Polyominoes: an Exploration in Problem Solving (with Python)

A "Polyomino" is a geometric figure of adjacent squares. But just how many "n" polyominoes are there?

Transcript

1. Kiwi PyCon 2013 Hamish Campbell @polemic
2. Polyominoes an exploration in problem solving
3. Software Development Art Science Engineering
4. What is a Polyonomino? Monomino Domino Tromino
5. Let's be specific Fixed Distinct when none is a translation of another... One-sided And none is a rotation of another... Free And none is a reflection / glide reflection of another.
6. 4-Ominoes “In the wild”
7. How Many One-Sided N-ominos?
8. Define a Polyomino class Polyomino: def __init__(self, iterable): self.squares = tuple(iterable) p = Polyomino(((0,0), (0,1), (1,1)))
9. Identifying Candidates Enumerating search space – or – Building from lower order
10. Enumerating from Search Space (n = 6)
11. Enumerating from Search Space n2 = 36 squares “From n choose r “ import gmpy gmpy.comb(36, 6) >> 1947792 GMPY: https://code.google.com/p/gmpy/ Multiple-precision arithmetic for Python
12. Enumerating from Search Space n(n+1)/2 = 21 gmpy.comb(21, 6) >> 54264
13. Enumerating from Search Space 1 +
14. Enumerating from Search Space 1 + (2 x 5) nCr 6 +
15. Enumerating from Search Space 1 + (2 x 5) nCr 6 + (3 x 4) nCr 6 1 + \ gmpy.comb(2*5, 6) +\ gmpy.comb(3*4, 6) >> 1135
16. Enumeration def count_enumerations(d): import gmpy return 1 + sum( gmpy.comb(i * (1 + d ­ i), d) for i in range(1 + d // 2, d) )
17. Enumeration for i in range(5, 10): print i, count_enumerations(i) 5 183 6 1,135 7 18,668 8 172,732 9 3,655,850
18. Building from Lower Order (n = 1) 1
19. Building from Lower Order (n = 2) 1 x (n - 1) x 4 (worse case)
20. Building from Lower Order (n = 3) (n - 2) x 4 x (n -1) x 4 n = 4 (n – 3) x 4 x (n – 2) x 4 x (n – 1) x 4
21. Building from Lower Order Assumes 4 edges! In reality we can pick a lower number, e.g. 1 import math for i in range(5, 10): print i, math.factorial(i) 5 120 8 40,320 6 720 9 362,880 7 5,040
22. We have a winner! 3,655,850 vs 409,114
23. Identifying a Polyomino Tuples are hashable, that's handy. def __hash__(self): return hash(self.squares)
24. Identifying a Polyomino Consistently hashable. Order matters. def __init__(self, iterable): self.squares = tuple( sorted(iterable) )
25. Identifying a Polyomino But remember the rules def __hash__(self): p = self, k = self.key() for _ in range(3): p = self.rotate().translate() k = min(k, p.key()) return k def key(self): return hash(self.squares)
26. Eliminating Duplicates What's the easiest way to store an array of unique items?
27. Using the set helpers How does the set `in` operator work? hash(a) == hash(b) and (a is b or id(a) == id(b))
28. Identifying a Polyomino def __eq__(self, other): return hash(self) == hash(other)
29. Unique Polyominoes len(set([ Polyomino(((0,0), (0,1))), Polyomino(((1,0), (0,0))), Polyomino(((1,1), (1,1))), ])) == 1
30. What's left? Raising the order Building the result set. We'll stop here.