VITALI SEPETNITSKY 22/05/2013 Research Current Status
Jan 02, 2016
VITALI SEPETNITSKY
22/05 /2013
Research Current Status
Background
Classical WA* algorithm was takenDifferent reopening policies (currently, the
radical): Always Reopen (AR) No Reopen (NR)
It sounds reasonable that any solution found by the “AR” policy it at least “good”(*) (or even better) as any solution found by the “NR” policy
(*) Measured by cost of the found path and number of expanded states
Experiments
Korf’s 100 instances of 15-puzzle were takenKorf’s example weights were takenWA* with “AR” and “NR” policies was ran in
order to solve each instance (using the weights)
In the results we can see a lot of runs in which WA* with “NR” policy outperforms WA* with “AR” policy!
This contradicts our assumption!
More detailed analysis
By running the same test on: 15-puzzle 9-puzzle 3x2-puzzle
1. The phenomenon described above can appear with any instance – there are no specific instances
2. The phenomenon appears mostly in around 4-5
3. As the weight grows, the improvement of “NR” over “AR” grows too
A toy example
Strange!
Moreover, let’s look on this graph:
Sh=2
4
Bh=2
Ch=4
Dh=3
Eh=4
Gh=0
4
4
40 5
Kh=4
4
4
61
D3h=4
1
S1h=4
S2h=4
S3h=4
S4h=4
S5h=4
6
6
6
6
6
A toy example (1)
We will show 4 different cases by simply changing the weight of WA*
Found solution cost
Lower for AR
Lower for NR
# of expand
ed states
Lower for AR
Lower for NR
Sh=2
4
Bh=2
Ch=4
Dh=3
Eh=4
Gh=0
4
4
40 5
Kh=4
4
4
61
D3h=4
1
S1h=4
S2h=4
S3h=4
S4h=4
S5h=4
6
6
6
6
6
A toy example (2): Case 1
“NR” produces a better solution cost“NR” generates and expands LESS states
Solving using “AR” : Solving using “NR” : Path found : [S,C,D,G] Path found : [S,B,K,G]Path cost : 45 Path cost : 12Generated : 28 Generated : 25Expanded : 12 Expanded : 11
See Run
A toy example (3): Case 2
“NR” produces a better solution cost“NR” generates and expands MORE states
Solving using “AR” : Solving using “NR” : Path found : [S,C,D,G] Path found : [S,B,K,G]Path cost : 45 Path cost : 12Generated : 22 Generated : 25Expanded : 6 Expanded : 11
A toy example (4): Case 3
“AR” produces a better solution cost“AR” generates and expands LESS states
Solving using “AR” : Solving using “NR” : Path found : [S,C,D,G] Path found : [S,B,D,G]Path cost : 45 Path cost : 48Generated : 22 Generated : 23Expanded : 6 Expanded : 10
A toy example (5): Case 4
“AR” produces a better solution cost“AR” generates and expands MORE states
Solving using “AR” : Solving using “NR” : Path found : [S,C,D,G] Path found : [S,B,D,G]Path cost : 45 Path cost : 48Generated : 22 Generated : 18Expanded : 6 Expanded : 5
Some Results
9-puzzle15-puzzle
(2x3-puzzle yields the same results)
Distribution - the instances set
0 10 20 30 40 50 60 70 80 90 1000
2
4
6
8
10
12
14
NR better than AR in different instances
Instance Number
Nu
mb
er
of
inst
an
ces
wit
h
NR
bett
er
than
AR
9-puzzle 15-puzzle
0 10 20 30 40 50 60 70 80 90 1000
5
10
15
20
25
NR better than AR in different instances
Instance NumberNu
mb
er
of
inst
an
ces
wit
h
NR
bett
er
than
AR
Distribution - different weights
0 5 10 15 20 25 30 35 40 45 500
2
4
6
8
10
12
NR better than AR with different weights
Wh/WgNu
mb
er
of
inst
an
ces
wit
h N
R
Bett
er
than
AR
9-puzzle 15-puzzle
0 5 10 15 20 25 30 35 40 45 500
5
10
15
20
25
30
35
NR better than AR with different weights
Wh/Wg
Nu
mb
er
of
inst
an
ces
wit
h
NR
Bett
er
than
AR
Distribution – depth improvement
9-puzzle 15-puzzle
0 5 10 15 20 25 30 35 40 45 500
5
10
15
Average difference be-tween AR depth and NR
depth with different weights
wh/wg
Ave
rag
e d
iffere
nce
betw
een
A
R d
ep
th a
nd
NR
dep
th
0 5 10 15 20 25 30 35 40 45 500
5
10
15
20
Average difference be-tween AR depth and NR
depth with different weights
wh/wg
Ave
rag
e d
iffere
nce
betw
een
A
R d
ep
th a
nd
NR
dep
th
Distribution over 4-cases
Number of different runs(run = instance (#) + weight)
2x3-puzzle 9-puzzle 15-puzzle(Case 1)
NR-dep < AR-depNR-exp+gen < AR-exp+gen
28 97 413
(Case 2)
NR-dep < AR-depNR-exp+gen > AR-exp+gen
16 66 406
(Case 3)
NR-dep > AR-depNR-exp+gen < AR-exp+gen
104 187 568
(Case 4)
NR-dep > AR-depNR-exp+gen > AR-exp+gen
99 147 579
avg: 61.75sdev: 46.20
avg: 124.25sdev: 54.51
avg: 491.5sdev: 94.83