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Implementation: General Tree Search

Implementation: General Tree Search. Single-State Problem Formulation. A problem is defined by four items: initial state successor function (which actually defines all reachable states) goal test path cost (additive). Your chance to review. initial state Si = 1 successor function

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Implementation: General Tree Search

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  1. Implementation: General Tree Search

  2. Single-State Problem Formulation • A problem is defined by four items: • initial state • successor function (which actually defines all reachable states) • goal test • path cost (additive)

  3. Your chance to review • initial state • Si = 1 • successor function • f(s) = 2s , f(s) = 2s+1 • goal test • Sg = 11 • path cost • Each transition costs one unit

  4. Your chance to review • Consider a state space where the start state is number 1 and the successor function for state n returns two states, numbers 2n and 2n+1. • Draw the portion of the state space for states 1 to 15. • Suppose the goal state is 11. • List the order in which nodes will be visited for breadth-first search.

  5. Solution • Breadth First • 1, 2, 3, 4, 5, 6, 7, 8 , 9, 10, 11

  6. Search Strategies • A 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/expanded space complexity – maximum number of nodes in memory optimality – does it always find a least-cost solution

  7. Search Strategies • Time and space complexity are measured in terms of b – maximum branching factor of the search treed – depth of the least-cost solutionm – maximum depth of the state space (may be infinite)

  8. Properties of Breadth-First Search • Complete??

  9. Properties of Breadth-First Search • Complete?? Yes (if b is finite) • Time??

  10. Properties of Breadth-First Search • Complete?? Yes (if b is finite) • Time?? 1 + b + b2 + b3 + … + bd + b(bd – 1)= O( bd+1 ), ie, exp. in d • Space??

  11. Properties of Breadth-First Search • Complete?? Yes (if b is finite) • Time?? 1 + b + b2 + b3 + … + bd + b(bd – 1)= O( bd+1 ), ie, exp. in d • Space?? O( bd+1 ) (keep every node in memory) • Optimal??

  12. Properties of Breadth-First Search • Complete?? Yes (if b is finite) • Time?? 1 + b + b2 + b3 + … + bd + b(bd – 1)= O( bd+1 ), ie, exp. in d • Space?? O( bd+1 ) (keep every node in memory) • Optimal?? Yes (if cost = 1 per step); not optimal in general • Space is the big problem: can easily generate nodes at 10MB/sec, so 24hours = 860GB.

  13. Your chance to review • Consider a state space where the start state is number 1 and the successor function for state n returns two states, numbers 2n and 2n+1. • Draw the portion of the state space for states 1 to 15. • Suppose the goal state is 11. • List the order in which nodes will be visited for depth-first search.

  14. Solution • Breadth First • 1, 2, 3, 4, 5, 6, 7, 8 , 9, 10, 11 • Depth First • Trick Question • 1, 2, 4, 8, 16, 32, ….

  15. Properties of Depth-first Search • Complete??

  16. Properties of Depth-first Search • Complete?? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path complete in finite spaces • Time??

  17. Properties of Depth-first Search • Complete?? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path complete in finite spaces • Time?? O(bm): terrible if m is much larger than dbut if solutions are dense, may be much faster than breadth-first • Space??

  18. Properties of Depth-first Search • Complete?? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path complete in finite spaces • Time?? O(bm): terrible if m is much larger than dbut if solutions are dense, may be much faster than breadth-first • Space?? O(bm), i.e., linear space! • Optimal??

  19. Properties of Depth-first Search • Complete?? No: fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path complete in finite spaces • Time?? O(bm): terrible if m is much larger than dbut if solutions are dense, may be much faster than breadth-first • Space?? O(bm), I.e., linear space! • Optimal?? No.

  20. Implementation: General Tree Search

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

  22. Properties of Breadth-First Search • Complete?? Yes (if b is finite) • Time?? 1 + b + b2 + b3 + … + bd + b(bd – 1)= O( bd+1 ), ie, exp. in d • Space?? O( bd+1 ) (keep every node in memory) • Optimal?? Yes (if cost = 1 per step); not optimal in general • Space is the big problem: can easily generate nodes at 10MB/sec, so 24hours = 860GB.

  23. Properties of Breadth-First Search • Complete?? Yes (if b is finite) • Time?? 1 + b + b2 + b3 + … + bd + b(bd – 1)= O( bd+1 ), ie, exp. in d • Space?? O( bd+1 ) (keep every node in memory) • Optimal?? Yes (if cost = 1 per step); not optimal in general • Space is the big problem: can easily generate nodes at 10MB/sec, so 24hours = 860GB.

  24. Problem Solving Agents

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