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CS621 : Artificial Intelligence. Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture 4: Search. Graph Search. S. 1. 10. 5. 3. A. B. C. 10. 9. 4. 5. 6. 3. I. H. D. E. 2. 6. 3. 2. 7. J. F. G(oal). 8-puzzle problem. 1. 2. 3. 4. 3. 6. 1. 4. 2. 8. 5. 6.

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Cs621 artificial intelligence

CS621 : Artificial Intelligence

Pushpak BhattacharyyaCSE Dept., IIT Bombay

Lecture 4: Search


Graph search
Graph Search

S

1

10

5

3

A

B

C

10

9

4

5

6

3

I

H

D

E

2

6

3

2

7

J

F

G(oal)


8-puzzle problem

1

2

3

4

3

6

1

4

2

8

5

6

5

8

7

7

S0

G

Tile movement represented as the movement of the blank space.

Operators:

L : Blank moves left

R : Blank moves right

U : Blank moves up

D : Blank moves down

C(L) = C(R) = C(U) = C(D) = 1


8 puzzle heuristics
8-puzzle: heuristics

Example: 8 puzzle

s

n

g

h*(n) = actual no. of moves to transform n to g

h1(n) = no. of tiles displaced from their destined position.

h2(n) = sum of Manhattan distances of tiles from their destined position.

h1(n) ≤ h*(n) and h1(n) ≤ h*(n)

h*

h2

h1

Comparison



Theorem

A version A2* of A* that has a “better” heuristic than another version A1* of A* performs at least “as well as” A1*

Meaning of “better”

h2(n) > h1(n) for all n

Meaning of “as well as”

A1* expands at least all the nodes of A2*

h*(n)

h2*(n)

h1*(n)

For all nodes n, except the goal node


Proof by induction on the search tree of A2*.

A* on termination carves out a tree out of G

Induction

on the depth k of the search tree of A2*. A1* before termination expands all the nodes of depth k in the search tree of A2*.

k=0. True since start node S is expanded by both

Suppose A1* terminates without expanding a node n at depth (k+1) of A2* search tree.

Since A1* has seen all the parents of n seen by A2*

g1(n) <= g2(n) (1)


Since A1* has terminated without expanding n,

f1(n) >= f*(S) (2)

Any node whose f value is strictly less than f*(S) has to be expanded.

Since A2* has expanded n

f2(n) <= f*(S) (3)

S

k+1

G

From (1), (2), and (3)

h1(n) >= h2(n) which is a contradiction. Therefore, A1* has to expand all nodes that A2* has expanded.

Exercise

If better means h2(n) > h1(n) for some n and h2(n) = h1(n) for others, then Can you prove the result ?


Monotone Restriction

  • Very reasonable assumption

  • That is the estimated cost of reaching the goal node for a particular node is no more than the cost of reaching a child and the estimated cost of reaching the goal from the child

  • This obviates the need for redirecting parent pointer for a less costly path in the closed list

  • This means that for any node taken up for expansion, the optimal path is already found

n1

C(n1,n2)

n2

h(n1)

h(n2)

g



Beam search
Beam Search

  • Run A*, but with a “Beam width K”

  • That is, for each iteration keep K most promising nodes and throw away other nodes

  • Incomplete and non-optimal, but fast


Genetic algorithm based
Genetic Algorithm Based

  • Use the operators of mutation and cross over

  • Mutation: flip a bit randomly to produce a new state

  • Cross over: produce by crossing portion of two states a new state


Ga for 8 puzzle

Blank 0000

1 0001

2 0010

3 0011

4 0100

5 0101

6 0110

7 0111

8 1000

GA for 8-puzzle

4

3

6

1

2

8

5

7

Represented as

0100 0011 0110 0010 0001 1000 0111 0000 0101

1

6

2

3

4

7

8

5

9


Mutation and crossover
Mutation and Crossover

Mutation: flip any bit randomly

A

0100 0011 0110 0010 0001 1000 0111 0000 0101

B

0100 0111 0110 0010 0001 1000 0111 0000 0101

Crossover: cross a set of bit

1000 0111 0000 0101 0001 1000 0111 0000 0101

First 16 bits of

A crossed With

last 16 bits

of B

0100 0111 0110 0010 0001 0100 0011 0110 0010


Care to be exercised
Care to be exercised

  • The bit strings are called chromosomes

  • Mutation and crossover produce bit strings which are state descriptions

  • But care has to be exercised to see that the produced state is

    • Legal

    • Reachable


Lab assignment
Lab assignment

  • Implement A* algorithm for the following problems:

    • 8 puzzle

  • Specifications:

    • Try different heuristics and compare with baseline case, i.e., the breadth first search.

    • Violate the condition h ≤ h*. See if the optimal path is still found. Observe the speedup.

  • Now implement for the same problem

    • Beam Search

    • Genetic Algorithm


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