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Problem Solving and Search

Problem Solving and Search. Andrea Danyluk September 11, 2013. Announcements. Progamming Assignment 0: Python Tutorial Due now Thanks for your patience with account set-up, turnin , etc. Assignment 1: Search Now posted on web site Due Fri, Sep 20 Team project! (35 students)

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Problem Solving and Search

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  1. Problem Solving and Search Andrea Danyluk September 11, 2013

  2. Announcements • Progamming Assignment 0: Python Tutorial • Due now • Thanks for your patience with account set-up, turnin, etc. • Assignment 1: Search • Now posted on web site • Due Fri, Sep 20 • Team project! (35 students) • First few parts straightforward. Don’t be fooled. It gets harder.

  3. Today’s Lecture • More on Uninformed search • Breadth-first • Depth-first • Uniform-cost search  New • Informed (Heuristic) search • Greedy best-first

  4. Formalizing State Space Search • A node in a search tree typically contains: • A state description • A reference to the parent node • The name of the operator that generated it from its parent • The cost of the path from the initial state to itself • Might also include its depth in the tree • The node that is the root of the search tree typically represents the initial state

  5. Formalizing State Space Search • Child nodes are generated by applying legal operators to a node • The process of expanding a node means to generate all of its successor nodes and to add them to the frontier. • A goal test is a function applied to a state to determine whether its associated node is a goal node

  6. Formalizing State Space Search • A solution is either • A sequence of operators that is associated with a path from start state to goal or • A state that satisfies the goal test • The cost of a solution is the sum of the arc costs on the solution path • If all arcs have the same (unit) cost, then the solution cost is just the length of the solution (i.e., the length of the path)

  7. Evaluating Search Strategies • Completeness • Is the strategy guaranteed to find a solution when there is one? • Optimality • Does the strategy find the highest-quality solution when there are several solutions? • Time Complexity • How long does it take (in the worst case) to find a solution? • Space Complexity • How much memory is required (in the worst case)?

  8. Evaluating BFS • Complete? • Yes (if the number of possible actions is finite) • Optimal? • Not in general. When is it optimal? • When costs of all actions are the same • Time Complexity? • How do we measure it? • Space Complexity?

  9. Time and Space Complexity Let • b = branching factor (i.e., max number of successors) • m = maximum depth of the search tree • d = depth of shallowest solution For BFS Time: O(bd) If we check for goal as we generate a node! Not if we check as we get ready to expand! Space: O(bd)

  10. Evaluating DFS • Complete? • Not in general • Yes if state space is finite and we modify tree search to account for loops • Optimal? • No • Time Complexity? • O(bm) • Space Complexity? • O(mb)

  11. Depth-Limited DFS? Let lbe the depth limit • Complete? • No • Optimal? • No • Time Complexity? • O(bl) • Space Complexity? • O(ml)

  12. Iterative Deepening? • Complete? • Yes (if b is finite) • Optimal? • Yes (if costs of all actions are the same) • Time Complexity? • O(bd) • Space Complexity? • O(bd)

  13. Costs on Actions

  14. Costs on Actions Cost of moving a truck = 2x the cost of moving a car that isn’t a sports car. Cost of moving a sports car is 4x the cost of moving any other vehicle.

  15. Uniform Cost Search f Similar to another algorithm you know? 3.5 a 3 b 1 3 1 4 2 1 g 1 e 2 c 5 3.75 3 Fringe is a Priority Queue Priority = cost so far d h 3 i

  16. Evaluating Uniform Cost Search • Complete? • Yes (if b is finite and step cost ≥ efor positive e ) • Optimal? • Yes • Time Complexity? • O(b1+C*/e)  Can’t check for goal until coming out of PQ! • Space Complexity? • O(b1+C*/e)

  17. So we’re done, right? Our search spaces are big.

  18. Informed (Heuristic) Search Strategies • Use problem-specific knowledge to find solutions more efficiently

  19. Best-first Search • Choose a node from the frontier based on its “desirability” • Frontier is a priority queue • Requires a search heuristic • Any estimate of how close a state is to a goal • Examples: • Euclidean distance • Manhattan distance • For the “rush hour” problem?

  20. Greedy Best-first Search h(n) = estimate of cost from n to the closest goal Expand the node with lowest h value - i.e., the node that appears to be closest to the goal

  21. Greedy Search with hSLD This time g is the goal. f 3.5 a 3 b 1 3 1 4 2 1 g 1 e 2 c 5 3.75 3 d h 3 i

  22. Problems with Greedy Search? Can be quite good with a high-quality heuristic, but • Not optimal • Time and space complexity: O(bm)

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