Tree Search

7/31/2019 Tree Search

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Tree Searching Strategies

Updated: 2010/12/27


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The procedure of solving manyproblems may be represented by trees.

Therefore the solving of these problemsbecomes a tree searching problem.


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Satisfiability problemx1 x2 x3

F F F

F F TF T F

F T T

T F F

T F TT T F

T T TTree 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 Determinethe 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:

2 3

1 8 4

7 6 5

Goal State:

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 In breadth-first search, all the

nodes on one level of the tree are

examined before the nodes on thenext level are examined.

It can be accomplished with thehelp of the queue.


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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. Step3: Remove the first element from the

queue. Add all descendants of the firstelement, if any, to the end of the queue one

by one. 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

2 31 8 4

7 6 5

1 2 3

8 47 6 5

2 8 31 4

7 6 5

2 3

1 8 47 6 5

1 2 37 8 4

6 5

1 2 38 4

7 6 5

1

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4

5

6

7Goal Node


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Depth-First Search Scheme The depth-first search always

selects the deepest node for

expansion.

It can be accomplished with the

help of the stack.


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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

stack is a goal node. If it is, stop. Step3: Remove the top element from the

stack. Add all descendants of the firstelement, if any, to the top of the stack one by

one. Step4: If the stack is empty, then signal



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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 climbingA variant ofdepth-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 misplacedtiles 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

stack is a goal node. If it is, stop. Step3: Remove the top element from the

stack. Add the first elements descendants, ifany, to the top of the stack according to

order computed by the evaluation function. Step4: If the stack is empty, then signal



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19An 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 estimatedcost 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-elementheap consisting of the root node.

Step2:Test to see if the root element in theheap is a goal node. If it is, stop.

Step3:Remove the root element from theheap and expand the element, i.e., add all

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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22An 8-Puzzle Problem Solved by the Best-First Search Scheme

Goal Node


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Q&A23

Tree Search

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