Coverage for local/lib/python2.7/site-packages/sage/matroids/set_system.pyx : 82%

""" 

Set systems 

Many matroid methods return a collection of subsets. In this module a class 

:class:`SetSystem <sage.matroids.set_system.SetSystem>` is defined to do 

just this. The class is intended for internal use, so all you can do as a user 

is iterate over its members. 

The class is equipped with partition refinement methods to help with matroid 

isomorphism testing. 

AUTHORS: 

- Rudi Pendavingh, Stefan van Zwam (2013-04-01): initial version 

Methods 

======= 

""" 

#***************************************************************************** 

# Copyright (C) 2013 Rudi Pendavingh <rudi.pendavingh@gmail.com> 

# Copyright (C) 2013 Stefan van Zwam <stefanvanzwam@gmail.com> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# as published by the Free Software Foundation; either version 2 of 

# the License, or (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from __future__ import print_function 

from cysignals.memory cimport check_allocarray, check_reallocarray, sig_free 

include 'sage/data_structures/bitset.pxi' 

# SetSystem 

cdef class SetSystem: 

""" 

A ``SetSystem`` is an enumerator of a collection of subsets of a given 

fixed and finite ground set. It offers the possibility to enumerate its 

contents. One is most likely to encounter these as output from some 

Matroid methods:: 

sage: M = matroids.named_matroids.Fano() 

sage: M.circuits() 

Iterator over a system of subsets 

To access the sets in this structure, simply iterate over them. The 

simplest way must be:: 

sage: from sage.matroids.set_system import SetSystem 

sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]]) 

sage: T = list(S) 

Or immediately use it to iterate:: 

sage: from sage.matroids.set_system import SetSystem 

sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]]) 

sage: [min(X) for X in S] 

[1, 3, 1] 

Note that this class is intended for runtime, so no loads/dumps mechanism 

was implemented. 

.. WARNING:: 

The only guaranteed behavior of this class is that it is iterable. It 

is expected that M.circuits(), M.bases(), and so on will in the near 

future return actual iterators. All other methods (which are already 

hidden by default) are only for internal use by the Sage matroid code. 

""" 

def __cinit__(self, groundset, subsets=None, capacity=1): 

""" 

Init internal data structures. 

EXAMPLES:: 

sage: from sage.matroids.set_system import SetSystem 

sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]]) 

sage: S 

Iterator over a system of subsets 

""" 

cdef long i 

if not isinstance(groundset, tuple): 

self._groundset = tuple(groundset) 

else: 

self._groundset = groundset 

self._idx = {} 

for i in xrange(len(groundset)): 

self._idx[groundset[i]] = i 

self._groundset_size = len(groundset) 

self._bitset_size = max(self._groundset_size, 1) 

self._capacity = capacity 

self._subsets = <bitset_t*>check_allocarray(self._capacity, sizeof(bitset_t)) 

bitset_init(self._temp, self._bitset_size) 

self._len = 0 

def __init__(self, groundset, subsets=None, capacity=1): 

""" 

Create a SetSystem. 

INPUT: 

- ``groundset`` -- a list or tuple of finitely many elements. 

- ``subsets`` -- (default: ``None``) an enumerator for a set of 

subsets of ``groundset``. 

- ``capacity`` -- (default: ``1``) Initial maximal capacity of the set 

system. 

EXAMPLES:: 

sage: from sage.matroids.set_system import SetSystem 

sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]]) 

sage: S 

Iterator over a system of subsets 

sage: sorted(S[1]) 

[3, 4] 

sage: for s in S: print(sorted(s)) 

[1, 2] 

[3, 4] 

[1, 2, 4] 

""" 

if subsets is not None: 

for e in subsets: 

self.append(e) 

def __dealloc__(self): 

cdef long i 

for i in xrange(self._len): 

bitset_free(self._subsets[i]) 

sig_free(self._subsets) 

bitset_free(self._temp) 

def __len__(self): 

""" 

Return the number of subsets in this SetSystem. 

EXAMPLES:: 

sage: from sage.matroids.set_system import SetSystem 

sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]]) 

sage: S 

Iterator over a system of subsets 

sage: len(S) 

3 

""" 

return self._len 

def __iter__(self): 

""" 

Return an iterator for the subsets in this SetSystem. 

EXAMPLES:: 

sage: from sage.matroids.set_system import SetSystem 

sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]]) 

sage: for s in S: print(sorted(s)) 

[1, 2] 

[3, 4] 

[1, 2, 4] 

""" 

return SetSystemIterator(self) 

def __getitem__(self, k): 

""" 

Return the `k`-th subset in this SetSystem. 

INPUT: 

- ``k`` -- an integer. The index of the subset in the system. 

OUTPUT: 

The subset at index `k`. 

EXAMPLES:: 

sage: from sage.matroids.set_system import SetSystem 

sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]]) 

sage: sorted(S[0]) 

[1, 2] 

sage: sorted(S[1]) 

[3, 4] 

sage: sorted(S[2]) 

[1, 2, 4] 

""" 

if k < len(self): 

return self.subset(k) 

else: 

raise ValueError("out of range") 

def __repr__(self): 

""" 

Return a string representation of ``self``. 

EXAMPLES:: 

sage: from sage.matroids.set_system import SetSystem 

sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]]) 

sage: repr(S) # indirect doctest 

'Iterator over a system of subsets' 

""" 

return "Iterator over a system of subsets" 

cdef copy(self): 

cdef SetSystem S 

S = SetSystem(self._groundset, capacity=len(self)) 

for i in xrange(len(self)): 

S._append(self._subsets[i]) 

return S 

cdef _relabel(self, l): 

""" 

Relabel each element `e` of the ground set as `l(e)`, where `l` is a 

given injective map. 

INPUT: 

- ``l`` -- a python object such that `l[e]` is the new label of e. 

OUTPUT: 

``None``. 

""" 

cdef long i 

E = [] 

for i in range(self._groundset_size): 

if self._groundset[i] in l: 

E.append(l[self._E[i]]) 

else: 

E.append(self._E[i]) 

self._groundset = E 

self._idx = {} 

for i in xrange(self._groundset_size): 

self._idx[self._groundset[i]] = i 

cpdef _complements(self): 

""" 

Return a SetSystem containing the complements of each element in the 

groundset. 

EXAMPLES:: 

sage: from sage.matroids.set_system import SetSystem 

sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]]) 

sage: T = S._complements() 

sage: for t in T: print(sorted(t)) 

[3, 4] 

[1, 2] 

[3] 

""" 

cdef SetSystem S 

if self._groundset_size == 0: 

return self 

S = SetSystem(self._groundset, capacity=len(self)) 

for i in xrange(len(self)): 

bitset_complement(self._temp, self._subsets[i]) 

S._append(self._temp) 

return S 

cdef inline resize(self, k=None): 

""" 

Change the capacity of the SetSystem. 

""" 

if k is None: 

k = self._len 

for i in xrange(k, self._len): 

bitset_free(self._subsets[i]) 

self._len = min(self._len, k) 

k2 = max(k, 1) 

self._subsets = <bitset_t*>check_reallocarray(self._subsets, k2, sizeof(bitset_t)) 

self._capacity = k2 

cdef inline _append(self, bitset_t X): 

""" 

Append subset in internal, bitset format 

""" 

if self._capacity == self._len: 

self.resize(self._capacity * 2) 

bitset_init(self._subsets[self._len], self._bitset_size) 

bitset_copy(self._subsets[self._len], X) 

self._len += 1 

cdef inline append(self, X): 

""" 

Append subset. 

""" 

if self._capacity == self._len: 

self.resize(self._capacity * 2) 

bitset_init(self._subsets[self._len], self._bitset_size) 

bitset_clear(self._subsets[self._len]) 

for x in X: 

bitset_add(self._subsets[self._len], self._idx[x]) 

self._len += 1 

cdef inline _subset(self, long k): 

""" 

Return the k-th subset, in index format. 

""" 

return bitset_list(self._subsets[k]) 

cdef subset(self, k): 

""" 

Return the k-th subset. 

""" 

cdef long i 

F = set() 

i = bitset_first(self._subsets[k]) 

while i >= 0: 

F.add(self._groundset[i]) 

i = bitset_next(self._subsets[k], i + 1) 

return frozenset(F) 

cpdef _get_groundset(self): 

""" 

Return the ground set of this SetSystem. 

EXAMPLES:: 

sage: from sage.matroids.set_system import SetSystem 

sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]]) 

sage: sorted(S._get_groundset()) 

[1, 2, 3, 4] 

""" 

return frozenset(self._groundset) 

cpdef is_connected(self): 

""" 

Test if the :class:`SetSystem` is connected. 

A :class:`SetSystem` is connected if there is no nonempty proper subset 

``X`` of the ground set so the each subset is either contained in ``X`` 

or disjoint from ``X``. 

EXAMPLES:: 

sage: from sage.matroids.set_system import SetSystem 

sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]]) 

sage: S.is_connected() 

True 

sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4]]) 

sage: S.is_connected() 

False 

sage: S = SetSystem([1], []) 

sage: S.is_connected() 

True 

""" 

if self._groundset_size <= 1: 

return True 

cdef long i 

bitset_clear(self._temp) 

cdef bitset_t active 

bitset_init(active, self._len) 

bitset_complement(active, active) 

# We compute the union of all sets containing 0, and deactivate them. 

for i in xrange(self._len): 

if bitset_in(self._subsets[i], 0): 

bitset_union(self._temp, self._subsets[i], self._temp) 

bitset_discard(active, i) 

cdef bint closed = False 

while not closed: 

closed = True 

# We update _temp with all active sets that intersects it. If there 

# is no such set, then _temp is closed (i.e. a connected component). 

i = bitset_first(active) 

while i>=0: 

if not bitset_are_disjoint(self._temp, self._subsets[i]): 

bitset_union(self._temp, self._subsets[i], self._temp) 

bitset_discard(active, i) 

closed = False 

i = bitset_next(active, i+1) 

bitset_free(active) 

bitset_complement(self._temp, self._temp) 

return bitset_isempty(self._temp) 

# isomorphism 

cdef list _incidence_count(self, E): 

""" 

For the sub-collection indexed by ``E``, count how often each element 

occurs. 

""" 

cdef long i, e 

cdef list cnt 

cnt = [0 for v in xrange(self._groundset_size)] 

for e in E: 

i = bitset_first(self._subsets[e]) 

while i >= 0: 

cnt[i] += 1 

i = bitset_next(self._subsets[e], i + 1) 

return cnt 

cdef SetSystem _groundset_partition(self, SetSystem P, list cnt): 

""" 

Helper method for partition methods below. 

""" 

cdef dict C 

cdef long i, j, v, t0, t 

cdef bint split 

C = {} 

for i in xrange(len(P)): 

v = bitset_first(P._subsets[i]) 

if v < 0: 

continue 

t0 = cnt[v] 

v = bitset_next(P._subsets[i], v + 1) 

split = False 

while v >= 0: 

t = cnt[v] 

if t != t0: 

split = True 

if t < t0: 

t0 = t 

v = bitset_next(P._subsets[i], v + 1) 

if split: 

C.clear() 

v = bitset_first(P._subsets[i]) 

while v >= 0: 

t = cnt[v] 

if t != t0: 

if t in C: 

C[t].add(v) 

else: 

C[t] = set([v]) 

v = bitset_next(P._subsets[i], v + 1) 

for t in sorted(C): 

bitset_clear(self._temp) 

for v in C[t]: 

bitset_add(self._temp, v) 

bitset_discard(P._subsets[i], v) 

P._append(self._temp) 

cdef long subset_characteristic(self, SetSystem P, long e): 

""" 

Helper method for partition methods below. 

""" 

cdef long c 

c = 0 

for p in xrange(len(P)): 

c <<= bitset_len(P._subsets[p]) 

bitset_intersection(self._temp, P._subsets[p], self._subsets[e]) 

c += bitset_len(self._temp) 

return c 

cdef subsets_partition(self, SetSystem P=None, E=None): 

""" 

Helper method for partition methods below. 

""" 

S = {} 

if P is None: 

P = self.groundset_partition() 

if E is None: 

E = xrange(self._len) 

if len(E) == 0: 

return [E] 

ED = [(self.subset_characteristic(P, e), e) for e in E] 

ED.sort() 

EP = [] 

ep = [] 

d = ED[0][0] 

eh = [d] 

for ed in ED: 

if ed[0] != d: 

EP.append(ep) 

ep = [ed[1]] 

d = ed[0] 

else: 

ep.append(ed[1]) 

eh.append(ed[0]) 

EP.append(ep) 

return EP, hash(tuple(eh)) 

cdef _distinguish(self, v): 

""" 

Helper method for partition methods below. 

""" 

cdef SetSystem S 

S = SetSystem(self._groundset, capacity=len(self) + 1) 

bitset_clear(self._temp) 

bitset_add(self._temp, v) 

for i in xrange(len(self)): 

bitset_difference(S._temp, self._subsets[i], self._temp) 

S._append(S._temp) 

S._append(self._temp) 

return S 

# partition functions 

cdef initial_partition(self, SetSystem P=None, E=None): 

""" 

Helper method for partition methods below. 

""" 

if E is None: 

E = xrange(self._len) 

if P is None: 

if self._groundset: 

P = SetSystem(self._groundset, [self._groundset], capacity=self._groundset_size) 

else: 

P = SetSystem([], []) 

cnt = self._incidence_count(E) 

self._groundset_partition(P, cnt) 

return P 

cpdef _equitable_partition(self, SetSystem P=None, EP=None): 

""" 

Return an equitable ordered partition of the ground set of the 

hypergraph whose edges are the subsets in this SetSystem. 

Given any ordered partition `P = (p_1, ..., p_k)` of the ground set of 

a hypergraph, any edge `e` of the hypergraph has a characteristic 

intersection number sequence `i(e)=(|p_1\cap e|, ... , |p_k\cap e|))`. 

There is an ordered partition `EP` of the edges that groups the edges 

according to this intersection number sequence. Given this an ordered 

partition of the edges, we may similarly refine `P` to a new ordered 

partition `P'`, by considering the incidence numbers of ground set 

elements with each partition element of `EP`. 

The ordered partition `P` is equitable when `P' = P`. 

INPUT: 

- ``P``, an equitable ordered partition of the ground set, stored as 

a SetSystem. 

- ``EP``, the corresponding equitable partition of the edges, stored 

as a list of lists of indices of subsets of this SetSystem. 

OUTPUT: 

- ``P``, an equitable ordered partition of the ground set, stored as a 

SetSystem. 

- ``EP``, the corresponding equitable partition of the edges, stored 

as a list of lists of indices of subsets of this SetSystem. 

- ``h``, an integer invariant of the SetSystem. 

EXAMPLES:: 

sage: from sage.matroids.set_system import SetSystem 

sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]]) 

sage: for p in S._equitable_partition()[0]: print(sorted(p)) 

[3] 

[4] 

[1, 2] 

sage: T = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 3, 4]]) 

sage: for p in T._equitable_partition()[0]: print(sorted(p)) 

[2] 

[1] 

[3, 4] 

.. NOTE:: 

We do not maintain any well-defined order when refining a 

partition. We do maintain that the resulting order of the 

partition elements is an invariant of the isomorphism class of the 

hypergraph. 

""" 

cdef long h, l 

cdef list EP2, H 

if P is None: 

P = self.initial_partition() 

else: 

P = P.copy() 

if EP is None: 

EP = self.subsets_partition(P)[0] 

h = len(EP) 

pl = 0 

while len(P) > pl: 

H = [h] 

pl = len(P) 

EP2 = [] 

for ep in EP: 

SP, h = self.subsets_partition(P, ep) 

H.append(h) 

for p in SP: 

cnt = self._incidence_count(p) 

self._groundset_partition(P, cnt) 

if len(p) > 1: 

EP2.append(p) 

EP = EP2 

h = hash(tuple(H)) 

return P, EP, h 

cpdef _heuristic_partition(self, SetSystem P=None, EP=None): 

""" 

Return an heuristic ordered partition into singletons of the ground 

set of the hypergraph whose edges are the subsets in this SetSystem. 

This partition obtained as follows: make an equitable 

partition ``P``, and while ``P`` has a partition element ``p`` with 

more than one element, select an arbitrary ``e`` from the first such 

``p`` and split ``p`` into ``p-e``. Then replace ``P`` with 

the equitable refinement of this partition. 

INPUT: 

- ``P`` -- (default: ``None``) an ordered partition of the ground set. 

- ``EP`` -- (default: ``None``) the corresponding partition of the 

edges, stored as a list of lists of indices of subsets of this 

SetSystem. 

OUTPUT: 

- ``P`` -- an ordered partition of the ground set into singletons, 

stored as a SetSystem. 

- ``EP`` -- the corresponding partition of the edges, stored as a list 

of lists of indices of subsets of this SetSystem. 

- ``h`` -- an integer invariant of the SetSystem. 

EXAMPLES:: 

sage: from sage.matroids.set_system import SetSystem 

sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]]) 

sage: for p in S._heuristic_partition()[0]: print(sorted(p)) 

[3] 

[4] 

[2] 

[1] 

sage: T = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 3, 4]]) 

sage: for p in T._heuristic_partition()[0]: print(sorted(p)) 

[2] 

[1] 

[4] 

[3] 

""" 

P, EP, h = self._equitable_partition(P, EP) 

for i in xrange(len(P)): 

if bitset_len(P._subsets[i]) > 1: 

return self._heuristic_partition(P._distinguish(bitset_first(P._subsets[i])), EP) 

return P, EP, h 

cpdef _isomorphism(self, SetSystem other, SetSystem SP=None, SetSystem OP=None): 

""" 

Return a groundset isomorphism between this SetSystem and an other. 

INPUT: 

- ``other`` -- a SetSystem 

- ``SP`` (optional) -- a SetSystem storing an ordered partition of the 

ground set of ``self`` 

- ``OP`` (optional) -- a SetSystem storing an ordered partition of the 

ground set of ``other`` 

OUTPUT: 

``morphism`` -- a dictionary containing an isomorphism respecting the 

given ordered partitions, or ``None`` if no such isomorphism exists. 

EXAMPLES:: 

sage: from sage.matroids.set_system import SetSystem 

sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]]) 

sage: T = SetSystem(['a', 'b', 'c', 'd'], [['a', 'b'], ['c', 'd'], 

....: ['a', 'c', 'd']]) 

sage: S._isomorphism(T) 

{1: 'c', 2: 'd', 3: 'b', 4: 'a'} 

sage: S = SetSystem([], []) 

sage: S._isomorphism(S) 

{} 

""" 

cdef long l, p 

if SP is None or OP is None: 

SP, SEP, sh = self._equitable_partition() 

OP, OEP, oh = other._equitable_partition() 

if sh != oh: 

return None 

if len(SP) != len(OP): 

return None 

p = SP._groundset_size + 1 

for i in xrange(len(SP)): 

l = bitset_len(SP._subsets[i]) 

if l != bitset_len(OP._subsets[i]): 

return None 

if l != 1 and l < p: 

p = l 

for i in xrange(len(SP)): 

if bitset_len(SP._subsets[i]) == p: 

SP2, SEP, sh = self._equitable_partition(SP._distinguish(bitset_first(SP._subsets[i]))) 

v = bitset_first(OP._subsets[i]) 

while v >= 0: 

OP2, OEP, oh = other._equitable_partition(OP._distinguish(v)) 

if sh == oh: 

m = self._isomorphism(other, SP2, OP2) 

if m is not None: 

return m 

v = bitset_next(OP._subsets[i], v + 1) 

return None 

if sorted([self.subset_characteristic(SP, i) for i in xrange(len(self))]) != sorted([other.subset_characteristic(OP, i) for i in xrange(len(other))]): 

return None 

return dict([(self._groundset[bitset_first(SP._subsets[i])], other._groundset[bitset_first(OP._subsets[i])]) for i in xrange(len(SP))]) 

cpdef _equivalence(self, is_equiv, SetSystem other, SetSystem SP=None, SetSystem OP=None): 

""" 

Return a groundset isomorphism that is an equivalence between this 

SetSystem and an other. 

INPUT: 

- ``is_equiv`` -- a function that determines if a given groundset 

isomorphism is a valid equivalence 

- ``other`` -- a SetSystem 

- ``SP`` (optional) -- a SetSystem storing an ordered partition of the 

groundset of ``self`` 

- ``OP`` (optional) -- a SetSystem storing an ordered partition of the 

groundset of ``other`` 

OUTPUT: 

``morphism``, a dictionary containing an isomorphism respecting the 

given ordered partitions, so that ``is_equiv(self, other, morphism)`` 

is ``True``; or ``None`` if no such equivalence exists. 

EXAMPLES:: 

sage: from sage.matroids.set_system import SetSystem 

sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]]) 

sage: T = SetSystem(['a', 'b', 'c', 'd'], [['a', 'b'], ['c', 'd'], 

....: ['a', 'c', 'd']]) 

sage: S._equivalence(lambda self, other, morph:True, T) 

{1: 'c', 2: 'd', 3: 'b', 4: 'a'} 

Check that :trac:`15189` is fixed:: 

sage: M = Matroid(ring=GF(5), reduced_matrix=[[1,0,3],[0,1,1],[1,1,0]]) 

sage: N = Matroid(ring=GF(5), reduced_matrix=[[1,0,1],[0,1,1],[1,1,0]]) 

sage: M.is_field_isomorphic(N) 

False 

sage: any(M.is_field_isomorphism(N, p) for p in Permutations(range(6))) 

False 

""" 

if SP is None or OP is None: 

SP, SEP, sh = self._equitable_partition() 

OP, OEP, oh = other._equitable_partition() 

if sh != oh: 

return None 

if len(SP) != len(OP): 

return None 

for i in xrange(len(SP)): 

if bitset_len(SP._subsets[i]) != bitset_len(OP._subsets[i]): 

return None 

for i in xrange(len(SP)): 

if bitset_len(SP._subsets[i]) > 1: 

SP2, SEP, sh = self._equitable_partition(SP._distinguish(bitset_first(SP._subsets[i]))) 

v = bitset_first(OP._subsets[i]) 

while v >= 0: 

OP2, OEP, oh = other._equitable_partition(OP._distinguish(v)) 

if sh == oh: 

m = self._equivalence(is_equiv, other, SP2, OP2) 

if m is not None: 

return m 

v = bitset_next(OP._subsets[i], v + 1) 

return None 

morph = dict([(self._groundset[bitset_first(SP._subsets[i])], other._groundset[bitset_first(OP._subsets[i])]) for i in xrange(len(SP))]) 

if is_equiv(self, other, morph): 

return morph 

cdef class SetSystemIterator: 

def __init__(self, H): 

""" 

Create an iterator for a SetSystem. 

Called internally when iterating over the contents of a SetSystem. 

EXAMPLES:: 

sage: from sage.matroids.set_system import SetSystem 

sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]]) 

sage: type(S.__iter__()) 

<... 'sage.matroids.set_system.SetSystemIterator'> 

""" 

self._H = H 

self._pointer = -1 

self._len = len(H) 

def __next__(self): 

""" 

Return the next subset of a SetSystem. 

EXAMPLES:: 

sage: from sage.matroids.set_system import SetSystem 

sage: S = SetSystem([1, 2, 3, 4], [[1, 2], [3, 4], [1, 2, 4]]) 

sage: I = S.__iter__() 

sage: sorted(I.__next__()) 

[1, 2] 

sage: sorted(I.__next__()) 

[3, 4] 

sage: sorted(I.__next__()) 

[1, 2, 4] 

""" 

self._pointer += 1 

if self._pointer == self._len: 

self._pointer = -1 

raise StopIteration 

else: 

return self._H.subset(self._pointer)