L4 Problem Solving n Search p3

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241-320 Design Architecture and Engineeringfor Intelligent System

Suntorn Witosurapot

Contact Address:Phone: 074 287369 or

Email: [email protected]

November 2009


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Lecture 4:

Problem Solving and Search part 3

Informed Search


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241-320 Design Architecture &Engineering for Intelligent System

Problem Solving and Search part 3 3

Informed (Heuristic) Search

We have seen that uninformed (trial-and-error)methodsof search are capable of systematically exploring thestate-space in finding a goal state

However, uninformed search methods are veryinefficient in most cases

In the worst case, most searches take exponential time

With the aid of problem-specific knowledge, informedmethods of search are more efficient


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Overview

Heuristics Informed Search Methods

Greedy Best-First Search

A Search

Iterative Deepening A Search

Conclusion


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Heuristics

Heuristics: ()A rule of thumb, simplication, or educated guess

that reduces or limits the search for solutions

in domains that are difficult and poorly understood.

Depending on what heuristic you use, you won'tnecessarily find an optimal solution, or even asolution at all.


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Heuristics - The 8-Puzzle Example

Using the number of tiles out of placein each statewhen it is compared with the goal

Therefore, h(n) = 5 ( 5)


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Heuristics - The 8-Puzzle Example

UsingManhattan distance(Sum of distances that tiles are out of place)

Therefore, h(n) = 1 + 2 + 0 + 1 + 1 + 0 + 1 + 0 =6

1 2 0

1 1 0

1 0


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Heuristics Example

Another common heuristic is the straight-line distance

(as the crow flies) from node to goal

Therefore, h(n) = distance from n to g


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Heuristics - Deployment

Normally, the Heuristic will be used as a function (ie.h(n) in previous examples)for supporting the decision

how the algorithm should proceed

Heuristic h(n) (Resource) n () (Goal Node)

Q: Whatre the resources that we used in theprevious slides?


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How the Heuristic function will be used?

The heuristic function, h(n), will be used in a part ofevaluation function,f(n), of an search algorithm toestimate how good each search node is Traditionally, f(N) is an estimated cost; so, the smaller f(N), the more

promising N

Typically, f(n) can be estimated from:

either the cost of a solution path through N

Then f(N) = g(N) + h(N), where

g(N) is the cost of the path from the initial node to N

h(N) is an estimate of the cost of a path from N to a goal node

or the cost of a path from N to a goal node

Then f(N) = h(N) => Greedy best-search


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How the Heuristic function will be used?

The heuristic function, h(n), will be used in a part ofevaluation function,f(n), of an search algorithm toestimate how good each search node is Traditionally, f(N) is an estimated cost; so, the smaller f(N), the more

promising N Typically, f(n) can be estimated from:

either the cost of a solution path through N

Then f(N) = g(N) + h(N), where

g(N) is the cost of the path from the initial node to N

h(N) is an estimate of the cost of a path from N to a goal node

or the cost of a path from N to a goal node

Then f(N) = h(N) Heuristic function


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So, in our previous examples

h1(N) = number of misplaced numbered tiles = 5

h2(N) = sum of the (Manhattan) distance of

every numbered tile to its goal position= 1 + 2 + 0 + 1 + 1 + 0 + 1 + 0 = 6

1 4

7

2

5

6

38

STATE(N)

6

4

7

1

5

2

8

3

Goal state


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8-Puzzle

4

5

5

3

3

4

3 4

4

2 1

2

0

3

4

3

f(N) = h(N) = number of misplaced numbered tiles

The white tile is the empty tile


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0+4

1+5

1+5

1+3

3+3

3+4

3+4

3+2 4+1

5+2

5+0

2+3

2+4

2+3

8-Puzzle

f(N) = g(N) + h(N)with h(N) = number of misplaced numbered tiles


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Overview

Heuristics

Informed Search Methods

Greedy Best-First Search A Search

Iterative Deepening A Search

Conclusion


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Idea:expand node with the smallest estimated costto reach the goal

Evaluation function f(n) = h(n)

() ()


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Ex: Greedy Search

of the 8-Puzzle


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241-320 Design Architecture &Engineering for Intelligent System Problem Solving and Search part 3 18

Overview

Heuristics



A Search Iterative Deepening A Search

Conclusion


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A Search(most popular algorithm in AI)

Idea:avoid expanding paths that are already expensive Evaluation function f(n) = g(n) + h(n)

g(n) = cost so far to reach n

h(n) = estimated cost to goal from n

f(n) estimated total cost of path through n to goal

Expand node from frontier with smallest f-value

Essentially combines uniform-cost search and greedy search


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Ex: A Search of

the 8-Puzzle


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Robot Navigation


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Robot Navigation

0 211

58 7

7

3

4

7

6

7

6 3 2

8

6

45

23 3

36 5 24 43 5

54 6

5

6

4

5

f(N) = h(N), with h(N) = Manhattan distance to the goal(not A*)


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Robot Navigation

0 211

58 7

7

3

4

7

6

7

6 3 2

8

6

45

23 3

36 5 24 43 5

54 6

5

6

4

5

f(N) = h(N), with h(N) = Manhattan distance to the goal(not A*)

7

0


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Robot Navigation

f(N) = g(N)+h(N), with h(N) = Manhattan distance to goal(A*)

0 211

58 7

7

3

4

7

6

7

6 3 2

8

6

45

23 3

36 5 24 43 5

54 6

5

6

4

57+0

6+1

6+1

8+1

7+0

7+2

6+1

7+2

6+1

8+1

7+2

8+3

7+2 6+36+3 5+45+4 4+54+5 3+63+6 2+7

8+3 7+47+4 6+5

5+6

6+3 5+6

2+7 3+8

4+7

5+6 4+7

3+8

4+7 3+83+8 2+92+9 3+10

2+9

3+8

2+9 1+101+10 0+110+11


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Overview

Heuristics



A Search

Iterative Deepening A Search Conclusion


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Iterative Deepening A* (IDA*)

Idea: Reduce memory requirement of A* by applyingcutoff on values of f

Consistent heuristic function h

Algorithm IDA*:

1. Initialize cutofftof(initial-node)

2. Repeat:

a. Perform depth-first search by expanding all nodesN

such thatf(N)cutoff

b. Reset cutoff to smallest valuefof non-expanded(leaf) nodes


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8-Puzzle

4

6

f(N) = g(N) + h(N)

with h(N) = number of misplaced tiles

Cutoff=4


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8-Puzzle

4

4

6

Cutoff=4

6

f(N) = g(N) + h(N)



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8-Puzzle

4

4

6

Cutoff=4

6

5

f(N) = g(N) + h(N)



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8-Puzzle

4

4

6

Cutoff=4

6

5

5

f(N) = g(N) + h(N)



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4

8-Puzzle

4

6

Cutoff=4

6

5

56

f(N) = g(N) + h(N)



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8-Puzzle

4

6

Cutoff=5

f(N) = g(N) + h(N)



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8-Puzzle

4

4

6

Cutoff=5

6

f(N) = g(N) + h(N)



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8-Puzzle

4

4

6

Cutoff=5

6

5

f(N) = g(N) + h(N)



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8-Puzzle

4

4

6

Cutoff=5

6

5

7

f(N) = g(N) + h(N)



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8-Puzzle

4

4

6

Cutoff=5

6

5

7

5

f(N) = g(N) + h(N)



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8-Puzzle

4

4

6

Cutoff=5

6

5

7

5 5

f(N) = g(N) + h(N)



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8-Puzzle

4

4

6

Cutoff=5

6

5

7

5 5

f(N) = g(N) + h(N)


5


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Advantages/Drawbacks of IDA*

Advantages:

Still complete and optimal

Requires less memory than A*

Avoid the overhead to sort the fringe

Drawbacks:

Cant avoid revisiting states not on the current path

Available memory is poorly used


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241 320 D i A hit t & P bl S l i d S h t 3 40

Conclusion

Informed search makes use of problem-specificknowledge to guide progress of search

This can lead to a significant improvement in theperformance of search

If our problem can be derived as a graph or tree,heuristic search (e.g. Greedy best-first and A* (andvariants of A*)) can be applied

Note: the other technique (known as Local search) is availablefor some problems where the path is irrelevant (e.g., 8-queen

puzzle) & hence the evaluation function is only performed onthecurrent state.

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L4 Problem Solving n Search p3