Artificial Intelligence Problem solving by searching.
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Transcript of Artificial Intelligence Problem solving by searching.
Artificial IntelligenceProblem solving by searching
Problem Solving by Searching
Search Methods : informed (Heuristic)
search
3
Using problem specific knowledge to aid searching
Without incorporating knowledge into searching, one can have no bias (i.e. a preference) on the search space.
Without a bias, one is forced to look everywhere to find the answer. Hence, the complexity of uninformed search is intractable.
Search everywhere!!
Informed Search Strategies
Best First Search
Informed Search Strategies
Greedy Searcheval-fn: f(n) = h(n)
6
Greedy Search
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
f(n) = h (n) = straight-line distance heuristic
State Heuristic: h(n)
A 366
B 374
C 329
D 244
E 253
F 178
G 193
H 98
I 0
140
7
Greedy Search
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
f(n) = h (n) = straight-line distance heuristic
State Heuristic: h(n)
A 366
B 374
C 329
D 244
E 253
F 178
G 193
H 98
I 0
140
8
Greedy Search
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
f(n) = h (n) = straight-line distance heuristic
State Heuristic: h(n)
A 366
B 374
C 329
D 244
E 253
F 178
G 193
H 98
I 0
140
9
Greedy Search
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
f(n) = h (n) = straight-line distance heuristic
State Heuristic: h(n)
A 366
B 374
C 329
D 244
E 253
F 178
G 193
H 98
I 0
140
10
Greedy Search
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
f(n) = h (n) = straight-line distance heuristic
State Heuristic: h(n)
A 366
B 374
C 329
D 244
E 253
F 178
G 193
H 98
I 0
140
11
Greedy Search
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
f(n) = h (n) = straight-line distance heuristic
State Heuristic: h(n)
A 366
B 374
C 329
D 244
E 253
F 178
G 193
H 98
I 0
140
12
Greedy Search
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
f(n) = h (n) = straight-line distance heuristic
State Heuristic: h(n)
A 366
B 374
C 329
D 244
E 253
F 178
G 193
H 98
I 0
140
13
Greedy Search
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
f(n) = h (n) = straight-line distance heuristic
State Heuristic: h(n)
A 366
B 374
C 329
D 244
E 253
F 178
G 193
H 98
I 0
140
14
Greedy Search
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
f(n) = h (n) = straight-line distance heuristic
State Heuristic: h(n)
A 366
B 374
C 329
D 244
E 253
F 178
G 193
H 98
I 0
140
15
Greedy Search
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
f(n) = h (n) = straight-line distance heuristic
State Heuristic: h(n)
A 366
B 374
C 329
D 244
E 253
F 178
G 193
H 98
I 0
140
16
Greedy Search: Tree Search
AStart
17
Greedy Search: Tree Search
A
BC
E
Start75118
140 [374][329]
[253]
18
Greedy Search: Tree Search
A
BC
E
F
99
GA
80
Start75118
140 [374][329]
[253]
[193]
[366][178]
19
Greedy Search: Tree Search
A
BC
E
F
I
99
211
GA
80
Start
Goal
75118
140 [374][329]
[253]
[193]
[366][178]
E[0][253]
20
Greedy Search: Tree Search
A
BC
E
F
I
99
211
GA
80
Start
Goal
75118
140 [374][329]
[253]
[193]
[366][178]
E[0][253]
Path cost(A-E-F-I) = 253 + 178 + 0 = 431
dist(A-E-F-I) = 140 + 99 + 211 = 450
21
Greedy Search: Optimal ?
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
f(n) = h (n) = straight-line distance heuristic
dist(A-E-G-H-I) =140+80+97+101=418
State Heuristic: h(n)
A 366
B 374
C 329
D 244
E 253
F 178
G 193
H 98
I 0
140
22
Greedy Search: Complete ?
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
f(n) = h (n) = straight-line distance heuristic
State Heuristic: h(n)
A 366
B 374
** C 250
D 244
E 253
F 178
G 193
H 98
I 0
140
23
Greedy Search: Tree Search
AStart
24
Greedy Search: Tree Search
A
BC
E
Start75118
140 [374][250]
[253]
25
Greedy Search: Tree Search
A
BC
E
D
111
Start75118
140 [374][250]
[253]
[244]
26
Greedy Search: Tree Search
A
BC
E
D
111
Start75118
140 [374][250]
[253]
[244]
C[250]Infinite Branch !
27
Greedy Search: Tree Search
A
BC
E
D
111
Start75118
140 [374][250]
[253]
[244]
C
D
[250]
[244]
Infinite Branch !
28
Greedy Search: Tree Search
A
BC
E
D
111
Start75118
140 [374][250]
[253]
[244]
C
D
[250]
[244]
Infinite Branch !
29
Greedy Search: Time and Space Complexity ?
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
140
• Greedy search is not optimal.
• Greedy search is incomplete without systematic checking of repeated states.
• In the worst case, the Time and Space Complexity of Greedy Search are both O(bm)
Where b is the branching factor and m the maximum path length
Informed Search Strategies
A* Search
eval-fn: f(n)=g(n)+h(n)
31
A* (A Star) A* uses a heuristic function which
combines g(n) and h(n): f(n) = g(n) + h(n)
g(n) is the exact cost to reach node n from the initial state. Cost so far up to node n.
h(n) is an estimation of the remaining cost to reach the goal.
32
A* (A Star)
n
g(n)
h(n)
f(n) = g(n)+h(n)
33
A* Search
f(n) = g(n) + h (n)
g(n): is the exact cost to reach node n from the initial state.
State Heuristic: h(n)
A 366
B 374
C 329
D 244
E 253
F 178
G 193
H 98
I 0
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
140
34
A* Search: Tree SearchA Start
35
A* Search: Tree SearchA
BC E
Start
75118140
[393]
[449][447
]
36
A* Search: Tree SearchA
BC E
F
99
G
80
Start
75118140
[393]
[449][447
]
[417]
[413]
37
A* Search: Tree SearchA
BC E
F
99
G
80
Start
75118140
[393]
[449][447
]
[417]
[413]
H
97
[415]
38
A* Search: Tree SearchA
BC E
F
I
99
G
H
80
Start
97
101
75118140
[393]
[449][447
]
[417]
[413]
[415]
Goal [418]
39
A* Search: Tree SearchA
BC E
F
I
99
G
H
80
Start
97
101
75118140
[393]
[449][447
]
[417]
[413]
[415]
Goal [418]
I [450]
40
A* Search: Tree SearchA
BC E
F
I
99
G
H
80
Start
97
101
75118140
[393]
[449][447
]
[417]
[413]
[415]
Goal [418]
I [450]
41
A* Search: Tree SearchA
BC E
F
I
99
G
H
80
Start
97
101
75118140
[393]
[449][447
]
[417]
[413]
[415]
Goal [418]
I [450]
A* with h() not Admissible
h() overestimates the cost to reach the goal state
43
A* Search: h not admissible !
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
f(n) = g(n) + h (n) – (H-I) Overestimated
g(n): is the exact cost to reach node n from the initial state.
State Heuristic: h(n)
A 366
B 374
C 329
D 244
E 253
F 178
G 193
H 138
I 0
140
44
A* Search: Tree SearchA Start
45
A* Search: Tree SearchA
BC E
Start
75118140
[393]
[449][447
]
46
A* Search: Tree SearchA
BC E
F
99
G
80
Start
75118140
[393]
[449][447
]
[417]
[413]
47
A* Search: Tree SearchA
BC E
F
99
G
80
Start
75118140
[393]
[449][447
]
[417]
[413]
H
97
[455]
48
A* Search: Tree SearchA
BC E
F
99
G
H
80
Start
97
75118140
[393]
[449][447
]
[417]
[413]
[455]
Goal I [450]
49
A* Search: Tree SearchA
BC E
F
99
G
H
80
Start
97
75118140
[393]
[449][447
]
[417]
[413]
[455]
Goal I [450]
D[473]
50
A* Search: Tree SearchA
BC E
F
99
G
H
80
Start
97
75118140
[393]
[449][447
]
[417]
[413]
[455]
Goal I [450]
D[473]
51
A* Search: Tree SearchA
BC E
F
99
G
H
80
Start
97
75118140
[393]
[449][447
]
[417]
[413]
[455]
Goal I [450]
D[473]
52
A* Search: Tree SearchA
BC E
F
99
G
H
80
Start
97
75118140
[393]
[449][447
]
[417]
[413]
[455]
Goal I [450]
D[473]
A* not optimal !!!
A* Algorithm
A* with systematic checking for repeated states …
54
A* Algorithm1. Search queue Q is empty.2. Place the start state s in Q with f value h(s).3. If Q is empty, return failure.4. Take node n from Q with lowest f value. (Keep Q sorted by f values and pick the first element).5. If n is a goal node, stop and return solution.6. Generate successors of node n.7. For each successor n’ of n do:
a) Compute f(n’) = g(n) + cost(n,n’) + h(n’).b) If n’ is new (never generated before), add n’ to Q. c) If node n’ is already in Q with a higher f value,
replace it with current f(n’) and place it in sorted order in Q.
End for8. Go back to step 3.
55
A* Search: Analysis
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
140
•A* is complete except if there is an infinity of nodes with f < f(G).
•A* is optimal if heuristic h is admissible.
•Time complexity depends on the quality of heuristic but is still exponential.
•For space complexity, A* keeps all nodes in memory. A* has worst case O(bd) space complexity, but an iterative deepening version is possible (IDA*).
A* Algorithm
A* with systematic checking for repeated states
An Example: Map Searching
57
SLD Heuristic: h()Straight Line Distances to Bucharest
Town SLDArad 366
Bucharest
0
Craiova 160
Dobreta 242
Eforie 161
Fagaras 178
Giurgiu 77
Hirsova 151
Iasi 226
Lugoj 244
Town SLDMehadai 241
Neamt 234
Oradea 380
Pitesti 98
Rimnicu 193
Sibiu 253
Timisoara 329
Urziceni 80
Vaslui 199
Zerind 374
We can use straight line distances as an admissible heuristic as they will never overestimate the cost to the goal. This is because there is no shorter distance between two cities than the straight line distance. Press space to continue with the slideshow.
58
Arad
Bucharest
OradeaZerind
Faragas
Neamt
Iasi
Vaslui
Hirsova
Eforie
Urziceni
Giurgui
Pitesti
Sibiu
Dobreta
Craiova
Rimnicu
Mehadia
Timisoara
Lugoj
87
92
142
86
98
86
211
101
90
99
151
71
75
140118
111
70
75
120
138
146
97
80
140
80
97
101
Sibiu
Rimnicu
Pitesti
Distances Between Cities
Greedy Search in Action …
Map Searching
60
Press space to see an Greedy search of the Romanian map featured in the previous slide. Note: Throughout the animation all nodes are labelled with f(n) = h(n). However,we will be using the abbreviations f, and h to make the notation simpler
OradeaZerind
Fagaras
Sibiu
RimnicuTimisoara
Arad
We begin with the initial state of Arad. The straight line distance from Arad to Bucharest (or h value) is 366 miles. This gives us a total value of ( f = h ) 366 miles. Press space to expand the initial state of Arad.
F= 366
F= 366
F= 374
F= 374
F= 253
F=253F= 329
F= 329
Once Arad is expanded we look for the node with the lowest cost. Sibiu has the lowest value for f. (The straight line distance from Sibiu to the goal state is 253 miles. This gives a total of 253 miles). Press space to move to this node and expand it.
We now expand Sibiu (that is, we expand the node with the lowest value of f ). Press space to continue the search.
F= 178
F= 178
F= 380
F= 380
F= 193
F= 193
We now expand Fagaras (that is, we expand the node with the lowest value of f ). Press space to continue the search.
BucharestF= 0
F= 0
A* in Action …
Map Searching
62
Press space to see an A* search of the Romanian map featured in the previous slide. Note: Throughout the animation all nodes are labelled with f(n) = g(n) + h(n). However,we will be using the abbreviations f, g and h to make the notation simpler
OradeaZerind
Fagaras
Pitesti
Sibiu
Craiova
RimnicuTimisoara
Bucharest
Arad
We begin with the initial state of Arad. The cost of reaching Arad from Arad (or g value) is 0 miles. The straight line distance from Arad to Bucharest (or h value) is 366 miles. This gives us a total value of ( f = g + h ) 366 miles. Press space to expand the initial state of Arad.
F= 0 + 366
F= 366
F= 75 + 374
F= 449
F= 140 + 253
F= 393F= 118 + 329
F= 447
Once Arad is expanded we look for the node with the lowest cost. Sibiu has the lowest value for f. (The cost to reach Sibiu from Arad is 140 miles, and the straight line distance from Sibiu to the goal state is 253 miles. This gives a total of 393 miles). Press space to move to this node and expand it.
We now expand Sibiu (that is, we expand the node with the lowest value of f ). Press space to continue the search.
F= 239 + 178
F= 417
F= 291 + 380
F= 671
F= 220 + 193
F= 413
We now expand Rimnicu (that is, we expand the node with the lowest value of f ). Press space to continue the search.
F= 317 + 98
F= 415F= 366 + 160
F= 526
Once Rimnicu is expanded we look for the node with the lowest cost. As you can see, Pitesti has the lowest value for f. (The cost to reach Pitesti from Arad is 317 miles, and the straight line distance from Pitesti to the goal state is 98 miles. This gives a total of 415 miles). Press space to move to this node and expand it.
We now expand Pitesti (that is, we expand the node with the lowest value of f ). Press space to continue the search.
We have just expanded a node (Pitesti) that revealed Bucharest, but it has a cost of 418. If there is any other lower cost node (and in this case there is one cheaper node, Fagaras, with a cost of 417) then we need to expand it in case it leads to a better solution to Bucharest than the 418 solution we have already found. Press space to continue.
F= 418 + 0
F= 418
In actual fact, the algorithm will not really recognise that we have found Bucharest. It just keeps expanding the lowest cost nodes (based on f ) until it finds a goal state AND it has the lowest value of f. So, we must now move to Fagaras and expand it. Press space to continue.
We now expand Fagaras (that is, we expand the node with the lowest value of f ). Press space to continue the search.
Bucharest(2)F= 450 + 0
F= 450
Once Fagaras is expanded we look for the lowest cost node. As you can see, we now have two Bucharest nodes. One of these nodes ( Arad – Sibiu – Rimnicu – Pitesti – Bucharest ) has an f value of 418. The other node (Arad – Sibiu – Fagaras – Bucharest(2) ) has an f value of 450. We therefore move to the first Bucharest node and expand it. Press space to continue
BucharestBucharestBucharest
We have now arrived at Bucharest. As this is the lowest cost node AND the goal state we can terminate the search. If you look back over the slides you will see that the solution returned by the A* search pattern ( Arad – Sibiu – Rimnicu – Pitesti – Bucharest ), is in fact the optimal solution. Press space to continue with the slideshow.
63
Consistent HeuristicConsistent Heuristic
The admissible heuristic h is consistent (or satisfies the monotone restriction) if for every node N and every successor N’ of N:
h(N) c(N,N’) + h(N’)
(triangular inequality) A consistent heuristic is admissible.
N
N’ h(N)
h(N’)
c(N,N’)
64
When to Use Search Techniques The search space is small, and
There are no other available techniques, or
It is not worth the effort to develop a more efficient technique
The search space is large, and There is no other available techniques,
and There exist “good” heuristics