisited arbitrary many times) 1 2 3 4 1 2 3 4 5 6 7 8 ? 5 6 7 8 9 10 11 12 9 10 11 12 13 14 15 13 15 14
  • 35. Bidirectional Strategy 2 fringe queues: FRINGE1 and FRINGE2 s Time and space complexity is O(bd/2) << O(bd) if both trees have the same branching factor b
  • 36. Bidirectional Strategy• What happens if the branching factor is different in each direction?Major difficulties:• Need to have a predecessor function• What about multiple goals?• What about implicitly defined goals? (e.g. n-queens problem)
  • 37. Depth-First StrategyNew nodes are inserted at the front of FRINGE 1 2 3 FRINGE = (1) 4 5
  • 38. Depth-First StrategyNew nodes are inserted at the front of FRINGE 1 2 3 FRINGE = (2, 3) 4 5
  • 39. Depth-First StrategyNew nodes are inserted at the front of FRINGE 1 2 3 FRINGE = (4, 5, 3) 4 5
  • 40. Depth-First StrategyNew nodes are inserted at the front of FRINGE 1 2 3 4 5
  • 41. Depth-First StrategyNew nodes are inserted at the front of FRINGE 1 2 3 4 5
  • 42. Depth-First StrategyNew nodes are inserted at the front of FRINGE 1 2 3 4 5
  • 43. Depth-First StrategyNew nodes are inserted at the front of FRINGE 1 2 3 4 5
  • 44. Depth-First StrategyNew nodes are inserted at the front of FRINGE 1 2 3 4 5
  • 45. Depth-First StrategyNew nodes are inserted at the front of FRINGE 1 2 3 4 5
  • 46. Depth-First StrategyNew nodes are inserted at the front of FRINGE 1 2 3 4 5
  • 47. Depth-First StrategyNew nodes are inserted at the front of FRINGE 1 2 3 4 5
  • 48. Evaluation b: branching factor d: depth of shallowest goal node m: maximal depth of a leaf node Depth-first search is:  Complete only for finite search tree  Not optimal Number of nodes generated: 1 + b + b2 + … + bm = O(bm) Time complexity is O(bm) Space complexity is O(bm) [or O(m)][Reminder: Breadth-first requires O(bd) time and space]
  • 49. Depth-Limited Search Depth-first with depth cutoff k (depth below which nodes are not expanded) Three possible outcomes: • Solution • Failure (no solution) • Cutoff (no solution within cutoff)
  • 50. Iterative Deepening Search Provides the best of both breadth-first and depth-first search Main idea: IDS For k = 0, 1, 2, … do: Perform depth-first search with depth cutoff k
  • 51. Iterative Deepening
  • 52. Iterative Deepening
  • 53. Iterative Deepening
  • 54. Performance Iterative deepening search is: • Complete • Optimal if step cost =1 Time complexity is: (d+1)(1) + db + (d-1)b2 + … + (1) bd = O(bd) Space complexity is: O(bd) or O(d)
  • 55. Calculationdb + (d-1)b2 + … + (1) bd = bd + 2bd-1 + 3bd-2 +… + db = (1 + 2b-1 + 3b-2 + … + db-d)×bd ≤ (Σi=1,…,∞ ib(1-i))×bd = bd (b/(b-1))2
  • 56. Number of Generated Nodes(Breadth-First & Iterative Deepening) d = 5 and b = 2 BF ID 1 1x6=6 2 2 x 5 = 10 4 4 x 4 = 16 8 8 x 3 = 24 16 16 x 2 = 32 32 32 x 1 = 32 120/63 ~ 2 63 120
  • 57. Number of Generated Nodes(Breadth-First & Iterative Deepening) d = 5 and b = 10 BF ID 1 6 10 50 100 400 1,000 3,000 10,000 20,000 100,000 100,000 111,111 123,456 123,456/111,111 ~ 1.111
  • 58. Comparison of Strategies Breadth-first is complete and optimal, but has high space complexity Depth-first is space efficient, but is neither complete, nor optimal Iterative deepening is complete and optimal, with the same space complexity as depth-first and almost the same time complexity as breadth-first
  • 59. Revisited States No Few Many 1 2 3 search tree is finite search tree is infinite 4 5 7 8 6 8-queens assembly 8-puzzle and robot navigation planning
  • 60. Avoiding Revisited States Requires comparing state descriptions Breadth-first search: • Store all states associated with generated nodes in CLOSED • If the state of a new node is in CLOSED, then discard the node
  • 61. Avoiding Revisited States Depth-first search:Solution 1: – Store all states associated with nodes in current path in CLOSED – If the state of a new node is in CLOSED, then discard the node  Only avoids loopsSolution 2: – Store of all generated states in CLOSED – If the state of a new node is in CLOSED, then discard the node  Same space complexity as breadth-first !
  • 62. Uniform-Cost Search (Optimal) Each arc has some cost c ≥ ε > 0 The cost of the path to each fringe node N is g(N) = Σ costs of arcs The goal is to generate a solution path of minimal cost The queue FRINGE is sorted in increasing cost A S 0 1 10 S 5 B 5 G A B C 1 5 15 5 15 C G G 11 10 Need to modify search algorithm
  • 63. Search Algorithm If GOAL?(initial-state) then return initial-state INSERT(initial-node,FRINGE) Repeat:  If empty(FRINGE) then return failure  n  REMOVE(FRINGE)  s  STATE(n)  For every state s’ in SUCCESSORS(s)  Create a new node n’ as a child of n  If GOAL?(s’) then return path or goal state  INSERT(n’,FRINGE)
  • 64. Modified Search Algorithm INSERT(initial-node,FRINGE) Repeat:  If empty(FRINGE) then return failure  n  REMOVE(FRINGE)  s  STATE(n)  If GOAL?(s) then return path or goal state  For every state s’ in SUCCESSORS(s)  Create a node n’ as a successor of n  INSERT(n’,FRINGE)
  • 65. Avoiding Revisited States inUniform-Cost Search When a node N is expanded the path to N is also the best path from the initial state to STATE(N) if it is the first time STATE(N) is encountered. So: • When a node is expanded, store its state into CLOSED • When a new node N is generated: – If STATE(N) is in CLOSED, discard N – If there exits a node N’ in the fringe such that STATE(N’) = STATE(N), discard the node – N or N’ – with the highest-cost path