Coverage for partitionsets/partition.py: 100.00%

1#! /usr/bin/env python

2""" Minimal implementation of a partitioning class for sets based

3on [PartImplPy] and feedback from [PyLint, Flake8].

5All python sources are licensed under the MIT License.

6As entry point for further research cf. [PartOfASet_WP]

8References:

10[Flake8]: https://pypi.python.org/pypi/flake8

12[PartImplPy]: http://code.activestate.com/recipes/577211/ (r1)

14[PartOfASet_WP]: Wikipedia entry Partition_of_a_set at

15 http://en.wikipedia.org/wiki/Partition_of_a_set

17[PyLint]: https://pypi.python.org/pypi/pylint

20"""

21from __future__ import print_function

23from collections import defaultdict

26class Partition:

27 """Sets of only a few items already have many ways they can be partitioned

28 into subsets. Therefore it can be useful to generate these partitions by

29 index, like the partition class were some large list where one can just

30 access element "i". Of course one should not compute the whole list in

31 advance but compute the partitions on the fly. This recipe was originally

32 extracted form a book by Kreher and Stinson. Over the years I came back to

33 my code from time to time creating a new and hopefully more pythonic

34 version each time I understood it better. My current take on it is that the

35 algorithm looks a lot like creating a Pascals triangle in order to compute

36 combinations. One just tries to find a way down the triangle to a specific

37 element, each time subtracting the amounts the different positions in the

38 triangle account for. It is also similar to finding indexed permutations of

39 a set with elements occurring more than once. One of these days I will

40 perhaps understand how all of this fits together. Until then I'll post code

41 solving specific situations. cf. [PartImplPy]"""

43 def __init__(self, ordered_set):

44 self.data = list(ordered_set)

45 self.n_data = len(ordered_set)

46 self.table = self.rgf_table()

48 def __getitem__(self, i):

49 """Generates set partitions by index"""

50 if i > len(self) - 1:

51 raise IndexError

52 a_list = self.unrank_rgf(i)

53 result = self.as_set_partition(a_list)

54 return result

56 def __len__(self):

57 return self.table[self.n_data, 0]

59 def __delitem__(self, index):

60 pass

62 def __setitem__(self, key, value):

63 pass

65 def as_set_partition(self, a_list):

66 """Transform a restricted growth function into a partition"""

67 n_list = max(a_list[1:] + [1])

68 n_data = self.n_data

69 data = self.data

70 partition = [[] for _ in range(n_list)]

71 for i in range(n_data):

72 partition[a_list[i + 1] - 1].append(data[i])

73 return partition

75 def rgf_table(self):

76 """Compute the table values, but: The key line in the algorithm is

77 D[i,j] = j * D[i-1,j] + D[i-1,j+1] which identifies D[i,j] as the

78 number of ways to add i new items to a partition that already has

79 j blocks. So e.g. D[1, j] = j+1 because a new

80 item can be added to a partition having j blocks by adding it to any

81 existing block or by having it start its own new block. Thus D[i,0] is

82 the number of ways to partition the set {1, ..., i}.

84 To solve this problem for combinations (i.e. get the i-th subset of a

85 set), just take the binary representation of the index i, right? The

86 places where 1's occur tell you which items to keep -- no need to

87 generate Pascal's triangle"""

88 n_data = self.n_data

89 a_dict = defaultdict(lambda: 1)

90 for i in range(1, n_data + 1):

91 for j in range(0, n_data - i + 1):

92 a_dict[i, j] = j * a_dict[i - 1, j] + a_dict[i - 1, j + 1]

93 return a_dict

95 def unrank_rgf(self, r_saught):

96 """Unrank a restricted growth function"""

97 n_data = self.n_data

98 a_list = [1 for _ in range(n_data + 1)]

99 j = 1

100 a_dict = self.table

101 for i in range(2, n_data + 1):

102 v_j = a_dict[n_data - i, j]

103 cr_j = j * v_j

104 if cr_j <= r_saught:

105 a_list[i] = j + 1

106 r_saught -= cr_j

107 j += 1

108 else:

109 a_list[i] = r_saught // v_j + 1

110 r_saught %= v_j

111 return a_list