Review of Graphs and Trees

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Review of Graphs and Trees

• Graph
• Directed graph
• Cycle

2
Review of Graphs and Trees

• Tree
• Subtree
• Branch
• Root
• Leaf Node
• Intermediate Node
• Ply
• Branching Factor

3
Two General Views of Problem Representations

• State-Space Representation
• Problem-Reduction Representation

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State-Space Representation
(S, I, G, O, R)

• S - the set of states or snapshots
  of the problem at various stages of
  solution
• I Ì S -
• G Ì S -
• O -
• R -

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• Search Space
• Search Graph

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Problem
Three sons and their father wish to cross a
river. The father weighs 200 lbs and each son
weighs 100 lbs. They have a boat that can only
carry 200 lbs (none can swim and the river has
alligators). Son 3 cannot row the boat, but
everyone else can. What are the steps required
for them to cross the river?
States Operators
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Finding a Solution
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What is the Search Space?
123F
F132
13F2
F123
F312
23F1
F213
F231
F123
1F23
12F3
F123
3F12
2F13
F123
F123
9
What is a Solution Path?
123F
F132
13F2
F123
F312
23F1
F213
F231
F123
1F23
12F3
F123
3F12
2F13
F123
F123
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Problem Reduction Representation
(G, I, P, O, R)

• G - the set of goals or problem
  descriptions
• I Î G -
• P Ì G -
• O -
• R -

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Problem

• The Tower of Hanoi Puzzle
• There are four disks (A, B, C, D) of graduated
  sizes initially stacked on Peg 1 of the three
  pegs. A, the smallest, is on the top and D, the
  largest, is at the bottom. The disks are to be
  transferred to Peg 3 observing the following
  rules
• (1)
• (2)

Initial State
Final State
A B C D
A B C D
1
2
3
1
2
3
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• The only operator
• The only primitive problem

• Given three pegs i, j, and k, the problem of
  moving a stack of size ngt1 from peg i to peg k
  can be replaced by the three problems
• 1.
• 2.
• 3.

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• One possible representation of a problem
  description is
• The only operator is then
• And the primitive problem is

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• The solution tree is then

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• In general, the search graph is called an AND/OR
  Graph
• Rules for And/Or Graphs

• 1. Each node represents either a single problem
  or a set of problems
• 2. The root node is the original problem
• 3. A leaf node is called a terminal node if it
  represents a primitive problem and, therefore,
  has no descendants
• 4. When an operator is applied to a problem P
  transforming it into a set of subproblems, A, B,
  C, any of which will solve the original
  problem, then these nodes are called OR nodes
• 5. When an operator is applied to a problem P
  transforming it into a set of subproblems, A, B,
  C, all of which must be solved to solve the
  original problem, then these nodes are called AND
  nodes
• 6. Typically, the graphs that are grown
  alternate an OR ply with an AND ply, but if only
  a single OR node exists, it may be omitted in the
  constructed tree

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Sample And/Or Tree
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
When generating a solution, you only need to
expand as much of this tree (or graph) as is
needed to have a solution. This is known as the
solution tree (or graph).
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• Developing a solution to a problem involves
  determining if the root node has a solution
• For any node in the tree, the node has a solution
  if

• 1.
• 2.
• 3.

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• Developing a solution to a problem involves
  determining if the root node has a solution
• For any node in the tree, the node has no
  solution if

• 1.
• 2.
• 3.

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Given the following And/Or Tree
6
12
1
2
3
4
5
7
8
9
10
11

• What are possible combinations of leaf nodes
  that will generate a solution to the original
  problem?

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Given the following And/Or Tree
6
12
1
2
3
4
5
7
8
9
10
11

• What combinations of leaf nodes will cause the
  original problem to not have a solution?

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NOTE!

• The state-space and problem-reduction
  representations are equivalent and algorithms
  exist for recasting a problem description from
  one representation into the other
• However, often one of these representations will
  appear more natural for representing a particular
  problem than the other

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Forward vs. Backward Reasoning

• If we think of our problem using the state-space
  representation, what we want is a path through
  the problem state from an initial state to a goal
  state
• Two approaches can be used
• Which method is better?

• Forward Reasoning --
• Backward Reasoning --
• 1.
• 2.
• 3.