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Heuristic Optimisation
Part 6: A* search
Sandor Zoltan Nemeth
http://web.mat.bham.ac.uk/S.Z.Nemeth
s.nemeth@bham.ac.uk
University of Birmingham
S Z Nemeth (s.nemeth@bham.ac.uk) 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
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 2 / 17
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.
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 3 / 17
Best-first vs hill climbing
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 4 / 17
Best-first vs hill climbing
(5,2)(4,1)
(5,1) 7
6 6
Hill climbing
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 4 / 17
Best-first vs hill climbing
(5,2)
(5,1) 7
6
Hill climbing
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 4 / 17
Best-first vs hill climbing
(4,2)
(5,2)
(5,1) 7
6
5
Hill climbing
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 4 / 17
Best-first vs hill climbing
(4,3)(4,1)
(4,2)
(5,2)
(5,1) 7
6
5
6 4
Hill climbing
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 4 / 17
Best-first vs hill climbing
(4,3)
(4,2)
(5,2)
(5,1) 7
6
5
4
Hill climbing
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 4 / 17
Best-first vs hill climbing
(4,4)(3,3)
(4,3)
(4,2)
(5,2)
(5,1) 7
6
5
4
3 3
Hill climbing
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 4 / 17
Best-first vs hill climbing
(4,4)
(4,3)
(4,2)
(5,2)
(5,1) 7
6
5
4
3
Hill climbing
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 4 / 17
Best-first vs 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
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 4 / 17
Best-first vs hill climbing
(4,4)
(4,3)
(4,2)
(5,2)
(5,1) 7
6
5
4
3
(3,4)2
Hill climbing
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 4 / 17
Best-first vs 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
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 4 / 17
Best-first vs 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
(3,5)
(5,4)(3,4)
(3,3)
2 4
3 3
Greedy
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 4 / 17
Best-first vs 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
(1,4)
(1,5)
(2,5)
(3,5)
(5,4)(3,4)
(3,3)
2 4
3 3
2
1
Greedy
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 4 / 17
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.
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 5 / 17
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?)
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 5 / 17
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
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 6 / 17
Example 1 – A* solution
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
P
CLOSED=[]
f(P)=0+7=7
OPEN=[P]
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
Example 1 – A* solution
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
2 3
NHG
P
BaS
f(NHG)=2+7=9 f(BaS)=3+5=8
OPEN=[BaS,NHG]
CLOSED=[P]
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
Example 1 – A* solution
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
2 3
NHG
P
BaS
f(NHG)=2+7=9 f(BaS)=3+5=8
OPEN=[BaS,NHG]
CLOSED=[P]
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
Example 1 – A* solution
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
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
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
Example 1 – A* solution
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
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
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
Example 1 – A* solution
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
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]
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
Example 1 – A* solution
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
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]
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
Example 1 – A* solution
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
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]
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
Example 1 – A* solution
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
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
Example 1 – A* solution
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
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]
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
Example 1 – A* solution
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
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]
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
Example 1 – A* solution
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
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]
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
Example 1 – A* solution
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
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]
CLOSED=[P,BaS,NHG,BoS,WaS]
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
Example 1 – A* solution
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
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]
CLOSED=[P,BaS,NHG,BoS,WaS]
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
Example 1 – A* solution
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
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)
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
Example 1 – A* solution
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
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]
CLOSED=[P,BaS,NHG,BoS,WaS,OC]
1
f(TCR)=6+3=9
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
Example 1 – A* solution
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
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]
CLOSED=[P,BaS,NHG,BoS,WaS,OC]
1
f(TCR)=6+3=9
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
Example 1 – A* solution
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
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
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
Example 1 – A* solution
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
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
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
Example 1 – A* solution
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
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
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 7 / 17
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
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 8 / 17
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
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 8 / 17
Example 2 – A* solution
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 9 / 17
Algorithm of A*
Let OPEN be the list of generated, but not yet examined nodes.
1. Start with OPEN containing the initial state.
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 10 / 17
Algorithm of A*
Let OPEN be the list of generated, but not yet examined nodes.
1. Start with OPEN containing the initial state.
2. Do until a goal state is found or no nodes are left in OPEN:
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 10 / 17
Algorithm of A*
Let OPEN be the list of generated, but not yet examined nodes.
1. Start with OPEN containing the initial state.
2. Do until a goal state is found or no nodes are left in OPEN:
(a) Take the best node(lowest f value) in OPEN and generate its
successors.
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 10 / 17
Algorithm of A*
Let OPEN be the list of generated, but not yet examined nodes.
1. Start with OPEN containing the initial state.
2. Do until a goal state is found or no nodes are left in OPEN:
(a) Take the best node(lowest f value) in OPEN and generate its
successors.
(b) For each successor do:
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 10 / 17
Algorithm of A*
Let OPEN be the list of generated, but not yet examined nodes.
1. Start with OPEN containing the initial state.
2. Do until a goal state is found or no nodes are left in OPEN:
(a) Take the best node(lowest f value) in OPEN and generate its
successors.
(b) For each successor do:
i. if it has not been generated before, evaluate it and add it to OPEN.
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 10 / 17
Algorithm of A*
Let OPEN be the list of generated, but not yet examined nodes.
1. Start with OPEN containing the initial state.
2. Do until a goal state is found or no nodes are left in OPEN:
(a) Take the best node(lowest f value) in OPEN and generate its
successors.
(b) For each successor do:
i. if it has not been generated before, evaluate it and add it to OPEN.
What does evaluation mean?
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 10 / 17
Algorithm of A*
Let OPEN be the list of generated, but not yet examined nodes.
1. Start with OPEN containing the initial state.
2. Do until a goal state is found or no nodes are left in OPEN:
(a) Take the best node(lowest f value) in OPEN and generate its
successors.
(b) For each successor do:
i. if it has not been generated before, evaluate it and add it to OPEN.
ii. otherwise change the parent, if this path is better and modify accordingly
the costs for the depending nodes.
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 10 / 17
Algorithm of A*
Let OPEN be the list of generated, but not yet examined nodes.
1. Start with OPEN containing the initial state.
2. Do until a goal state is found or no nodes are left in OPEN:
(a) Take the best node(lowest f value) in OPEN and generate its
successors.
(b) For each successor do:
i. if it has not been generated before, evaluate it and add it to OPEN.
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?
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 10 / 17
How to perform step 2.(b)ii.
Let CLOSED be the list of generated and examined nodes.
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 11 / 17
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.
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 11 / 17
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.
A node that has a copy in the CLOSED list has already beenexamined.
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 11 / 17
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.
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 12 / 17
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.
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 12 / 17
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.
Each branch ends if there are no more successors or the present gvalue of a node is better.
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 12 / 17
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.
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 13 / 17
Particular cases of A*
g = 0 ⇒ greedy best-first search
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 14 / 17
Particular cases of A*
g = 0 ⇒ greedy best-first search
h = 0 and g = 0 ⇒ random search
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 14 / 17
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.
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 15 / 17
Search tree or search graph?
More general form ⇒ search graph
More simple form ⇒ search tree
Search trees generate nodes faster, BUT duplicate nodes.
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 16 / 17
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?
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 16 / 17
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
S Z Nemeth (s.nemeth@bham.ac.uk) Heuristic Optimisation University of Birmingham 17 / 17
Recommended