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 1Horizontal’s turn, who wins? • Safe moves for vertical = 3. • Total moves for horizontal = 3. • Vertical wins.

7. Example 2Horizontal’s turn, who wins?

8. 2 x 1 unoccupied region of the board. Opponent unable to overlap with a tile. Safe Area

9. 2 x 2 unoccupied region of the board. Placing a tile within creates another safe area. Two protective areas can not be adjacent. Protective Area

10. 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. Vulnerable Area

11. Packing Example

12. Calculating Lower Bound

13. Opponent’s Moves • Have good lower bound on number of moves for one player (moves(α)). • Need upper bound on number of moves for other player. • Count squares available for opponent to play on divided by 2.

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

15. A 1 x 1 unoccupied region of the board. Not included in α’s board covering. Not available to α’s opponent. Unavailable Squares

16. 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. Option Area

17. Vulnerable area. Contains a square which is unavailable for the opponent. By not playing creates one more unavailable square for opponent. Vulnerable Area With A Protected Square

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 horizontal’s turn, horizontal must run out of moves before vertical.

24. Solving 8x8 Domineering

25. Solving 8x8 Domineering

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

27. Comparisons to DOMI

28. New Results

29. 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.