Heuristic Optimisation - University of Birminghamweb.mat.bham.ac.uk/S.Z.Nemeth/presentations_ho_part_06.pdf · 2016-01-29 · Best-ﬁrst search We follow only one path at a time,

Heuristic Optimisation

Part 6: A* search

Sandor Zoltan Nemeth

http://web.mat.bham.ac.uk/S.Z.Nemeth

[email protected]

University of Birmingham

S Z Nemeth ([email protected]) Heuristic Optimisation University of Birmingham 1 / 17

Overview

Best-first search

Minimizing the total cost ⇒ A*

Two examples of A*

The algorithm of A*

Observations


Best-first search

We follow only one path at a time, but when a competing pathlooks more promising, we switch to that path.

We apply a heuristic function to each of the generated nodes.


Best-first vs hill climbing



(5,2)(4,1)

(5,1) 7

6 6

Hill climbing



(5,2)

(5,1) 7

6

Hill climbing



(4,2)

(5,2)

(5,1) 7

6

5

Hill climbing



(4,3)(4,1)

(4,2)

(5,2)

(5,1) 7

6

5

6 4

Hill climbing



(4,3)

(4,2)

(5,2)

(5,1) 7

6

5

4

Hill climbing



(4,4)(3,3)

(4,3)

(4,2)

(5,2)

(5,1) 7

6

5

4

3 3

Hill climbing



(4,4)

(4,3)

(4,2)

(5,2)

(5,1) 7

6

5

4

3

Hill climbing



(4,4)

(4,3)

(4,2)

(5,2)

(5,1) 7

6

5

4

3

(5,4)(3,4)2 4

Hill climbing



(4,4)

(4,3)

(4,2)

(5,2)

(5,1) 7

6

5

4

3

(3,4)2

Hill climbing



(4,4)

(4,3)

(4,2)

(5,2)

(5,1) 7

6

5

4

3

(3,4)2

Hill climbing

(4,4)(3,3)

(4,3)(4,1)

(4,2)

(5,2)(4,1)

(5,1) 7

6 6

5

6 4

3 3

(5,4)(3,4)2 4

Greedy



(4,4)

(4,3)

(4,2)

(5,2)

(5,1) 7

6

5

4

3

(3,4)2

Hill climbing

(4,4)(3,3)

(4,3)(4,1)

(4,2)

(5,2)(4,1)

(5,1) 7

6 6

5

6 4

3 3

(3,5)

(5,4)(3,4)

(3,3)

2 4

3 3

Greedy



(4,4)

(4,3)

(4,2)

(5,2)

(5,1) 7

6

5

4

3

(3,4)2

Hill climbing

(4,4)(3,3)

(4,3)(4,1)

(4,2)

(5,2)(4,1)

(5,1) 7

6 6

5

6 4

3 3

(1,4)

(1,5)

(2,5)

(3,5)

(5,4)(3,4)

(3,3)

2 4

3 3

2

1

Greedy


The underlying heuristic function

f(n) = g(n) + h(n)

g(n) is a measure of the cost of getting from the initial state to thecurrent state n.

h(n) is an estimate of the cost of getting from the current state n tothe goal state.


The underlying heuristic function

f(n) = g(n) + h(n)

g(n) is a measure of the cost of getting from the initial state to thecurrent state n.

h(n) is an estimate of the cost of getting from the current state n tothe goal state.

What is a good value of h?(large or low?)


Example 1

Underground routes in London (fragment of the map)

Paddington

Notting HillGate

Victoria Westminster Embankment

OxfordCircusGreen

Park

BondStreet

Baker Street Warren Street

TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

The estimated distances from Monument (h):

P BaS WaS NHG BoS OC TCR GP V We E M

7 5 4 7 5 5 3 5 5 4 3 0


Example 1 – A* solution

Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

P

CLOSED=[]

f(P)=0+7=7

OPEN=[P]



Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

2 3

NHG

P

BaS

f(NHG)=2+7=9 f(BaS)=3+5=8

OPEN=[BaS,NHG]

CLOSED=[P]



Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

2 3

NHG

P

BaS

f(NHG)=2+7=9 f(BaS)=3+5=8

OPEN=[BaS,NHG]

CLOSED=[P]



Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

2 3

12

2

NHG

P

BaS

OCBoS WaS

f(NHG)=9

OPEN=[NHG, BoS,WaS,OC]

CLOSED=[P,BaS]

f(BoS)=4+5=9 f(OC)=5+5=10 f(WaS)=5+4=9



Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

2 3

12

2

NHG

P

BaS

OCBoS WaS

f(NHG)=9

OPEN=[NHG, BoS,WaS,OC]

CLOSED=[P,BaS]

f(BoS)=4+5=9 f(OC)=5+5=10 f(WaS)=5+4=9



Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

2

5

3

12

2

V

NHG

P

BaS

OCBoS WaS

f(BoS)=9 f(OC)=10 f(WaS)=9

f(V)=7+5=12 g’(BoS)=6>4=g(BoS)

4

OPEN=[BoS,WaS,OC,V]

CLOSED=[P,BaS,NHG]



Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

2

5

3

12

2

V

NHG

P

BaS

OCBoS WaS

f(BoS)=9 f(OC)=10 f(WaS)=9

f(V)=7+5=12

OPEN=[BoS,WaS,OC,V]

CLOSED=[P,BaS,NHG]



Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

2

5

3

12

2

V

NHG

P

BaS

OCBoS

f(V)=12

WaS

f(OC)=10 f(WaS)=9f(BoS)=9

OPEN=[BoS,WaS,OC,V]

CLOSED=[P,BaS,NHG]



Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1



Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

2

5

3

12

2

1V

NHG

P

BaS

OC

GP

BoS

f(V)=12

WaS

f(GP)=5+5=10

f(OC)=10 f(WaS)=9

OPEN=[WaS,OC,GP,V]

CLOSED=[P,BaS,NHG,BoS]



Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

2

5

3

12

2

1V

NHG

P

BaS

GP

BoS

f(GP)=10

f(V)=12

WaSOCf(WaS)=9f(OC)=10

CLOSED=[P,BaS,NHG,BoS]

OPEN=[WaS,OC,GP,V]



Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

2

5

3

12

2

1V

NHG

P

BaS

TCRGP

BoS

f(GP)=10

f(V)=12

WaS

f(TCR)=7+3=10

2

1OC

g’(OC)=6>5=g(OC)f(OC)=10

OPEN=[OC,GP,TCR,V]

CLOSED=[P,BaS,NHG,BoS,WaS]



Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

2

5

3

12

2

1V

NHG

P

BaS

TCRGP

BoS

f(GP)=10

f(V)=12

WaS

f(TCR)=7+3=10

2

OCf(OC)=10

OPEN=[OC,GP,TCR,V]




Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

2

5

3

12

2

1V

NHG

P

BaS

TCRGP

BoS

f(GP)=10

f(V)=12

WaS

f(TCR)=7+3=10

2

OCf(OC)=10

OPEN=[OC,GP,TCR,V]




Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

2

5

3

12

2

1 3V

NHG

P

BaS

TCR

EGP

BoS

f(V)=12

WaS

f(E)=8+3=11 g’(TCR)=6<7=g(TCR)f(TCR)=10

211

g’(GP)=6>5=g(GP)f(GP)=10

OPEN=[TCR,GP,V,E]

CLOSED=[P,BaS,NHG,BoS,WaS,OC]

OC1 1

g’(BoS)=6>4=g(BoS) g’(WaS)=6>5=g(WaS)



Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

2

5

3

12

2

1 3V

NHG

P

BaS

OC

TCR

EGP

BoS

f(V)=12

WaS

f(E)=8+3=11f(GP)=10

OPEN=[TCR,GP,V,E]


1

f(TCR)=6+3=9



Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

2

5

3

12

2

1 3V

NHG

P

BaS

OC

TCR

EGP

BoS

f(V)=12

WaS

f(E)=8+3=11f(GP)=10

OPEN=[TCR,GP,V,E]


1

f(TCR)=6+3=9



Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

2

5

3

12

2

1 3 1V

NHG

P

BaS

OC

TCR

M

EGP

BoS

f(GP)=10

f(V)=12

f(E)=11

f(M)=10

WaS

3

g’(E)=9>8=g(E)

OPEN=[M,GP,V,E]

CLOSED=[P,BaS,NHG,BoS,WaS,OC,TCR]

4



Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

2

5

3

12

2

1 3 1V

NHG

P

BaS

OC

TCR

M

EGP

BoS

f(GP)=10

f(V)=12

f(E)=11

f(M)=10

WaS

OPEN=[M,GP,V,E]

CLOSED=[P,BaS,NHG,BoS,WaS,OC,TCR]

4



Paddington

Notting HillGate


OxfordCircusGreen

Park

BondStreet


TottenhamCourt Road

Monument

2

3

5

4

1

1

2

1

1

1

2

2

1

1

2

3

15

3

4

1

2

5

3

12

2

1 3 1V

NHG

P

BaS

OC

TCR

M

EGP

BoS

f(GP)=10

f(V)=12

f(E)=11

f(M)=10

WaS

4


Example 2

The 8-puzzle:

2 3 1 2 3

1 4 5 8 4

8 7 6 7 6 5

Start state Goal state

Possible estimated costs:

The number of tiles in wrong position:h(start) = 6


Example 2

The 8-puzzle:

2 3 1 2 3

1 4 5 8 4

8 7 6 7 6 5

Start state Goal state

Possible estimated costs:

The number of tiles in wrong position:h(start) = 6

The Manhattan distance:h(start) = 1 + 0 + 0 + 1 + 1 + 1 + 1 + 1 = 6




Algorithm of A*

Let OPEN be the list of generated, but not yet examined nodes.

1. Start with OPEN containing the initial state.


Algorithm of A*



2. Do until a goal state is found or no nodes are left in OPEN:


Algorithm of A*




(a) Take the best node(lowest f value) in OPEN and generate its

successors.


Algorithm of A*





successors.

(b) For each successor do:


Algorithm of A*





successors.


i. if it has not been generated before, evaluate it and add it to OPEN.


Algorithm of A*





successors.



What does evaluation mean?


Algorithm of A*





successors.



ii. otherwise change the parent, if this path is better and modify accordingly

the costs for the depending nodes.


Algorithm of A*





successors.



ii. otherwise change the parent, if this path is better and modify accordingly

the costs for the depending nodes.

When is a path better then the other?


How to perform step 2.(b)ii.

Let CLOSED be the list of generated and examined nodes.




For a node that has a copy in the OPEN list we only have to modifythe parent and the g, f values.




For a node that has a copy in the OPEN list we only have to modifythe parent and the g, f values.

A node that has a copy in the CLOSED list has already beenexamined.


How to perform step 2.(b)ii. – 2

So, we need to modify the g values (and the f values, too) of thenodes below this node.




We can do a depth-first traversal of the tree rooted at this node.




We can do a depth-first traversal of the tree rooted at this node.

Each branch ends if there are no more successors or the present gvalue of a node is better.


Observations

f(n) is the shortest estimated path from the start state to the goalstate that passes through state n.

If the cost of each step is 1, A* will find the path consisting of thefewest number of steps.


Particular cases of A*

g = 0 ⇒ greedy best-first search


Particular cases of A*

g = 0 ⇒ greedy best-first search

h = 0 and g = 0 ⇒ random search


Properties of h

If h is a perfect estimator of the distance from the current nodeto the goal node, A* will never leave the optimal path.

The better h estimates the real distance, the closer A* is to the”direct” path.

If h never overestimates the real distance, A* is guaranteed tofind the optimal solution.

If h may overestimate the real distance, the optimal path can befound only if all paths in the search graph longer than theoptimal solution are expanded.


Search tree or search graph?

More general form ⇒ search graph

More simple form ⇒ search tree

Search trees generate nodes faster, BUT duplicate nodes.


Search tree or search graph?

More general form ⇒ search graph

More simple form ⇒ search tree

Search trees generate nodes faster, BUT duplicate nodes.

Is node duplication a problem?


Recommended reading

S. Russell and P. Norvig: Artificial Intelligence, A Modern Approach

Section 4.1

E. Rich and K. Knight: Artificial Intelligence

Section 3.3


Documents

Heuristic Optimisation - University of Birminghamweb.mat.bham.ac.uk/S.Z.Nemeth/presentations_ho_part_06.pdf · 2016-01-29 · Best-ﬁrst search We follow only one path at a time,