AI heuristic search

Renas R. Rekany Artificial Intelligence Nawroz University
Keep Reading as long as you breathComSci: Renas R. Rekany Nov2016
Problem Solving by Searching
Search Methods :
informed (Heuristic) search

2
Traditional informed search
strategies
 Greedy Best first search
 “Always chooses the successor node with the best f value”
where f(n) = h(n)
 We choose the one that is nearest to the final state among
all possible choices
 A* search
 Best first search using an “admissible” heuristic function f
that takes into account the current cost g
 Always returns the optimal solution path

Informed Search Strategies
 Best First Search
Greedy Search
eval-fn: f(n) = h(n)

4
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

5
Greedy Search
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
A 366
B 374
C 329
D 244
E 253
F 178
G 193
H 98
I 0
140

6
Greedy Search
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
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
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
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
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
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
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
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
A 366
B 374
C 329
D 244
E 253
F 178
G 193
H 98
I 0
140

14
Greedy Search: Tree Search
A
Start

15
A
B
C
E
Start
75118
140 [374][329]
[253]

16
A
B
C
E
F
99
G
A
80
Start
75118
140 [374][329]
[253]
[193]
[366]
[178]

17
A
B
C
E
F
I
99
211
G
A
80
Start
Goal
75118
140 [374][329]
[253]
[193]
[366]
[178]
E
[0][253]

18
A
B
C
E
F
I
99
211
G
A
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

19
Greedy Search: Optimal ?
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
dist(A-E-G-H-I) =140+80+97+101=418
A 366
B 374
C 329
D 244
E 253
F 178
G 193
H 98
I 0
140

20
Greedy Search: Complete ?
A
B
D
C
E
F
I
99
211
G
H
80
Start
Goal
97
101
75118
111
A 366
B 374
** C 250
D 244
E 253
F 178
G 193
H 98
I 0
140

21
A
Start

22
A
B
C
E
Start
75118
140 [374][250]
[253]

23
A
B
C
E
D
111
Start
75118
140 [374][250]
[253]
[244]

24
A
B
C
E
D
111
Start
75118
140 [374][250]
[253]
[244]
C[250]
Infinite Branch !

25
A
B
C
E
D
111
Start
75118
140 [374][250]
[253]
[244]
C
D
[250]
[244]
Infinite Branch !

26
A
B
C
E
D
111
Start
75118
140 [374][250]
[253]
[244]
C
D
[250]
[244]
Infinite Branch !

27
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

 A* Search
 eval-fn: f(n)=g(n)+h(n)

29
 Greedy Search minimizes a heuristic h(n) which is an
estimated cost from a node n to the goal state. However,
although greedy search can considerably cut the search
time (efficient), it is neither optimal nor complete.
 Uniform Cost Search minimizes the cost g(n) from the
initial state to n. UCS is optimal and complete but not
efficient.
 New Strategy: Combine Greedy Search and UCS to get an
efficient algorithm which is complete and optimal.
A* Search

30
 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.
A* Search

31
A* (A Star)
n
g(n)
h(n)
f(n) = g(n)+h(n)

32
A* Search
f(n) = g(n) + h (n)
g(n): is the exact cost to reach node n from the initial state.
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

33
A* Search: Tree Search
A Start

34
A
BC E
Start
75118
140
[393] [449]
[447]

35
A
BC E
F
99
G
80
Start
75118
140
[393] [449]
[447]
[417][413]

36
A
BC E
F
99
G
80
Start
75118
140
[393] [449]
[447]
[417][413]
H
97
[415]

37
A
BC E
F
I
99
G
H
80
Start
97
101
75118
140
[393] [449]
[447]
[417][413]
[415]
Goal [418]

38
A
BC E
F
I
99
G
H
80
Start
97
101
75118
140
[393] [449]
[447]
[417][413]
[415]
Goal [418]
I [450]

39
A
BC E
F
I
99
G
H
80
Start
97
101
75118
140
[393] [449]
[447]
[417][413]
[415]
Goal [418]
I [450]

40
A
BC E
F
I
99
G
H
80
Start
97
101
75118
140
[393] [449]
[447]
[417][413]
[415]
Goal [418]
I [450]

A* Search

A* Search Tree

Admissible and Consistency

Admissible
h(n) =< C(n,g)
Consistency
h(n) =< h(n’) + C(n,n’)

Admissible

Consistency
S------B

A* with h() not Admissible
 h() overestimates the cost to reach
the goal state

49
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.
A 366
B 374
C 329
D 244
E 253
F 178
G 193
H 138
I 0
140

50
A Start

51
A
BC E
Start
75118
140
[393] [449]
[447]

52
A
BC E
F
99
G
80
Start
75118
140
[393] [449]
[447]
[417][413]

53
A
BC E
F
99
G
80
Start
75118
140
[393] [449]
[447]
[417][413]
H
97
[455]

54
A
BC E
F
99
G
H
80
Start
97
75118
140
[393] [449]
[447]
[417][413]
[455] Goal I [450]

55
A
BC E
F
99
G
H
80
Start
97
75118
140
[393] [449]
[447]
[417][413]
[455] Goal I [450]
D[473]

56
A
BC E
F
99
G
H
80
Start
97
75118
140
[393] [449]
[447]
[417][413]
[455] Goal I [450]
D[473]

57
A
BC E
F
99
G
H
80
Start
97
75118
140
[393] [449]
[447]
[417][413]
[455] Goal I [450]
D[473]

58
A
BC E
F
99
G
H
80
Start
97
75118
140
[393] [449]
[447]
[417][413]
[455] Goal I [450]
D[473]
A* not optimal !!!

A* Algorithm
 A* with systematic checking for
repeated states …

60
A* Algorithm
1. 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 for
8. Go back to step 3.

61
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

63
SLD Heuristic: h()
Straight Line Distances to Bucharest
Town SLD
Arad 366
Bucharest 0
Craiova 160
Dobreta 242
Eforie 161
Fagaras 178
Giurgiu 77
Hirsova 151
Iasi 226
Lugoj 244
Town SLD
Mehadai 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.

64
Arad
Bucharest
Oradea
Zerind
Faragas
Neamt
Iasi
Vaslui
Hirsov
a
Eforie
Urziceni
Giurgui
Pitesti
Sibiu
Dobreta
Craiova
Rimnicu
Mehadi
a
Timisoara
Lugoj
87
92
142
86
98
86
211
101
90
99
151
71
75
140
118
111
70
75
120
138
146
97
80
140
80
97
101
Sibiu
Rimnicu
Pitesti
Distances Between Cities

Greedy Search in Action …
 Map Searching

66
OradeaZerind
Fagaras
Sibiu
Rimnicu
Timisoara
Arad
F= 366
F= 366
F= 374
F= 374
F= 253
F=253
F= 329
F= 329
F= 178
F= 178
F= 380
F= 380
F= 193
F= 193
Bucharest
F= 0
F= 0
Path cost(Arad-Sibiu-Fagaras-Bucharest) = 253 + 178 + 0 = 431
dist(Arad-Sibiu-Fagaras-Bucharest) = 140 + 99 + 211 = 450

A* in Action …
 Map Searching

68
OradeaZerind
Fagaras
Pitesti
Sibiu
Craiova
Rimnicu
Timisoara
Bucharest
Arad
F= 0 + 366
F= 366
F= 75 + 374
F= 449
F= 140 + 253
F= 393
F= 118 + 329
F= 447
F= 239 + 178
F= 417
F= 291 + 380
F= 671
F= 220 + 193
F= 413
F= 317 + 98
F= 415
F= 366 + 160
F= 526
F= 418 + 0
F= 418
Bucharest(2)
F= 450 + 0
F= 450
BucharestBucharestBucharest
F= 418 + 0
F= 418

 Iterative Deepening A*

70
Iterative Deepening A*:IDA*
 Use f(N) = g(N) + h(N) with admissible and
consistent h
 Each iteration is depth-first with cutoff on
the value of f of expanded nodes

71
Consistent 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’)

72
IDA* Algorithm
 In the first iteration, we determine a “f-cost limit” – cut-off value
f(n0) = g(n0) + h(n0) = h(n0), where n0 is the start node.
 We expand nodes using the depth-first algorithm and backtrack whenever
f(n) for an expanded node n exceeds the cut-off value.
 If this search does not succeed, determine the lowest f-value among the
nodes that were visited but not expanded.
 Use this f-value as the new limit value – cut-off value and do another
depth-first search.
 Repeat this procedure until a goal node is found.

73
8-Puzzle
4
6
f(N) = g(N) + h(N)
with h(N) = number of misplaced tiles
Cutoff=4

74
8-Puzzle
4
4
6
Cutoff=4
6
f(N) = g(N) + h(N)

75
8-Puzzle
4
4
6
Cutoff=4
6
5
f(N) = g(N) + h(N)

76
8-Puzzle
4
3
6
Cutoff=4
6
5
5
f(N) = g(N) + h(N)

77
4
8-Puzzle
4
6
Cutoff=4
6
5
56
f(N) = g(N) + h(N)

78
8-Puzzle
4
6
Cutoff=5
f(N) = g(N) + h(N)

79
8-Puzzle
4
4
6
Cutoff=5
6
f(N) = g(N) + h(N)

80
8-Puzzle
4
4
6
Cutoff=5
6
5
f(N) = g(N) + h(N)

81
8-Puzzle
4
4
6
Cutoff=5
6
5
7
f(N) = g(N) + h(N)

82
8-Puzzle
4
4
6
Cutoff=5
6
5
7
5
f(N) = g(N) + h(N)

83
8-Puzzle
4
4
6
Cutoff=5
6
5
7
5 5
f(N) = g(N) + h(N)

84
8-Puzzle
4
4
6
Cutoff=5
6
5
7
5 5
f(N) = g(N) + h(N)

85
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

86
Conclusions
 Frustration with uninformed search led to the idea
of using domain specific knowledge in a search so
that one can intelligently explore only the relevant
part of the search space that has a good chance of
containing the goal state. These new techniques
are called informed (heuristic) search strategies.
 Even though heuristics improve the performance
of informed search algorithms, they are still time
consuming especially for large size instances.

AI heuristic search

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