Efficient Nimber Calculation in Dots and Boxes

This is a talk from the 2013 Southern California Conference for Undergraduate Research (SCCUR). It introduces an algorithm for efficient Nimber calculation in Dots and Boxes, including an explanation of the theory needed for understanding it.

Transcript

1. Developing an Efficient Algorithm for Nimber Calculation in Dots and Boxes by Andrew Brinker
2. What is Dots and Boxes?
3. Picture courtesy of Tiger66 < wiki/File:Dots-and-boxes.svg
4. There are 4 levels of play
5. You need chain control
6. This is a chain
7. There are actually 2 games
8. Enter Nim
9. NIM http://hanno-rein.de/images/main/2009_01/dsc00962.jpg
10. http://hanno-rein.de/images/main/2009_01/dsc00962.jpg
11. It all relies on the nim-sum
12. And the Sprague–Grundy Theorem
13. Win the 1st game, win the 2nd
14. Presenting naive_nim()
15. def mex(values): counter = 0 for value in values: if counter != value: return counter counter += 1 return values[len(values) - 1] + 1 ! def naive_nim(position): if all following moves for position are loony: return 0 return mex(all following positions to position)
16. It’s O(n!) (really slow)
17. Time for memoization (it’s a real word)
18. It’s better in reverse
19. Presenting fast_nim()
20. function fast_nim(region): board = board with the same dimensions as region l = number of missing lines in region n = 1 saved = set of positions w/out 1 line and w/ value 0 while (n != l): G = set of antecedent positions w/out n lines for (each position in G): F = set of follower positions to position nim_values = [] for (each follower in F): {follower position will always be in saved} nim_values = saved[follower] saved[position] = mex(nim_values) return saved
21. It’s O(n3) (much faster)
22. It’s usable now!
23. There’s still more to do
24. Any questions?