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More advanced aspects of search

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More advanced aspects of search

Extensions of A*

Extensions of A*

Iterated deepening A*

Simplified Memory-bounded A*

Iterative-deepening A*

f1

A*

d = 1

Breadth-

first

f2

d = 2

d = 3

f3

d = 4

f4

Expand by depth-layers

Expands by f-contours

- A* is similar to breadth-first:

- Here: 2 extensions of A* that improve memory usage.

Depth-first

in each f-

contour

- Perform depth-first search LIMITED to some f-bound.
- If goal found: ok.
- Else: increase de f-bound and restart.

f1

f2

f3

f4

How to establish the f-bounds?

- initially: f(S)

generate all successors

record the minimalf(succ)>f(S)

Continue with minimalf(succ) instead of f(S)

f-new = 120

S

f=100

A

f=120

B

f=130

C

f=120

D

f=140

G

f=125

E

f=140

F

f=125

f-limited, f-bound = 100

f-new = 125

S

f=100

A

f=120

B

f=130

C

f=120

D

f=140

G

f=125

E

f=140

F

f=125

f-limited, f-bound = 120

S

f=100

A

f=120

B

f=130

C

f=120

D

f=140

G

f=125

E

f=140

F

f=125

SUCCESS

f-limited, f-bound = 125

1. QUEUE <-- path only containing the root;

f-bound <-- <some natural number>;

f-new <--

2. WHILEQUEUE is not empty AND goal is not reached

DO remove the first path from the QUEUE;

create new paths (to all children);

reject the new paths with loops;

add the new paths with f(path)f-bound

to front of QUEUE;

f-new <-- minimum of current f-new and

of the minimum of new f-values which are

larger than f-bound

3. IF goal reached THEN success; ELSE report f-new ;

1. f-bound <-- f(S)

2. WHILE goal is not reached

DO perform f-limited search;

f-bound <-- f-new

- Complete and optimal:
- under the same conditions as for A*

- Memory:
- Let be the minimal cost of an arc:

- == O( b* (cost(B) /))
- Speed:
- depends very strongly on the number of f-contours there are !!
- In the worst case: f(p) f(q) for every 2 paths:
- 1 + 2 + ….+ N = O(N2)

- In absence of Monotonicity:
- we can have search spaces like:

100

S

120

120

A

B

C

D

E

F

140

150

90

60

- If f can decrease,
- how can we be sure that the first goal reached is the optimal one ???

- IDA* is one of the very best optimal search techniques !
- Example: the 8-puzzle
- But: also for MANY other practical problems

- increase f-bound by a fixed number at each iteration:
- effects: less re-computations, BUT: optimality is lost: obtained solution can deviate up to

- Ifthere are only a reduced number of different contours:

- Else, the gain of the extended f-contour is not sufficient to compensate recalculating the previous
- In such cases:

Simplified Memory-bounded A*

- If memory is full and we need to generate an extra node (C):
- Remove thehighest f-value leaffrom QUEUE(A).
- Remember thef-value of the best ‘forgotten’ childin each parent node(15 in S).

13

S

13

15

A

B

B

18

C

memory of 3 nodes only

- Fairly complex algorithm.

- Optimizes A* to work within reduced memory.
- Key idea:

(15)

- When expanding a node (S), only add its children 1 at a time to QUEUE.
- we use left-to-right

- Avoids memory overflow and allows monitoring of whether we need to delete another node

13

S

A

A

B

B

First add A, later B

- If extending a node would produce a path longer than memory: give up on this path (C).
- Set the f-value of the node (C) to
- (to remember that we can’t find a path here)

13

S

13

B

B

18

C

C

D

memory of 3 nodes only

- If all children M of a node N have been explored and for all M:
- f(S...M)f(S...N)

- then reset:
- f(S…N)= min { f(S…M) | M child of N}
- A path through N needs to go through 1 of its children !

15

13

S

24

15

A

B

better estimate for f(S)

0+12=12

10

S

8

8+5=13

10+5=15

A

B

8

16

10

10

20+0=20

16+2=18

20+5=25

24+0=24

C

G1

D

G2

8

10

10

8

30+5=35

24+5=29

E

G3

30+0=30

G4

F

24+0=24

12

12

12

13

S

S

S

S

A

A

A

B

A

B

13

13

15

15

15

15

D

18

12

13

(15)

0+12=12

10

S

8

8+5=13

B

10+5=15

A

B

13

8

16

10

10

20+0=20

16+2=18

20+5=25

24+0=24

C

G1

D

G2

8

10

10

8

D

24+0=24

30+5=35

24+5=29

E

G3

30+0=30

G4

F

(15)

13

(15)

15

15

15 (24)

13 (15)

S

S

S

S

S

13

B

B

B

B

A

A

A

()

15

24

24

24

15

15

G2

G2

G2

C

C

C

G1

D

D

24

24

24

25

20

Example: continued

15

(15)

(24)

20

13

15

()

13

20

()

()

24

15

- Complete: If available memory allows to store the shortest path.
- Optimal: If available memory allows to store the best path.
- Otherwise: returns the best path that fits in memory.

- Memory: Uses whatever memory available.
- Speed: If enough memory to store entire tree: same as A*