from __future__ import print_function, division
from itertools import combinations
from sympy.core import Basic
from sympy.combinatorics.graycode import GrayCode
from sympy.core.compatibility import range
[docs]class Subset(Basic):
"""
Represents a basic subset object.
We generate subsets using essentially two techniques,
binary enumeration and lexicographic enumeration.
The Subset class takes two arguments, the first one
describes the initial subset to consider and the second
describes the superset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.next_binary().subset
['b']
>>> a.prev_binary().subset
['c']
"""
_rank_binary = None
_rank_lex = None
_rank_graycode = None
_subset = None
_superset = None
def __new__(cls, subset, superset):
"""
Default constructor.
It takes the subset and its superset as its parameters.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.subset
['c', 'd']
>>> a.superset
['a', 'b', 'c', 'd']
>>> a.size
2
"""
if len(subset) > len(superset):
raise ValueError('Invalid arguments have been provided. The superset must be larger than the subset.')
for elem in subset:
if elem not in superset:
raise ValueError('The superset provided is invalid as it does not contain the element %i' % elem)
obj = Basic.__new__(cls)
obj._subset = subset
obj._superset = superset
return obj
[docs] def iterate_binary(self, k):
"""
This is a helper function. It iterates over the
binary subsets by k steps. This variable can be
both positive or negative.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.iterate_binary(-2).subset
['d']
>>> a = Subset(['a', 'b', 'c'], ['a', 'b', 'c', 'd'])
>>> a.iterate_binary(2).subset
[]
See Also
========
next_binary, prev_binary
"""
bin_list = Subset.bitlist_from_subset(self.subset, self.superset)
n = (int(''.join(bin_list), 2) + k) % 2**self.superset_size
bits = bin(n)[2:].rjust(self.superset_size, '0')
return Subset.subset_from_bitlist(self.superset, bits)
[docs] def next_binary(self):
"""
Generates the next binary ordered subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.next_binary().subset
['b']
>>> a = Subset(['a', 'b', 'c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.next_binary().subset
[]
See Also
========
prev_binary, iterate_binary
"""
return self.iterate_binary(1)
[docs] def prev_binary(self):
"""
Generates the previous binary ordered subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset([], ['a', 'b', 'c', 'd'])
>>> a.prev_binary().subset
['a', 'b', 'c', 'd']
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.prev_binary().subset
['c']
See Also
========
next_binary, iterate_binary
"""
return self.iterate_binary(-1)
[docs] def next_lexicographic(self):
"""
Generates the next lexicographically ordered subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.next_lexicographic().subset
['d']
>>> a = Subset(['d'], ['a', 'b', 'c', 'd'])
>>> a.next_lexicographic().subset
[]
See Also
========
prev_lexicographic
"""
i = self.superset_size - 1
indices = Subset.subset_indices(self.subset, self.superset)
if i in indices:
if i - 1 in indices:
indices.remove(i - 1)
else:
indices.remove(i)
i = i - 1
while not i in indices and i >= 0:
i = i - 1
if i >= 0:
indices.remove(i)
indices.append(i+1)
else:
while i not in indices and i >= 0:
i = i - 1
indices.append(i + 1)
ret_set = []
super_set = self.superset
for i in indices:
ret_set.append(super_set[i])
return Subset(ret_set, super_set)
[docs] def prev_lexicographic(self):
"""
Generates the previous lexicographically ordered subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset([], ['a', 'b', 'c', 'd'])
>>> a.prev_lexicographic().subset
['d']
>>> a = Subset(['c','d'], ['a', 'b', 'c', 'd'])
>>> a.prev_lexicographic().subset
['c']
See Also
========
next_lexicographic
"""
i = self.superset_size - 1
indices = Subset.subset_indices(self.subset, self.superset)
while i not in indices and i >= 0:
i = i - 1
if i - 1 in indices or i == 0:
indices.remove(i)
else:
if i >= 0:
indices.remove(i)
indices.append(i - 1)
indices.append(self.superset_size - 1)
ret_set = []
super_set = self.superset
for i in indices:
ret_set.append(super_set[i])
return Subset(ret_set, super_set)
[docs] def iterate_graycode(self, k):
"""
Helper function used for prev_gray and next_gray.
It performs k step overs to get the respective Gray codes.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset([1, 2, 3], [1, 2, 3, 4])
>>> a.iterate_graycode(3).subset
[1, 4]
>>> a.iterate_graycode(-2).subset
[1, 2, 4]
See Also
========
next_gray, prev_gray
"""
unranked_code = GrayCode.unrank(self.superset_size,
(self.rank_gray + k) % self.cardinality)
return Subset.subset_from_bitlist(self.superset,
unranked_code)
[docs] def next_gray(self):
"""
Generates the next Gray code ordered subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset([1, 2, 3], [1, 2, 3, 4])
>>> a.next_gray().subset
[1, 3]
See Also
========
iterate_graycode, prev_gray
"""
return self.iterate_graycode(1)
[docs] def prev_gray(self):
"""
Generates the previous Gray code ordered subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset([2, 3, 4], [1, 2, 3, 4, 5])
>>> a.prev_gray().subset
[2, 3, 4, 5]
See Also
========
iterate_graycode, next_gray
"""
return self.iterate_graycode(-1)
@property
def rank_binary(self):
"""
Computes the binary ordered rank.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset([], ['a','b','c','d'])
>>> a.rank_binary
0
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.rank_binary
3
See Also
========
iterate_binary, unrank_binary
"""
if self._rank_binary is None:
self._rank_binary = int("".join(
Subset.bitlist_from_subset(self.subset,
self.superset)), 2)
return self._rank_binary
@property
def rank_lexicographic(self):
"""
Computes the lexicographic ranking of the subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.rank_lexicographic
14
>>> a = Subset([2, 4, 5], [1, 2, 3, 4, 5, 6])
>>> a.rank_lexicographic
43
"""
if self._rank_lex is None:
def _ranklex(self, subset_index, i, n):
if subset_index == [] or i > n:
return 0
if i in subset_index:
subset_index.remove(i)
return 1 + _ranklex(self, subset_index, i + 1, n)
return 2**(n - i - 1) + _ranklex(self, subset_index, i + 1, n)
indices = Subset.subset_indices(self.subset, self.superset)
self._rank_lex = _ranklex(self, indices, 0, self.superset_size)
return self._rank_lex
@property
def rank_gray(self):
"""
Computes the Gray code ranking of the subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c','d'], ['a','b','c','d'])
>>> a.rank_gray
2
>>> a = Subset([2, 4, 5], [1, 2, 3, 4, 5, 6])
>>> a.rank_gray
27
See Also
========
iterate_graycode, unrank_gray
"""
if self._rank_graycode is None:
bits = Subset.bitlist_from_subset(self.subset, self.superset)
self._rank_graycode = GrayCode(len(bits), start=bits).rank
return self._rank_graycode
@property
def subset(self):
"""
Gets the subset represented by the current instance.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.subset
['c', 'd']
See Also
========
superset, size, superset_size, cardinality
"""
return self._subset
@property
def size(self):
"""
Gets the size of the subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.size
2
See Also
========
subset, superset, superset_size, cardinality
"""
return len(self.subset)
@property
def superset(self):
"""
Gets the superset of the subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.superset
['a', 'b', 'c', 'd']
See Also
========
subset, size, superset_size, cardinality
"""
return self._superset
@property
def superset_size(self):
"""
Returns the size of the superset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.superset_size
4
See Also
========
subset, superset, size, cardinality
"""
return len(self.superset)
@property
def cardinality(self):
"""
Returns the number of all possible subsets.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> a = Subset(['c', 'd'], ['a', 'b', 'c', 'd'])
>>> a.cardinality
16
See Also
========
subset, superset, size, superset_size
"""
return 2**(self.superset_size)
@classmethod
[docs] def subset_from_bitlist(self, super_set, bitlist):
"""
Gets the subset defined by the bitlist.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> Subset.subset_from_bitlist(['a', 'b', 'c', 'd'], '0011').subset
['c', 'd']
See Also
========
bitlist_from_subset
"""
if len(super_set) != len(bitlist):
raise ValueError("The sizes of the lists are not equal")
ret_set = []
for i in range(len(bitlist)):
if bitlist[i] == '1':
ret_set.append(super_set[i])
return Subset(ret_set, super_set)
@classmethod
[docs] def bitlist_from_subset(self, subset, superset):
"""
Gets the bitlist corresponding to a subset.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> Subset.bitlist_from_subset(['c', 'd'], ['a', 'b', 'c', 'd'])
'0011'
See Also
========
subset_from_bitlist
"""
bitlist = ['0'] * len(superset)
if type(subset) is Subset:
subset = subset.args[0]
for i in Subset.subset_indices(subset, superset):
bitlist[i] = '1'
return ''.join(bitlist)
@classmethod
[docs] def unrank_binary(self, rank, superset):
"""
Gets the binary ordered subset of the specified rank.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> Subset.unrank_binary(4, ['a', 'b', 'c', 'd']).subset
['b']
See Also
========
iterate_binary, rank_binary
"""
bits = bin(rank)[2:].rjust(len(superset), '0')
return Subset.subset_from_bitlist(superset, bits)
@classmethod
[docs] def unrank_gray(self, rank, superset):
"""
Gets the Gray code ordered subset of the specified rank.
Examples
========
>>> from sympy.combinatorics.subsets import Subset
>>> Subset.unrank_gray(4, ['a', 'b', 'c']).subset
['a', 'b']
>>> Subset.unrank_gray(0, ['a', 'b', 'c']).subset
[]
See Also
========
iterate_graycode, rank_gray
"""
graycode_bitlist = GrayCode.unrank(len(superset), rank)
return Subset.subset_from_bitlist(superset, graycode_bitlist)
@classmethod
[docs] def subset_indices(self, subset, superset):
"""Return indices of subset in superset in a list; the list is empty
if all elements of subset are not in superset.
Examples
========
>>> from sympy.combinatorics import Subset
>>> superset = [1, 3, 2, 5, 4]
>>> Subset.subset_indices([3, 2, 1], superset)
[1, 2, 0]
>>> Subset.subset_indices([1, 6], superset)
[]
>>> Subset.subset_indices([], superset)
[]
"""
a, b = superset, subset
sb = set(b)
d = {}
for i, ai in enumerate(a):
if ai in sb:
d[ai] = i
sb.remove(ai)
if not sb:
break
else:
return list()
return [d[bi] for bi in b]
def ksubsets(superset, k):
"""
Finds the subsets of size k in lexicographic order.
This uses the itertools generator.
Examples
========
>>> from sympy.combinatorics.subsets import ksubsets
>>> list(ksubsets([1, 2, 3], 2))
[(1, 2), (1, 3), (2, 3)]
>>> list(ksubsets([1, 2, 3, 4, 5], 2))
[(1, 2), (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), \
(2, 5), (3, 4), (3, 5), (4, 5)]
See Also
========
class:Subset
"""
return combinations(superset, k)
