Domineering

1
Domineering

• Solving Large Combinatorial
• Search Spaces

2
Rules

• 2 player game (horizontal and vertical).
• n x m game board (or subset of).
• Players take turns placing 2 x 1 tiles (of their
  specified orientation) onto the board.
• First player unable to play, loses.

3
Previous Work

• Mathematicians have examined domineering using
  combinatorial game theory.
• AI research has focused on general alpha-beta
  techniques such as transposition tables and move
  ordering.

4
Why Study Domineering?

• Large amount of room for improvement over
  previous work.
• Nice mathematical properties.
• Simple rule set.
• Large search space.
• Interest has been shown from both math and
  computer science researchers.

5
Our Contributions

• A far superior evaluation function.
• Improved move ordering.
• Proof that we can ignore safe moves.
• Improved transposition table replacement scheme.

6
Example 1Horizontals turn, who wins?

• Safe moves for vertical 3.

• Total moves for horizontal 3.

• Vertical wins.

7
Example 2Horizontals turn, who wins?
8
Safe Area

• 2 x 1 unoccupied region of the board.
• Opponent unable to overlap with a tile.

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Protective Area

• 2 x 2 unoccupied region of the board.
• Placing a tile within creates another safe area.
• Two protective areas can not be adjacent.

10
Vulnerable Area

• 2 x 1 unoccupied region of the board.
• Type 1 vulnerable areas are not adjacent to any
  other area.
• Type 2 can be adjacent to any other areas.

11
Packing Example
12
Calculating Lower Bound
13
Opponents Moves

• Have good lower bound on number of moves for one
  player (moves(a)).
• Need upper bound on number of moves for other
  player.
• Count squares available for opponent to play on
  divided by 2.

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Unoccupied Squares

• Count the number of unoccupied squares.
• Subtract 2 moves(a) squares.
• Gives us total number of unoccupied squares after
  a has placed there tiles.

15
Unavailable Squares

• A 1 x 1 unoccupied region of the board.
• Not included in as board covering.
• Not available to as opponent.

16
Option Area

• 1 x 1 unoccupied region of the board attached to
  a safe area.
• Can not be adjacent to any other area.
• By playing creates unavailable squares for
  opponent.

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Vulnerable Area With A Protected Square

• Vulnerable area.
• Contains a square which is unavailable for the
  opponent.
• By not playing creates one more unavailable
  square for opponent.

18
Packing Example
19
Calculating Upper Bound
20
Unplayable Squares
21
Calculating Upper Bound
22
Packing Example
23
Vertical Wins

• Vertical can play at least 10 more tiles.
• Horizontal can play at most 10 more tiles.
• Since it is horizontals turn, horizontal must
  run out of moves before vertical.

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Solving 8x8 Domineering
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Solving 8x8 Domineering
Enhancements Nodes
All Enhancements. 2,023,031
All protective areas. 6,610,775
All unavailable squares. 2,566,004
All vulnerable areas with protected squares. 4,045,384
All vulnerable areas type 1. 2,972,216
All option areas. 4,525,704
No enhancements to evaluation. 84,034,856
26
Other Enhancements

• Improved move ordering.
• Proof that we can ignore safe moves.
• Improved transposition table replacement scheme.

27
Comparisons to DOMI
Board Size DOMI Obsequi
5x5 604 259
5x9 177,324 11,669
6x6 17,232 908
7x7 408,260 31,440
8x8 441,990,070 2,023,301
8x9 70 trillion 259 million
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New Results
Board Size Result Nodes
4x19 H 314,148,901
4x21 H 3,390,074,758
6x14 H 1,864,870,370
8x10 H 4,125,516,739
10x10 1 3,541,685,253,370
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Conclusion

• Enhanced evaluation function reduced the tree
  size by a factor of 80 (best case).
• All other improvements together created another 3
  to 4 times reduction in nodes.

30
Future Work

• Further refinements to evaluation function.
• Prove certain moves are always inferior.
• Better board packing algorithm.
• Directing the search to already examined board
  positions.
• Combining the benefits of the two transposition
  table replacement schemes.