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Minesweeper Solver
Marina Maznikova
Artificial Intelligence II Fall 2011
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Agenda
1. The Minesweeper Game
2. Theoretical Background
3. Solving Agents
4. Evaluation
5. Extension
6. Summary
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1. The Minesweeper Game
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1. The Minesweeper Game
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2. Theoretical Background
NP-completeness (8-SAT)
Solving approaches: ConstraintsReinforcement learning
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3. Solving Agents
Random agent (RA)
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3. Solving Agents
Random agent (RA)
Simple agent (SA)Only mines remain around
a square => mark as mineAll mines around a square
are marked => open neighbors
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3. Solving Agents
Greedy agent (GA)Start with (0, 0)Search for mine and safe squares If nothing found
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 Open the square with the minimum
“probability” of mine
2/3
2/3
1/5
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3. Solving Agents
Constraints agent (CA)Start with (0, 0)Consider only the relevant squaresModel the problem as constraints problem
Mine square => 1; opened square => 0 Sum of mines around opened squares Number of mines in the game
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3. Solving Agents
Constraints agent (CA)Backtracking for 3000 resultsCalculate how many times
each square is mineMark mine squares and open
safe squares If no such squares found,
open the square with the lowest sum of mines
0 0 1 0 0 0 0 0
…
0 1 1 1 1 1 0 1
1 1 1 1 1 1 0 1
1 2 3 2 2 2 0 2
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3. Solving Agents
Constraints agent (CA)Problem: Exponential algorithm
Impossible to generate always all the solutions Time
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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
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4. Evaluation
Observed variablesOverall resultAverage squares revealed per gameTime
Tests (20.000 games)Agent comparison on standard boardVariation of board sizeVariation of mine density
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4. Evaluation: Standard Game
Problem of too many solution possibilities
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4. Evaluation: Standard GameBacktracking is expensive
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4. Evaluation: Board SizeBetter performance
on larger and smaller boards
Exception: backtracking
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4. Evaluation: Board SizeLarger boards require more time, especially when backtracking is
used
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4. Evaluation: Mine DensityThe higher the mine density, the worse the performance
Greedy strategy is not good for
few mines
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4. Evaluation: Mine Density
RA, SA, and GA: More mines
require less time
CA and SCA: More mines
require more time
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5. Extension
Possibility to open 5% of the squares as jokerRA: At the beginning of the gameSA: In case no mine/safe squares are foundGA: 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”
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5. Extension: EvaluationJokers improve the performance
… and make unsafe strategies inefficient
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5. Extension: EvaluationTime increases because of the more moves
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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