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Tree Searching Strategies

Tree Searching Strategies. The procedure of solving many problems may be represented by trees. Therefore the solving of these problems becomes a tree searching problem. Satisfiability problem. Tree Representation of Eight Assignments.

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Tree Searching Strategies

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  1. Tree Searching Strategies

  2. The procedure of solving many problems may be represented by trees. • Therefore the solving of these problems becomes a tree searching problem.

  3. Satisfiability problem Tree Representation of Eight Assignments. If there are n variables x1, x2, …,xn, then there are 2n possible assignments.

  4. Satisfiability problem • An instance: -x1……..……(1) x1…………..(2) x2 v x5….….(3) x3…….…….(4) -x2…….…….(5) A Partial Tree to Determine the Satisfiability Problem. • We may not need to examine all possible assignments.

  5. Hamiltonian circuit problem • E.g. the Hamiltonian circuit problem A Graph Containing a Hamiltonian Circuit

  6. Fig. 6-8 The Tree Representation of Whether There Exists a Hamiltonian Circuit of the Graph in Fig. 6-6

  7. A tree showing the non-existence of any Hamiltonian circuit.

  8. 8-Puzzle Problem Initial State: Goal State:

  9. Tree Representation of the solution of 8-puzzle problem

  10. How to expand the tree ? • Breadth-First Search • Depth-First Search • Hill Climbing • Best-First Search • Branch-and-Bound Strategy (for optimization problems) • A* Algorithm

  11. Breadth-First Search Scheme • Step1: Form a one-element queue consisting of the root node. • Step2: Test to see if the first element in the queue is a goal node. If it is, stop. Otherwise, go to step 3. • Step3: Remove the first element from the queue. Add the first element’s descendants, if any, to the end of the queue. • Step4: If the queue is empty, then signal failure. Otherwise, go to Step 2.

  12. 1 2 3 4 6 5 7 Goal Node

  13. Depth-First Search Scheme • Step1: Form a one-element stack consisting of the root node. • Step2: Test to see if the top element in the queue is a goal node. If it is, stop. Otherwise, go to step 3. • Step3: Remove the top element from the stack. Add the first element’s descendants, if any, to the top of the stack. • Step4: If the stack is empty, then signal failure. Otherwise, go to Step 2.

  14. E.G.: the depth-first search • E.g. sum of subset problem Given a set S={7, 5, 1, 2, 10}, answer if  S’ S  sum of S’ = 9. The Sum of Subset Problem Solved by Depth-First Search.

  15. Hill climbing • A variant of depth-first search The method selects the locally optimal node to expand. • E.g. for the 8-puzzle problem, evaluation function f(n) = w(n), where w(n) is the number of misplaced tiles in node n.

  16. Hill Climbing Search Scheme • Step1: Form a one-element stack consisting of the root node. • Step2: Test to see if the top element in the queue is a goal node. If it is, stop. Otherwise, go to step 3. • Step3: Remove the top element from the stack. Add the first element’s descendants, if any, to the top of the stack according to order computed by the evaluation function. • Step4: If the stack is empty, then signal failure. Otherwise, go to Step 2.

  17. An 8-Puzzle Problem Solved by the Hill Climbing Method

  18. Best-first search strategy • Combing depth-first search and breadth-first search • Selecting the node with the best estimated cost among all nodes. • This method has a global view.

  19. Best-First Search Scheme • Step1:Consturct a heap by using the evaluation function. First, form a 1-element heap consisting of the root node. • Step2:Test to see if the root element in the heap is a goal node. If it is, stop; otherwise, go to Step 3. • Step3:Remove the root element from the heap and expand the element. Add the descendants of the element into the heap. • Step4:If the heap is empty, then signal failure. Otherwise, go to Step 2.

  20. An 8-Puzzle Problem Solved by the Best-First Search Scheme Goal Node

  21. Feasible Solution vs. Optimal Solution • DFS, BFS, hill climbing and best-first search can not be used to solve optimization problems. • They only point out a feasible solution.

  22. The branch-and-bound strategy • This strategy can be used to solve optimization problems without an exhaustive search.

  23. Global Maximum and Local Maximum global maximum local maximum

  24. A Multi-Stage Graph Searching Problem.

  25. E.G.:A Multi-Stage Graph Searching Problem

  26. Solved by branch-and-bound

  27. Solved by branch-and-bound

  28. Branch-and-bound strategy • 2 mechanisms: • A mechanism to generate branches • A mechanism to generate a bound so that many braches can be terminated. • Although it is usually very efficient, a very large tree may be generated in the worst case. • It is efficient in the sense of average case.

  29. Exercise • Use BFS, DFS, Hill-Climbing and Best-First Search schemes to solve the following 8-puzzle problem with the evaluation function being the number of misplaced tiles. Initial State: Goal State:

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