Tree Searching Strategies

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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
Initial State
Goal State
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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 elements 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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1
2
3
4
6
5
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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 elements 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 S7, 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 elements 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

• Step1Consturct a heap by using the evaluation
  function. First, form a 1-element heap consisting
  of the root node.
• Step2Test to see if the root element in the heap
  is a goal node. If it is, stop otherwise, go to
  Step 3.
• Step3Remove the root element from the heap and
  expand the element. Add the descendants of the
  element into the heap.
• Step4If the heap is empty, then signal failure.
  Otherwise, go to Step 2.

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Goal Node

• An 8-Puzzle Problem Solved by the Best-First
  Search Scheme

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• QA