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Tree Searching Strategies
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The procedure of solving many problems may be represented by trees.
Therefore the solving of these problems becomes a tree searching problem.
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Satisfiability problem
Tree Representation of Eight Assignments. If there are n variables x1, x2, …,xn, then there are 2n possible assignments.
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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.
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Hamiltonian circuit problem
E.g. the Hamiltonian circuit problem A Graph Containing a Hamiltonian Circuit
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Fig. 6-8 The Tree Representation of Whether There Exists a Hamiltonian Circuit of the Graph in Fig. 6-6
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A tree showing the non-existence of any Hamiltonian circuit.
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8-Puzzle Problem 2 3 1 8 4 7 6 5 1 2 3 8 4 7 6 5 Initial State:
Goal State: 2 3 1 8 4 7 6 5 1 2 3 8 4 7 6 5
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Tree Representation of the solution of 8-puzzle problem
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How to expand the tree ? Breadth-First Search Depth-First Search
Hill Climbing Best-First Search
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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.
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2 3 1 8 4 7 6 5 1 2 3 1 8 4 7 6 5 1 2 3 8 4 7 6 5 2 3 2 8 3 1 4 7 6 5 2 3 1 8 4 7 6 5 1 2 3 8 4 7 6 5 1 2 3 7 8 4 6 5 4 6 5 7 Goal Node
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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.
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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.
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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.
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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.
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An 8-Puzzle Problem Solved by the Hill Climbing Method
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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.
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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.
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An 8-Puzzle Problem Solved by the Best-First Search Scheme
Goal Node An 8-Puzzle Problem Solved by the Best-First Search Scheme
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Q&A
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