1. No category

Minesweeper Solver
Marina Maznikova
Artificial Intelligence II Fall 2011
Agenda
1.
2.
3.
4.
5.
6.
The Minesweeper Game
Theoretical Background
Solving Agents
Evaluation
Extension
Summary
2
1. The Minesweeper Game
3
1. The Minesweeper Game
4
2. Theoretical Background

NP-completeness (8-SAT)

Solving approaches:
 Constraints
 Reinforcement
learning
5
3. Solving Agents

Random agent (RA)
6
3. Solving Agents

Random agent (RA)

Simple agent (SA)
 Only
mines remain around
a square => mark as mine
 All mines around a square
are marked => open
neighbors
7
3. Solving Agents

Greedy agent (GA)
 Start with (0, 0)
 Search for mine and safe squares
 If nothing found
2/3
 Calculate for each square the
density of mines on the squares
around it
 For each square, take the maximum
of the densities calculated for this
 If no opened neighbors => ½
2/3
 Open the square with the minimum
“probability” of mine
1/5
8
3. Solving Agents

Constraints agent (CA)
 Start
with (0, 0)
 Consider only the relevant squares
 Model the problem as constraints problem
Mine square => 1; opened square => 0
 Sum of mines around opened squares
 Number of mines in the game

9
3. Solving Agents

Constraints agent (CA)
 Backtracking
for 3000 results
 Calculate how many times
each square is mine
 Mark mine squares and open
safe squares
 If no such squares found,
open the square with the
lowest sum of mines
00100000
…
01111101
11111101
12322202
10
3. Solving Agents

Constraints agent (CA)
 Problem:
Exponential algorithm
Impossible to generate always all the solutions
 Time

11
3. Solving Agents

Constraints agent (CA)
 Problem:
Exponential algorithm
Impossible to generate always all the solutions
 Time


“Simple” constraints agent (SCA)
 Start
with (0, 0)
 Search for mine and safe squares
 If nothing found, use constraints
12
4. Evaluation

Observed variables
 Overall
result
 Average squares revealed per game
 Time

Tests (20.000 games)
 Agent
comparison on standard board
 Variation of board size
 Variation of mine density
13
4. Evaluation: Standard Game
Overall Result
5 000
98
748
0
-6 192
-5 000
50
45
46
SA
GA
56
40
-10 000
-15 000
60
Average Moves
55
30
20
-18 572
-20 000
-20 000
10
-25 000
0
RA
SA
GA
CA
SCA
14
RA
Problem of too
many solution
possibilities
CA
SCA
14
4. Evaluation: Standard Game
Backtracking
is expensive
Time (minutes)
60
50.72
50
40
30
20
13.74
7.02
10
0.09
0.44
RA
SA
0
GA
CA
SCA
15
4. Evaluation: Board Size
Overall Result
Better performance
on larger and
smaller boards
Average Moves
10 000
70
5 000
60
0
50
-5 000
40
-10 000
30
-15 000
20
-20 000
10
-25 000
0
RA
SA
Standard
GA
Small
CA
Large
SCA
RA
SA
Standard
Exception:
backtracking
GA
CA
Small
Large
SCA
16
4. Evaluation: Board Size
Larger boards require
more time, especially
when backtracking is
used
Time (minutes)
180
160
140
120
100
80
60
40
20
0
RA
SA
GA
Standard
Small
CA
SCA
Large
17
4. Evaluation: Mine Density
The higher the mine
density, the worse
the performance
Greedy strategy is
not good for few
mines
Average Moves
Overall Result
15 000
90
10 000
80
5 000
70
60
0
50
-5 000
40
-10 000
30
-15 000
20
-20 000
10
-25 000
0
RA
40 mines
SA
GA
20 mines
CA
60 mines
SCA
RA
40 mines
SA
GA
20 mines
CA
SCA
60 mines
18
4. Evaluation: Mine Density
Time (minutes)
300
CA and SCA:
More mines
require more time
250
200
150
100
50
0
RA
RA, SA, and GA:
More mines
require less time
SA
GA
40 mines
20 mines
CA
SCA
60 mines
19
5. Extension

Possibility to open 5% of the squares as
joker
 RA: At
the beginning of the game
 SA: In case no mine/safe squares are found
 GA: In case of two squares with the same
mine “probability”
 CA and SCA: On start, in case of too many
solutions of the problem, and when two
squares are “safest”
20
5. Extension: Evaluation
Overall Result
Jokers improve
the performance
20 000
Average Moves
100
90
80
70
60
50
40
30
20
10
0
15 000
10 000
5 000
0
-5 000
-10 000
-15 000
-20 000
-25 000
RA
SA
GA
CA
Without joker
With joker
SCA
RA
SA
GA
Without joker
… and make unsafe
strategies inefficient
CA
SCA
With joker
21
5. Extension: Evaluation
Time increases
because of the
more moves
Time (minutes)
100
90
80
70
60
50
40
30
20
10
0
RA
SA
GA
Without joker
CA
SCA
With joker
22
6. Summary

Backtracking is expensive
and for many solution
possibilities inefficient
 Solution:



Sets
The Greedy strategy is not
a good strategy
The higher the mine
density, the difficult the
game
Jokers help and make
unsafe move strategies
inefficient
23