Problem Solving. Russell and Norvig: Chapter 3

Transcription

1 Problem Solving Russell and Norvig: Chapter 3

2 Example: Route finding

3 Example: 8-puzzle Initial state 7 8 Goal state

4 Example: 8-puzzle

5 Example: 8-puzzle Size of the state space = 9!/2 = 8,440 5-puzzle.65 x puzzle.5 x billion years 6 days 0.8 sec 0 million states/sec

6 Example: 8-queens Place 8 queens in a chessboard so that no two queens are in the same row, column, or diagonal. A solution Not a solution

7 Example: 8-queens problem Incremental formulation vs. complete-state formulation States?? Initial state?? Actions?? Goal test?? Path cost??

8 Example: 8-queens Formulation #: States: any arrangement of 0 to 8 queens on the board Initial state: 0 queens on the board Actions: add a queen in any square Goal test: 8 queens on the board, none attacked Path cost: none 64 8 states with 8 queens

9 Example: 8-queens 2,067 states Formulation #2: States: any arrangement of k = 0 to 8 queens in the k leftmost columns with none attacked Initial state: 0 queens on the board Successor function: add a queen to any square in the leftmost empty column such that it is not attacked by any other queen Goal test: 8 queens on the board

10 Search of State Space

11 Search of State Space

12 Search State Space

13 Search of State Space

14 Search of State Space

15 Search of State Space search tree

16 Breadth-First Strategy Expand shallowest unexpanded node Implementation: fringe is a FIFO queue New nodes are inserted at the end of the queue FRINGE = ()

17 Breadth-First Strategy Expand shallowest unexpanded node Implementation: fringe is a FIFO queue New nodes are inserted at the end of the queue FRINGE = (2, 3)

18 Breadth-First Strategy Expand shallowest unexpanded node Implementation: fringe is a FIFO queue New nodes are inserted at the end of the queue FRINGE = (3, 4, 5)

19 Breadth-First Strategy Expand shallowest unexpanded node Implementation: fringe is a FIFO queue New nodes are inserted at the end of the queue FRINGE = (4, 5, 6, 7)

20 Breadth-First Strategy Expand shallowest unexpanded node Implementation: fringe is a FIFO queue New nodes are inserted at the end of the queue FRINGE = (5, 6, 7, 8)

21 Breadth-First Strategy Expand shallowest unexpanded node Implementation: fringe is a FIFO queue New nodes are inserted at the end of the queue FRINGE = (6, 7, 8)

22 Breadth-First Strategy Expand shallowest unexpanded node Implementation: fringe is a FIFO queue New nodes are inserted at the end of the queue FRINGE = (7, 8, 9)

23 Breadth-First Strategy Expand shallowest unexpanded node Implementation: fringe is a FIFO queue New nodes are inserted at the end of the queue FRINGE = (8, 9)

24 Breadth-first search: evaluation lessons: Memory requirements are a bigger problem than execution time. Exponential complexity search problems cannot be solved by uninformed search methods for any but the smallest instances. DEPTH NODES TIME MEMORY seconds megabyte 4 00 seconds 06 megabytes minutes 0 gigabytes hours terabyte days 0 terabytes years 0 petabytes years exabyte Assumptions: b = 0; 0,000 nodes/sec; 000 bytes/node

25 Depth-First Strategy Expand deepest unexpanded node Implementation: fringe is a LIFO queue (=stack) FRINGE = ()

26 Depth-First Strategy Expand deepest unexpanded node Implementation: fringe is a LIFO queue (=stack) FRINGE = (2, 3)

27 Depth-First Strategy Expand deepest unexpanded node Implementation: fringe is a LIFO queue (=stack) FRINGE = (4, 5, 3)

28 Depth-First Strategy Expand deepest unexpanded node Implementation: fringe is a LIFO queue (=stack)

29 Depth-First Strategy Expand deepest unexpanded node Implementation: fringe is a LIFO queue (=stack)

30 Depth-First Strategy Expand deepest unexpanded node Implementation: fringe is a LIFO queue (=stack)

31 Depth-First Strategy Expand deepest unexpanded node Implementation: fringe is a LIFO queue (=stack)

32 Depth-First Strategy Expand deepest unexpanded node Implementation: fringe is a LIFO queue (=stack)

33 Depth-First Strategy Expand deepest unexpanded node Implementation: fringe is a LIFO queue (=stack)

34 Depth-First Strategy Expand deepest unexpanded node Implementation: fringe is a LIFO queue (=stack)

35 Depth-First Strategy Expand deepest unexpanded node Implementation: fringe is a LIFO queue (=stack)

36 Depth-First Strategy Expand deepest unexpanded node Implementation: fringe is a LIFO queue (=stack)

37 Depth-First Strategy Expand deepest unexpanded node Implementation: fringe is a LIFO queue (=stack)

38 Bidirectional Strategy 2 fringe queues: FRINGE and FRINGE2 Time and space complexity = O(b d/2 ) << O(b d ) The predecessor of each node should be efficiently computable.

39 Graph search algorithm Closed list stores all expanded nodes function GRAPH-SEARCH(problem,fringe) return a solution or failure closed an empty set fringe INSERT(MAKE-NODE(INITIAL-STATE[problem]), fringe) loop do if EMPTY?(fringe) then return failure node REMOVE-FIRST(fringe) if GOAL-TEST[problem] applied to STATE[node] succeeds then return SOLUTION(node) if STATE[node] is not in closed then add STATE[node] to closed fringe INSERT-ALL(EXPAND(node, problem), fringe)

40 Graph search, evaluation Optimality: GRAPH-SEARCH discard newly discovered paths. This may result in a sub-optimal solution YET: when uniform-cost search or BF-search with constant step cost Time and space complexity, proportional to the size of the state space (may be much smaller than O(b d )). DF- and ID-search with closed list no longer has linear space requirements since all nodes are stored in closed list!!