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Solving problems by searching

Solving problems by searching. Chapter 3. Example: The 8-puzzle. states? actions? goal test? path cost?. Example: The 8-puzzle. states? locations of tiles actions? move blank left, right, up, down goal test? = goal state (given) path cost? 1 per move

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Solving problems by searching

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  1. Solving problems by searching Chapter 3 Blind Search

  2. Example: The 8-puzzle • states? • actions? • goal test? • path cost? Blind Search

  3. Example: The 8-puzzle • states?locations of tiles • actions?move blank left, right, up, down • goal test?= goal state (given) • path cost? 1 per move [Note: optimal solution of n-Puzzle family is NP-hard] Blind Search

  4. Implementation: states vs. nodes • A state is a (representation of) a physical configuration • A node is a data structure constituting part of a search tree includes state, parent node, action, path costg(x), depth • The Expand function creates new nodes, filling in the various fields and using the SuccessorFn of the problem to create the corresponding states. Blind Search

  5. Search strategies • A search strategy is defined by picking the order of node expansion • Strategies are evaluated along the following dimensions: • completeness: does it always find a solution if one exists? • time complexity: number of nodes generated • space complexity: maximum number of nodes in memory • optimality: does it always find a least-cost solution? • Time and space complexity are measured in terms of • b: maximum branching factor of the search tree • d: depth of the least-cost solution • m: maximum depth of the state space (may be ∞) Blind Search

  6. Uninformed search strategies • Uninformed search strategies use only the information available in the problem definition • Breadth-first search • Uniform-cost search • Depth-first search • Depth-limited search • Iterative deepening search Blind Search

  7. Breadth-first search • Expand shallowest unexpanded node • Implementation: • fringe is a FIFO queue, i.e., new successors go at end Blind Search

  8. Breadth-first search • Expand shallowest unexpanded node • Implementation: • fringe is a FIFO queue, i.e., new successors go at end Blind Search

  9. Breadth-first search • Expand shallowest unexpanded node • Implementation: • fringe is a FIFO queue, i.e., new successors go at end Blind Search

  10. Breadth-first search • Expand shallowest unexpanded node • Implementation: • fringe is a FIFO queue, i.e., new successors go at end Blind Search

  11. Properties of breadth-first search • Complete?Yes (if b is finite) • Time?1+b+b2+b3+… +bd + b(bd-1) = O(bd+1) • Space?O(bd+1) (keeps every node in memory) • Optimal? Yes (if cost = 1 per step) • Space is the bigger problem (more than time) Blind Search

  12. Uniform-cost search • Expand least-cost unexpanded node • Implementation: • fringe = queue ordered by path cost • Equivalent to breadth-first if step costs all equal • Complete? Yes, if step cost ≥ ε • Time? # of nodes with g ≤ cost of optimal solution, O(bceiling(C*/ ε)) where C* is the cost of the optimal solution • Space? # of nodes with g≤ cost of optimal solution, O(bceiling(C*/ ε)) • Optimal? Yes – nodes expanded in increasing order of g(n) Blind Search

  13. Depth-first search • Expand deepest unexpanded node • Implementation: • fringe = LIFO queue, i.e., put successors at front Blind Search

  14. Depth-first search • Expand deepest unexpanded node • Implementation: • fringe = LIFO queue, i.e., put successors at front Blind Search

  15. Depth-first search • Expand deepest unexpanded node • Implementation: • fringe = LIFO queue, i.e., put successors at front Blind Search

  16. Depth-first search • Expand deepest unexpanded node • Implementation: • fringe = LIFO queue, i.e., put successors at front Blind Search

  17. Depth-first search • Expand deepest unexpanded node • Implementation: • fringe = LIFO queue, i.e., put successors at front Blind Search

  18. Depth-first search • Expand deepest unexpanded node • Implementation: • fringe = LIFO queue, i.e., put successors at front Blind Search

  19. Depth-first search • Expand deepest unexpanded node • Implementation: • fringe = LIFO queue, i.e., put successors at front Blind Search

  20. Depth-first search • Expand deepest unexpanded node • Implementation: • fringe = LIFO queue, i.e., put successors at front Blind Search

  21. Depth-first search • Expand deepest unexpanded node • Implementation: • fringe = LIFO queue, i.e., put successors at front Blind Search

  22. Depth-first search • Expand deepest unexpanded node • Implementation: • fringe = LIFO queue, i.e., put successors at front Blind Search

  23. Depth-first search • Expand deepest unexpanded node • Implementation: • fringe = LIFO queue, i.e., put successors at front Blind Search

  24. Depth-first search • Expand deepest unexpanded node • Implementation: • fringe = LIFO queue, i.e., put successors at front Blind Search

  25. Properties of depth-first search • Complete? No: infinite-depth, loops • Modify to avoid repeated states along path  complete in finite spaces • Time?O(bm): terrible if m is much larger than d • if solutions are dense, may be faster than BFS • Space?O(bm), i.e., linear space! • Optimal? No Blind Search

  26. Depth-limited search = depth-first search with depth limit l, i.e., nodes at depth l have no successors Blind Search

  27. Iterative deepening search Blind Search

  28. Iterative deepening search l =0 Blind Search

  29. Iterative deepening search l =1 Blind Search

  30. Iterative deepening search l =2 Blind Search

  31. Iterative deepening search l =3 Blind Search

  32. Iterative deepening search • Nodes generated in a DLS to d with branching factor b: NDLS = b0 + b1 + b2 + … + bd-2 + bd-1 + bd • Nodes generated in an IDS to d with branching factor b: NIDS = (d+1)b0 + d b1 + (d-1)b2 + … + 3bd-2 +2bd-1 + 1bd • For b = 10, d = 5, • NDLS = 1 + 10 + 100 + 1,000 + 10,000 + 100,000 = 111,111 • NIDS = 6 + 50 + 400 + 3,000 + 20,000 + 100,000 = 123,456 • Overhead = (123,456 - 111,111)/111,111 = 11% Blind Search

  33. Properties of iterative deepening search • Complete? Yes • Time? O(bd) • Space?O(bd) • Optimal? Yes, if step cost = 1 Blind Search

  34. Summary of algorithms Blind Search

  35. Repeated states • Failure to detect repeated states can turn a linear problem into an exponential one! Blind Search

  36. Summary • Problem formulation usually requires abstracting away real-world details to define a state space that can feasibly be explored • Variety of uninformed search strategies • Iterative deepening search uses only linear space and not much more time than other uninformed algorithms Blind Search

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