Mutable Poset¶
This module provides a class representing a finite partially ordered set (poset) for the purpose of being used as a data structure. Thus the posets introduced in this module are mutable, i.e., elements can be added and removed from a poset at any time.
To get in touch with Sage’s “usual” posets, start with the page
Posets
in the reference manual.
Examples¶
First Steps¶
We start by creating an empty poset. This is simply done by
sage: from sage.data_structures.mutable_poset import MutablePoset as MP
sage: P = MP()
sage: P
poset()
A poset should contain elements, thus let us add them with
sage: P.add(42)
sage: P.add(7)
sage: P.add(13)
sage: P.add(3)
Let us look at the poset again:
sage: P
poset(3, 7, 13, 42)
We see that they elements are sorted using \(\leq\) which exists on the integers \(\ZZ\). Since this is even a total order, we could have used a more efficient data structure. Alternatively, we can write
sage: MP([42, 7, 13, 3])
poset(3, 7, 13, 42)
to add several elements at once on construction.
A less boring Example¶
Let us continue with a less boring example. We define the class
sage: class T(tuple):
....: def __le__(left, right):
....: return all(l <= r for l, r in zip(left, right))
It is equipped with a \(\leq\)-operation such that \(a \leq b\) if all entries of \(a\) are at most the corresponding entry of \(b\). For example, we have
sage: a = T((1,1))
sage: b = T((2,1))
sage: c = T((1,2))
sage: a <= b, a <= c, b <= c
(True, True, False)
The last comparison gives False
, since the comparison of the
first component checks whether \(2 \leq 1\).
Now, let us add such elements to a poset:
sage: Q = MP([T((1, 1)), T((3, 3)), T((4, 1)),
....: T((3, 2)), T((2, 3)), T((2, 2))]); Q
poset((1, 1), (2, 2), (2, 3), (3, 2), (3, 3), (4, 1))
In the representation above, the elements are sorted topologically, smallest first. This does not (directly) show more structural information. We can overcome this and display a “wiring layout” by typing:
sage: print(Q.repr_full(reverse=True))
poset((3, 3), (2, 3), (3, 2), (2, 2), (4, 1), (1, 1))
+-- oo
| +-- no successors
| +-- predecessors: (3, 3), (4, 1)
+-- (3, 3)
| +-- successors: oo
| +-- predecessors: (2, 3), (3, 2)
+-- (2, 3)
| +-- successors: (3, 3)
| +-- predecessors: (2, 2)
+-- (3, 2)
| +-- successors: (3, 3)
| +-- predecessors: (2, 2)
+-- (2, 2)
| +-- successors: (2, 3), (3, 2)
| +-- predecessors: (1, 1)
+-- (4, 1)
| +-- successors: oo
| +-- predecessors: (1, 1)
+-- (1, 1)
| +-- successors: (2, 2), (4, 1)
| +-- predecessors: null
+-- null
| +-- successors: (1, 1)
| +-- no predecessors
Note that we use reverse=True
to let the elements appear from
largest (on the top) to smallest (on the bottom).
If you look at the output above, you’ll see two additional elements,
namely oo
(\(\infty\)) and null
(\(\emptyset\)). So what are these
strange animals? The answer is simple and maybe you can guess it
already. The \(\infty\)-element is larger than every other element,
therefore a successor of the maximal elements in the poset. Similarly,
the \(\emptyset\)-element is smaller than any other element, therefore a
predecessor of the poset’s minimal elements. Both do not have to scare
us; they are just there and sometimes useful.
AUTHORS:
Daniel Krenn (2015)
ACKNOWLEDGEMENT:
Daniel Krenn is supported by the Austrian Science Fund (FWF): P 24644-N26.
Classes and their Methods¶
- class sage.data_structures.mutable_poset.MutablePoset(data=None, key=None, merge=None, can_merge=None)¶
Bases:
sage.structure.sage_object.SageObject
A data structure that models a mutable poset (partially ordered set).
INPUT:
data
– data from which to construct the poset. It can be any of the following:None
(default), in which case an empty poset is created,a
MutablePoset
, which will be copied during creation,an iterable, whose elements will be in the poset.
key
– a function which maps elements to keys. IfNone
(default), this is the identity, i.e., keys are equal to their elements.Two elements with the same keys are considered as equal; so only one of these two elements can be in the poset.
This
key
is not used for sorting (in contrast to sorting-functions, e.g.sorted
).merge
– a function which merges its second argument (an element) to its first (again an element) and returns the result (as an element). If the return value isNone
, the element is removed from the poset.This hook is called by
merge()
. Moreover it is used duringadd()
when an element (more precisely its key) is already in this poset.merge
isNone
(default) is equivalent tomerge
returning its first argument. Note that it is not allowed that the key of the returning element differs from the key of the first input parameter. This meansmerge
must not change the position of the element in the poset.can_merge
– a function which checks whether its second argument can be merged to its first.This hook is called by
merge()
. Moreover it is used duringadd()
when an element (more precisely its key) is already in this poset.can_merge
isNone
(default) is equivalent tocan_merge
returningTrue
in all cases.
OUTPUT:
A mutable poset.
You can find a short introduction and examples
here
.EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP
We illustrate the different input formats
No input:
sage: A = MP(); A poset()
A
MutablePoset
:sage: B = MP(A); B poset() sage: B.add(42) sage: C = MP(B); C poset(42)
An iterable:
sage: C = MP([5, 3, 11]); C poset(3, 5, 11)
See also
- add(element)¶
Add the given object as element to the poset.
INPUT:
element
– an object (hashable and supporting comparison with the operator<=
).
OUTPUT:
Nothing.
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: class T(tuple): ....: def __le__(left, right): ....: return all(l <= r for l, r in zip(left, right)) sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)), ....: T((4, 4)), T((1, 2))]) sage: print(P.repr_full(reverse=True)) poset((4, 4), (1, 3), (1, 2), (2, 1), (1, 1)) +-- oo | +-- no successors | +-- predecessors: (4, 4) +-- (4, 4) | +-- successors: oo | +-- predecessors: (1, 3), (2, 1) +-- (1, 3) | +-- successors: (4, 4) | +-- predecessors: (1, 2) +-- (1, 2) | +-- successors: (1, 3) | +-- predecessors: (1, 1) +-- (2, 1) | +-- successors: (4, 4) | +-- predecessors: (1, 1) +-- (1, 1) | +-- successors: (1, 2), (2, 1) | +-- predecessors: null +-- null | +-- successors: (1, 1) | +-- no predecessors sage: P.add(T((2, 2))) sage: reprP = P.repr_full(reverse=True); print(reprP) poset((4, 4), (1, 3), (2, 2), (1, 2), (2, 1), (1, 1)) +-- oo | +-- no successors | +-- predecessors: (4, 4) +-- (4, 4) | +-- successors: oo | +-- predecessors: (1, 3), (2, 2) +-- (1, 3) | +-- successors: (4, 4) | +-- predecessors: (1, 2) +-- (2, 2) | +-- successors: (4, 4) | +-- predecessors: (1, 2), (2, 1) +-- (1, 2) | +-- successors: (1, 3), (2, 2) | +-- predecessors: (1, 1) +-- (2, 1) | +-- successors: (2, 2) | +-- predecessors: (1, 1) +-- (1, 1) | +-- successors: (1, 2), (2, 1) | +-- predecessors: null +-- null | +-- successors: (1, 1) | +-- no predecessors
When adding an element which is already in the poset, nothing happens:
sage: e = T((2, 2)) sage: P.add(e) sage: P.repr_full(reverse=True) == reprP True
We can influence the behavior when an element with existing key is to be inserted in the poset. For example, we can perform an addition on some argument of the elements:
sage: def add(left, right): ....: return (left[0], ''.join(sorted(left[1] + right[1]))) sage: A = MP(key=lambda k: k[0], merge=add) sage: A.add((3, 'a')) sage: A poset((3, 'a')) sage: A.add((3, 'b')) sage: A poset((3, 'ab'))
We can also deal with cancellations. If the return value of our hook-function is
None
, then the element is removed out of the poset:sage: def add_None(left, right): ....: s = left[1] + right[1] ....: if s == 0: ....: return None ....: return (left[0], s) sage: B = MP(key=lambda k: k[0], ....: merge=add_None) sage: B.add((7, 42)) sage: B.add((7, -42)) sage: B poset()
- clear()¶
Remove all elements from this poset.
INPUT:
Nothing.
OUTPUT:
Nothing.
- contains(key)¶
Test whether
key
is encapsulated by one of the poset’s elements.INPUT:
key
– an object.
OUTPUT:
True
orFalse
.See also
- copy(mapping=None)¶
Create a shallow copy.
INPUT:
mapping
– a function which is applied on each of the elements.
OUTPUT:
A poset with the same content as
self
.
- difference(*other)¶
Return a new poset where all elements of this poset, which are contained in one of the other given posets, are removed.
INPUT:
other
– a poset or an iterable. In the latter case the iterated objects are seen as elements of a poset. It is possible to specify more than oneother
as variadic arguments (arbitrary argument lists).
OUTPUT:
A poset.
Note
The key of an element is used for comparison. Thus elements with the same key are considered as equal.
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP([3, 42, 7]); P poset(3, 7, 42) sage: Q = MP([4, 8, 42]); Q poset(4, 8, 42) sage: P.difference(Q) poset(3, 7)
- difference_update(*other)¶
Remove all elements of another poset from this poset.
INPUT:
other
– a poset or an iterable. In the latter case the iterated objects are seen as elements of a poset. It is possible to specify more than oneother
as variadic arguments (arbitrary argument lists).
OUTPUT:
Nothing.
Note
The key of an element is used for comparison. Thus elements with the same key are considered as equal.
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP([3, 42, 7]); P poset(3, 7, 42) sage: Q = MP([4, 8, 42]); Q poset(4, 8, 42) sage: P.difference_update(Q) sage: P poset(3, 7)
- discard(key, raise_key_error=False)¶
Remove the given object from the poset.
INPUT:
key
– the key of an object.raise_key_error
– (default:False
) switch raisingKeyError
on and off.
OUTPUT:
Nothing.
If the element is not a member and
raise_key_error
is set (not default), raise aKeyError
.Note
As with Python’s
set
, the methodsremove()
anddiscard()
only differ in their behavior when an element is not contained in the poset:remove()
raises aKeyError
whereasdiscard()
does not raise any exception.This default behavior can be overridden with the
raise_key_error
parameter.EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: class T(tuple): ....: def __le__(left, right): ....: return all(l <= r for l, r in zip(left, right)) sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)), ....: T((4, 4)), T((1, 2)), T((2, 2))]) sage: P.discard(T((1, 2))) sage: P.remove(T((1, 2))) Traceback (most recent call last): ... KeyError: 'Key (1, 2) is not contained in this poset.' sage: P.discard(T((1, 2)))
- element(key)¶
Return the element corresponding to
key
.INPUT:
key
– the key of an object.OUTPUT:
An object.
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP() sage: P.add(42) sage: e = P.element(42); e 42 sage: type(e) <type 'sage.rings.integer.Integer'>
- elements(**kwargs)¶
Return an iterator over all elements.
INPUT:
kwargs
– arguments are passed toshells()
.
OUTPUT:
An iterator.
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP([3, 42, 7]) sage: [(v, type(v)) for v in sorted(P.elements())] [(3, <type 'sage.rings.integer.Integer'>), (7, <type 'sage.rings.integer.Integer'>), (42, <type 'sage.rings.integer.Integer'>)]
Note that
sage: it = iter(P) sage: sorted(it) [3, 7, 42]
returns all elements as well.
- elements_topological(**kwargs)¶
Return an iterator over all elements in topological order.
INPUT:
kwargs
– arguments are passed toshells_topological()
.
OUTPUT:
An iterator.
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: class T(tuple): ....: def __le__(left, right): ....: return all(l <= r for l, r in zip(left, right)) sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)), ....: T((4, 4)), T((1, 2)), T((2, 2))]) sage: [(v, type(v)) for v in P.elements_topological(key=repr)] [((1, 1), <class '__main__.T'>), ((1, 2), <class '__main__.T'>), ((1, 3), <class '__main__.T'>), ((2, 1), <class '__main__.T'>), ((2, 2), <class '__main__.T'>), ((4, 4), <class '__main__.T'>)]
- get_key(element)¶
Return the key corresponding to the given element.
INPUT:
element
– an object.
OUTPUT:
An object (the key of
element
).
- intersection(*other)¶
Return the intersection of the given posets as a new poset
INPUT:
other
– a poset or an iterable. In the latter case the iterated objects are seen as elements of a poset. It is possible to specify more than oneother
as variadic arguments (arbitrary argument lists).
OUTPUT:
A poset.
Note
The key of an element is used for comparison. Thus elements with the same key are considered as equal.
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP([3, 42, 7]); P poset(3, 7, 42) sage: Q = MP([4, 8, 42]); Q poset(4, 8, 42) sage: P.intersection(Q) poset(42)
- intersection_update(*other)¶
Update this poset with the intersection of itself and another poset.
INPUT:
other
– a poset or an iterable. In the latter case the iterated objects are seen as elements of a poset. It is possible to specify more than oneother
as variadic arguments (arbitrary argument lists).
OUTPUT:
Nothing.
Note
The key of an element is used for comparison. Thus elements with the same key are considered as equal;
A.intersection_update(B)
andB.intersection_update(A)
might result in different posets.EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP([3, 42, 7]); P poset(3, 7, 42) sage: Q = MP([4, 8, 42]); Q poset(4, 8, 42) sage: P.intersection_update(Q) sage: P poset(42)
- is_disjoint(other)¶
Return whether another poset is disjoint to this poset.
INPUT:
other
– a poset or an iterable. In the latter case the iterated objects are seen as elements of a poset.
OUTPUT:
Nothing.
Note
If this poset uses a
key
-function, then all comparisons are performed on the keys of the elements (and not on the elements themselves).EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP([3, 42, 7]); P poset(3, 7, 42) sage: Q = MP([4, 8, 42]); Q poset(4, 8, 42) sage: P.is_disjoint(Q) False sage: P.is_disjoint(Q.difference(P)) True
- is_subset(other)¶
Return whether another poset contains this poset, i.e., whether this poset is a subset of the other poset.
INPUT:
other
– a poset or an iterable. In the latter case the iterated objects are seen as elements of a poset.
OUTPUT:
Nothing.
Note
If this poset uses a
key
-function, then all comparisons are performed on the keys of the elements (and not on the elements themselves).EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP([3, 42, 7]); P poset(3, 7, 42) sage: Q = MP([4, 8, 42]); Q poset(4, 8, 42) sage: P.is_subset(Q) False sage: Q.is_subset(P) False sage: P.is_subset(P) True sage: P.is_subset(P.union(Q)) True
- is_superset(other)¶
Return whether this poset contains another poset, i.e., whether this poset is a superset of the other poset.
INPUT:
other
– a poset or an iterable. In the latter case the iterated objects are seen as elements of a poset.
OUTPUT:
Nothing.
Note
If this poset uses a
key
-function, then all comparisons are performed on the keys of the elements (and not on the elements themselves).EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP([3, 42, 7]); P poset(3, 7, 42) sage: Q = MP([4, 8, 42]); Q poset(4, 8, 42) sage: P.is_superset(Q) False sage: Q.is_superset(P) False sage: P.is_superset(P) True sage: P.union(Q).is_superset(P) True
- isdisjoint = <function MutablePoset.is_disjoint>¶
Alias of
is_disjoint()
.
- issubset = <function MutablePoset.is_subset>¶
Alias of
is_subset()
.
- issuperset = <function MutablePoset.is_superset>¶
Alias of
is_superset()
.
- keys(**kwargs)¶
Return an iterator over all keys of the elements.
INPUT:
kwargs
– arguments are passed toshells()
.
OUTPUT:
An iterator.
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP([3, 42, 7], key=lambda c: -c) sage: [(v, type(v)) for v in sorted(P.keys())] [(-42, <type 'sage.rings.integer.Integer'>), (-7, <type 'sage.rings.integer.Integer'>), (-3, <type 'sage.rings.integer.Integer'>)] sage: [(v, type(v)) for v in sorted(P.elements())] [(3, <type 'sage.rings.integer.Integer'>), (7, <type 'sage.rings.integer.Integer'>), (42, <type 'sage.rings.integer.Integer'>)] sage: [(v, type(v)) for v in sorted(P.shells(), ....: key=lambda c: c.element)] [(3, <class 'sage.data_structures.mutable_poset.MutablePosetShell'>), (7, <class 'sage.data_structures.mutable_poset.MutablePosetShell'>), (42, <class 'sage.data_structures.mutable_poset.MutablePosetShell'>)]
- keys_topological(**kwargs)¶
Return an iterator over all keys of the elements in topological order.
INPUT:
kwargs
– arguments are passed toshells_topological()
.
OUTPUT:
An iterator.
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP([(1, 1), (2, 1), (4, 4)], ....: key=lambda c: c[0]) sage: [(v, type(v)) for v in P.keys_topological(key=repr)] [(1, <type 'sage.rings.integer.Integer'>), (2, <type 'sage.rings.integer.Integer'>), (4, <type 'sage.rings.integer.Integer'>)] sage: [(v, type(v)) for v in P.elements_topological(key=repr)] [((1, 1), <... 'tuple'>), ((2, 1), <... 'tuple'>), ((4, 4), <... 'tuple'>)] sage: [(v, type(v)) for v in P.shells_topological(key=repr)] [((1, 1), <class 'sage.data_structures.mutable_poset.MutablePosetShell'>), ((2, 1), <class 'sage.data_structures.mutable_poset.MutablePosetShell'>), ((4, 4), <class 'sage.data_structures.mutable_poset.MutablePosetShell'>)]
- map(function, topological=False, reverse=False)¶
Apply the given
function
to each element of this poset.INPUT:
function
– a function mapping an existing element to a new element.topological
– (default:False
) if set, then the mapping is done in topological order, otherwise unordered.reverse
– is passed on to topological ordering.
OUTPUT:
Nothing.
Note
Since this method works inplace, it is not allowed that
function
alters the key of an element.Note
If
function
returnsNone
, then the element is removed.EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: class T(tuple): ....: def __le__(left, right): ....: return all(l <= r for l, r in zip(left, right)) sage: P = MP([T((1, 3)), T((2, 1)), ....: T((4, 4)), T((1, 2)), T((2, 2))], ....: key=lambda e: e[:2]) sage: P.map(lambda e: e + (sum(e),)) sage: P poset((1, 2, 3), (1, 3, 4), (2, 1, 3), (2, 2, 4), (4, 4, 8))
- mapped(function)¶
Return a poset where on each element the given
function
was applied.INPUT:
function
– a function mapping an existing element to a new element.topological
– (default:False
) if set, then the mapping is done in topological order, otherwise unordered.reverse
– is passed on to topological ordering.
OUTPUT:
A
MutablePoset
.Note
function
is not allowed to change the order of the keys, but changing the keys themselves is allowed (in contrast tomap()
).EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: class T(tuple): ....: def __le__(left, right): ....: return all(l <= r for l, r in zip(left, right)) sage: P = MP([T((1, 3)), T((2, 1)), ....: T((4, 4)), T((1, 2)), T((2, 2))]) sage: P.mapped(lambda e: str(e)) poset('(1, 2)', '(1, 3)', '(2, 1)', '(2, 2)', '(4, 4)')
- maximal_elements()¶
Return an iterator over the maximal elements of this poset.
INPUT:
Nothing.
OUTPUT:
An iterator.
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: class T(tuple): ....: def __le__(left, right): ....: return all(l <= r for l, r in zip(left, right)) sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)), ....: T((1, 2)), T((2, 2))]) sage: sorted(P.maximal_elements()) [(1, 3), (2, 2)]
See also
- merge(key=None, reverse=False)¶
Merge the given element with its successors/predecessors.
INPUT:
key
– the key specifying an element orNone
(default), in which case this method is called on each element in this poset.reverse
– (default:False
) specifies which direction to go first:False
searches towards'oo'
andTrue
searches towards'null'
. Whenkey=None
, then this also specifies which elements are merged first.
OUTPUT:
Nothing.
This method tests all (not necessarily direct) successors and predecessors of the given element whether they can be merged with the element itself. This is done by the
can_merge
-function ofMutablePoset
. If this merge is possible, then it is performed by callingMutablePoset
’smerge
-function and the corresponding successor/predecessor is removed from the poset.Note
can_merge
is applied in the sense of the condition of depth first iteration, i.e., oncecan_merge
fails, the successors/predecessors are no longer tested.Note
The motivation for such a merge behavior comes from asymptotic expansions: \(O(n^3)\) merges with, for example, \(3n^2\) or \(O(n)\) to \(O(n^3)\) (as \(n\) tends to \(\infty\); see Wikipedia article Big_O_notation).
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: class T(tuple): ....: def __le__(left, right): ....: return all(l <= r for l, r in zip(left, right)) sage: key = lambda t: T(t[0:2]) sage: def add(left, right): ....: return (left[0], left[1], ....: ''.join(sorted(left[2] + right[2]))) sage: def can_add(left, right): ....: return key(left) >= key(right) sage: P = MP([(1, 1, 'a'), (1, 3, 'b'), (2, 1, 'c'), ....: (4, 4, 'd'), (1, 2, 'e'), (2, 2, 'f')], ....: key=key, merge=add, can_merge=can_add) sage: Q = copy(P) sage: Q.merge(T((1, 3))) sage: print(Q.repr_full(reverse=True)) poset((4, 4, 'd'), (1, 3, 'abe'), (2, 2, 'f'), (2, 1, 'c')) +-- oo | +-- no successors | +-- predecessors: (4, 4, 'd') +-- (4, 4, 'd') | +-- successors: oo | +-- predecessors: (1, 3, 'abe'), (2, 2, 'f') +-- (1, 3, 'abe') | +-- successors: (4, 4, 'd') | +-- predecessors: null +-- (2, 2, 'f') | +-- successors: (4, 4, 'd') | +-- predecessors: (2, 1, 'c') +-- (2, 1, 'c') | +-- successors: (2, 2, 'f') | +-- predecessors: null +-- null | +-- successors: (1, 3, 'abe'), (2, 1, 'c') | +-- no predecessors sage: for k in sorted(P.keys()): ....: Q = copy(P) ....: Q.merge(k) ....: print('merging %s: %s' % (k, Q)) merging (1, 1): poset((1, 1, 'a'), (1, 2, 'e'), (1, 3, 'b'), (2, 1, 'c'), (2, 2, 'f'), (4, 4, 'd')) merging (1, 2): poset((1, 2, 'ae'), (1, 3, 'b'), (2, 1, 'c'), (2, 2, 'f'), (4, 4, 'd')) merging (1, 3): poset((1, 3, 'abe'), (2, 1, 'c'), (2, 2, 'f'), (4, 4, 'd')) merging (2, 1): poset((1, 2, 'e'), (1, 3, 'b'), (2, 1, 'ac'), (2, 2, 'f'), (4, 4, 'd')) merging (2, 2): poset((1, 3, 'b'), (2, 2, 'acef'), (4, 4, 'd')) merging (4, 4): poset((4, 4, 'abcdef')) sage: Q = copy(P) sage: Q.merge(); Q poset((4, 4, 'abcdef'))
See also
- minimal_elements()¶
Return an iterator over the minimal elements of this poset.
INPUT:
Nothing.
OUTPUT:
An iterator.
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: class T(tuple): ....: def __le__(left, right): ....: return all(l <= r for l, r in zip(left, right)) sage: P = MP([T((1, 3)), T((2, 1)), ....: T((4, 4)), T((1, 2)), T((2, 2))]) sage: sorted(P.minimal_elements()) [(1, 2), (2, 1)]
See also
- null¶
The shell \(\emptyset\) whose element is smaller than any other element.
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP() sage: z = P.null; z null sage: z.is_null() True
- oo¶
The shell \(\infty\) whose element is larger than any other element.
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP() sage: oo = P.oo; oo oo sage: oo.is_oo() True
- pop(**kwargs)¶
Remove and return an arbitrary poset element.
INPUT:
kwargs
– arguments are passed toshells_topological()
.
OUTPUT:
An object.
Note
The special elements
'null'
and'oo'
cannot be popped.EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP() sage: P.add(3) sage: P poset(3) sage: P.pop() 3 sage: P poset() sage: P.pop() Traceback (most recent call last): ... KeyError: 'pop from an empty poset'
- remove(key, raise_key_error=True)¶
Remove the given object from the poset.
INPUT:
key
– the key of an object.raise_key_error
– (default:True
) switch raisingKeyError
on and off.
OUTPUT:
Nothing.
If the element is not a member and
raise_key_error
is set (default), raise aKeyError
.Note
As with Python’s
set
, the methodsremove()
anddiscard()
only differ in their behavior when an element is not contained in the poset:remove()
raises aKeyError
whereasdiscard()
does not raise any exception.This default behavior can be overridden with the
raise_key_error
parameter.EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: class T(tuple): ....: def __le__(left, right): ....: return all(l <= r for l, r in zip(left, right)) sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)), ....: T((4, 4)), T((1, 2)), T((2, 2))]) sage: print(P.repr_full(reverse=True)) poset((4, 4), (1, 3), (2, 2), (1, 2), (2, 1), (1, 1)) +-- oo | +-- no successors | +-- predecessors: (4, 4) +-- (4, 4) | +-- successors: oo | +-- predecessors: (1, 3), (2, 2) +-- (1, 3) | +-- successors: (4, 4) | +-- predecessors: (1, 2) +-- (2, 2) | +-- successors: (4, 4) | +-- predecessors: (1, 2), (2, 1) +-- (1, 2) | +-- successors: (1, 3), (2, 2) | +-- predecessors: (1, 1) +-- (2, 1) | +-- successors: (2, 2) | +-- predecessors: (1, 1) +-- (1, 1) | +-- successors: (1, 2), (2, 1) | +-- predecessors: null +-- null | +-- successors: (1, 1) | +-- no predecessors sage: P.remove(T((1, 2))) sage: print(P.repr_full(reverse=True)) poset((4, 4), (1, 3), (2, 2), (2, 1), (1, 1)) +-- oo | +-- no successors | +-- predecessors: (4, 4) +-- (4, 4) | +-- successors: oo | +-- predecessors: (1, 3), (2, 2) +-- (1, 3) | +-- successors: (4, 4) | +-- predecessors: (1, 1) +-- (2, 2) | +-- successors: (4, 4) | +-- predecessors: (2, 1) +-- (2, 1) | +-- successors: (2, 2) | +-- predecessors: (1, 1) +-- (1, 1) | +-- successors: (1, 3), (2, 1) | +-- predecessors: null +-- null | +-- successors: (1, 1) | +-- no predecessors
- repr(include_special=False, reverse=False)¶
Return a representation of the poset.
INPUT:
include_special
– (default:False
) a boolean indicating whether to include the special elements'null'
and'oo'
or not.reverse
– (default:False
) a boolean. If set, then largest elements are displayed first.
OUTPUT:
A string.
See also
- repr_full(reverse=False)¶
Return a representation with ordering details of the poset.
INPUT:
reverse
– (default:False
) a boolean. If set, then largest elements are displayed first.
OUTPUT:
A string.
See also
- shell(key)¶
Return the shell of the element corresponding to
key
.INPUT:
key
– the key of an object.OUTPUT:
An instance of
MutablePosetShell
.Note
Each element is contained/encapsulated in a shell inside the poset.
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP() sage: P.add(42) sage: e = P.shell(42); e 42 sage: type(e) <class 'sage.data_structures.mutable_poset.MutablePosetShell'>
- shells(include_special=False)¶
Return an iterator over all shells.
INPUT:
include_special
– (default:False
) if set, then including shells containing a smallest element (\(\emptyset\)) and a largest element (\(\infty\)).
OUTPUT:
An iterator.
Note
Each element is contained/encapsulated in a shell inside the poset.
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP() sage: tuple(P.shells()) () sage: tuple(P.shells(include_special=True)) (null, oo)
- shells_topological(include_special=False, reverse=False, key=None)¶
Return an iterator over all shells in topological order.
INPUT:
include_special
– (default:False
) if set, then including shells containing a smallest element (\(\emptyset\)) and a largest element (\(\infty\)).reverse
– (default:False
) – if set, reverses the order, i.e.,False
gives smallest elements first,True
gives largest first.key
– (default:None
) a function used for sorting the direct successors of a shell (used in case of a tie). If this isNone
, no sorting occurs.
OUTPUT:
An iterator.
Note
Each element is contained/encapsulated in a shell inside the poset.
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: class T(tuple): ....: def __le__(left, right): ....: return all(l <= r for l, r in zip(left, right)) sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)), ....: T((4, 4)), T((1, 2)), T((2, 2))]) sage: list(P.shells_topological(key=repr)) [(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (4, 4)] sage: list(P.shells_topological(reverse=True, key=repr)) [(4, 4), (1, 3), (2, 2), (1, 2), (2, 1), (1, 1)] sage: list(P.shells_topological(include_special=True, key=repr)) [null, (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (4, 4), oo] sage: list(P.shells_topological( ....: include_special=True, reverse=True, key=repr)) [oo, (4, 4), (1, 3), (2, 2), (1, 2), (2, 1), (1, 1), null]
- symmetric_difference(other)¶
Return the symmetric difference of two posets as a new poset.
INPUT:
other
– a poset.
OUTPUT:
A poset.
Note
The key of an element is used for comparison. Thus elements with the same key are considered as equal.
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP([3, 42, 7]); P poset(3, 7, 42) sage: Q = MP([4, 8, 42]); Q poset(4, 8, 42) sage: P.symmetric_difference(Q) poset(3, 4, 7, 8)
- symmetric_difference_update(other)¶
Update this poset with the symmetric difference of itself and another poset.
INPUT:
other
– a poset.
OUTPUT:
Nothing.
Note
The key of an element is used for comparison. Thus elements with the same key are considered as equal;
A.symmetric_difference_update(B)
andB.symmetric_difference_update(A)
might result in different posets.EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP([3, 42, 7]); P poset(3, 7, 42) sage: Q = MP([4, 8, 42]); Q poset(4, 8, 42) sage: P.symmetric_difference_update(Q) sage: P poset(3, 4, 7, 8)
- union(*other)¶
Return the union of the given posets as a new poset
INPUT:
other
– a poset or an iterable. In the latter case the iterated objects are seen as elements of a poset. It is possible to specify more than oneother
as variadic arguments (arbitrary argument lists).
OUTPUT:
A poset.
Note
The key of an element is used for comparison. Thus elements with the same key are considered as equal.
Due to keys and a
merge
function (seeMutablePoset
) this operation might not be commutative.Todo
Use the already existing information in the other poset to speed up this function. (At the moment each element of the other poset is inserted one by one and without using this information.)
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP([3, 42, 7]); P poset(3, 7, 42) sage: Q = MP([4, 8, 42]); Q poset(4, 8, 42) sage: P.union(Q) poset(3, 4, 7, 8, 42)
- union_update(*other)¶
Update this poset with the union of itself and another poset.
INPUT:
other
– a poset or an iterable. In the latter case the iterated objects are seen as elements of a poset. It is possible to specify more than oneother
as variadic arguments (arbitrary argument lists).
OUTPUT:
Nothing.
Note
The key of an element is used for comparison. Thus elements with the same key are considered as equal;
A.union_update(B)
andB.union_update(A)
might result in different posets.Todo
Use the already existing information in the other poset to speed up this function. (At the moment each element of the other poset is inserted one by one and without using this information.)
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP([3, 42, 7]); P poset(3, 7, 42) sage: Q = MP([4, 8, 42]); Q poset(4, 8, 42) sage: P.union_update(Q) sage: P poset(3, 4, 7, 8, 42)
- update = <function MutablePoset.union_update>¶
Alias of
union_update()
.
- class sage.data_structures.mutable_poset.MutablePosetShell(poset, element)¶
Bases:
sage.structure.sage_object.SageObject
A shell for an element of a
mutable poset
.INPUT:
poset
– the poset to which this shell belongs.element
– the element which should be contained/encapsulated in this shell.
OUTPUT:
A shell for the given element.
Note
If the
element()
of a shell isNone
, then this element is considered as “special” (seeis_special()
). There are two special elements, namelya
'null'
(an element smaller than each other element; it has no predecessors) andan
'oo'
(an element larger than each other element; it has no successors).
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: P = MP() sage: P.add(66) sage: P poset(66) sage: s = P.shell(66) sage: type(s) <class 'sage.data_structures.mutable_poset.MutablePosetShell'>
See also
- element¶
The element contained in this shell.
See also
- eq(other)¶
Return whether this shell is equal to
other
.INPUT:
other
– a shell.
OUTPUT:
True
orFalse
.Note
This method compares the keys of the elements contained in the (non-special) shells. In particular, elements/shells with the same key are considered as equal.
See also
- is_null()¶
Return whether this shell contains the null-element, i.e., the element smaller than any possible other element.
OUTPUT:
True
orFalse
.See also
- is_oo()¶
Return whether this shell contains the infinity-element, i.e., the element larger than any possible other element.
OUTPUT:
True
orFalse
.See also
- is_special()¶
Return whether this shell contains either the null-element, i.e., the element smaller than any possible other element or the infinity-element, i.e., the element larger than any possible other element.
INPUT:
Nothing.
OUTPUT:
True
orFalse
.See also
- iter_depth_first(reverse=False, key=None, condition=None)¶
Iterate over all shells in depth first order.
INPUT:
reverse
– (default:False
) if set, reverses the order, i.e.,False
searches towards'oo'
andTrue
searches towards'null'
.key
– (default:None
) a function used for sorting the direct successors of a shell (used in case of a tie). If this isNone
, no sorting occurs.condition
– (default:None
) a function mapping a shell toTrue
(include in iteration) orFalse
(do not include).None
is equivalent to a function returning alwaysTrue
. Note that the iteration does not go beyond a not included shell.
OUTPUT:
An iterator.
Note
The depth first search starts at this (
self
) shell. Thus only this shell and shells greater than (in case ofreverse=False
) this shell are visited.ALGORITHM:
See Wikipedia article Depth-first_search.
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: class T(tuple): ....: def __le__(left, right): ....: return all(l <= r for l, r in zip(left, right)) sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)), ....: T((4, 4)), T((1, 2)), T((2, 2))]) sage: list(P.null.iter_depth_first(reverse=False, key=repr)) [null, (1, 1), (1, 2), (1, 3), (4, 4), oo, (2, 2), (2, 1)] sage: list(P.oo.iter_depth_first(reverse=True, key=repr)) [oo, (4, 4), (1, 3), (1, 2), (1, 1), null, (2, 2), (2, 1)] sage: list(P.null.iter_depth_first( ....: condition=lambda s: s.element[0] == 1)) [null, (1, 1), (1, 2), (1, 3)]
See also
- iter_topological(reverse=False, key=None, condition=None)¶
Iterate over all shells in topological order.
INPUT:
reverse
– (default:False
) if set, reverses the order, i.e.,False
searches towards'oo'
andTrue
searches towards'null'
.key
– (default:None
) a function used for sorting the direct predecessors of a shell (used in case of a tie). If this isNone
, no sorting occurs.condition
– (default:None
) a function mapping a shell toTrue
(include in iteration) orFalse
(do not include).None
is equivalent to a function returning alwaysTrue
. Note that the iteration does not go beyond a not included shell.
OUTPUT:
An iterator.
Note
The topological search will only find shells smaller than (in case of
reverse=False
) or equal to this (self
) shell. This is in contrast toiter_depth_first()
.ALGORITHM:
Here a simplified version of the algorithm found in [Tar1976] and [CLRS2001] is used. See also Wikipedia article Topological_sorting.
EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: class T(tuple): ....: def __le__(left, right): ....: return all(l <= r for l, r in zip(left, right)) sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)), ....: T((4, 4)), T((1, 2)), T((2, 2))])
sage: for e in P.shells_topological(include_special=True, ....: reverse=True, key=repr): ....: print(e) ....: print(list(e.iter_topological(reverse=True, key=repr))) oo [oo] (4, 4) [oo, (4, 4)] (1, 3) [oo, (4, 4), (1, 3)] (2, 2) [oo, (4, 4), (2, 2)] (1, 2) [oo, (4, 4), (1, 3), (2, 2), (1, 2)] (2, 1) [oo, (4, 4), (2, 2), (2, 1)] (1, 1) [oo, (4, 4), (1, 3), (2, 2), (1, 2), (2, 1), (1, 1)] null [oo, (4, 4), (1, 3), (2, 2), (1, 2), (2, 1), (1, 1), null]
sage: for e in P.shells_topological(include_special=True, ....: reverse=True, key=repr): ....: print(e) ....: print(list(e.iter_topological(reverse=False, key=repr))) oo [null, (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (4, 4), oo] (4, 4) [null, (1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (4, 4)] (1, 3) [null, (1, 1), (1, 2), (1, 3)] (2, 2) [null, (1, 1), (1, 2), (2, 1), (2, 2)] (1, 2) [null, (1, 1), (1, 2)] (2, 1) [null, (1, 1), (2, 1)] (1, 1) [null, (1, 1)] null [null]
sage: list(P.null.iter_topological( ....: reverse=True, condition=lambda s: s.element[0] == 1, ....: key=repr)) [(1, 3), (1, 2), (1, 1), null]
- key¶
The key of the element contained in this shell.
The key of an element is determined by the mutable poset (the parent) via the
key
-function (see construction of aMutablePoset
).See also
- le(other, reverse=False)¶
Return whether this shell is less than or equal to
other
.INPUT:
other
– a shell.reverse
– (default:False
) if set, then return whether this shell is greater than or equal toother
.
OUTPUT:
True
orFalse
.Note
The comparison of the shells is based on the comparison of the keys of the elements contained in the shells, except for special shells (see
MutablePosetShell
).See also
- lower_covers(shell, reverse=False)¶
Return the lower covers of the specified
shell
; the search is started at this (self
) shell.A lower cover of \(x\) is an element \(y\) of the poset such that \(y < x\) and there is no element \(z\) of the poset so that \(y < z < x\).
INPUT:
shell
– the shell for which to find the covering shells. There is no restriction ofshell
being contained in the poset. Ifshell
is contained in the poset, then use the more efficient methodspredecessors()
andsuccessors()
.reverse
– (default:False
) if set, then find the upper covers (see alsoupper_covers()
) instead of the lower covers.
OUTPUT:
A set of
shells
.Note
Suppose
reverse
isFalse
. This method starts at the calling shell (self
) and searches towards'oo'
. Thus, only shells which are (not necessarily direct) successors of this shell are considered.If
reverse
isTrue
, then the reverse direction is taken.EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: class T(tuple): ....: def __le__(left, right): ....: return all(l <= r for l, r in zip(left, right)) sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)), ....: T((4, 4)), T((1, 2)), T((2, 2))]) sage: e = P.shell(T((2, 2))); e (2, 2) sage: sorted(P.null.lower_covers(e), ....: key=lambda c: repr(c.element)) [(1, 2), (2, 1)] sage: set(_) == e.predecessors() True sage: sorted(P.oo.upper_covers(e), ....: key=lambda c: repr(c.element)) [(4, 4)] sage: set(_) == e.successors() True
sage: Q = MP([T((3, 2))]) sage: f = next(Q.shells()) sage: sorted(P.null.lower_covers(f), ....: key=lambda c: repr(c.element)) [(2, 2)] sage: sorted(P.oo.upper_covers(f), ....: key=lambda c: repr(c.element)) [(4, 4)]
See also
- merge(element, check=True, delete=True)¶
Merge the given element with the element contained in this shell.
INPUT:
element
– an element (of the poset).check
– (default:True
) if set, then thecan_merge
-function ofMutablePoset
determines whether the merge is possible.can_merge
isNone
means that this check is always passed.delete
– (default:True
) if set, thenelement
is removed from the poset after the merge.
OUTPUT:
Nothing.
Note
This operation depends on the parameters
merge
andcan_merge
of theMutablePoset
this shell is contained in. These parameters are defined when the poset is constructed.Note
If the
merge
function returnsNone
, then this shell is removed from the poset.EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: def add(left, right): ....: return (left[0], ''.join(sorted(left[1] + right[1]))) sage: def can_add(left, right): ....: return left[0] <= right[0] sage: P = MP([(1, 'a'), (3, 'b'), (2, 'c'), (4, 'd')], ....: key=lambda c: c[0], merge=add, can_merge=can_add) sage: P poset((1, 'a'), (2, 'c'), (3, 'b'), (4, 'd')) sage: P.shell(2).merge((3, 'b')) sage: P poset((1, 'a'), (2, 'bc'), (4, 'd'))
See also
- poset¶
The poset to which this shell belongs.
See also
- predecessors(reverse=False)¶
Return the predecessors of this shell.
INPUT:
reverse
– (default:False
) if set, then return successors instead.
OUTPUT:
A set.
See also
- successors(reverse=False)¶
Return the successors of this shell.
INPUT:
reverse
– (default:False
) if set, then return predecessors instead.
OUTPUT:
A set.
See also
- upper_covers(shell, reverse=False)¶
Return the upper covers of the specified
shell
; the search is started at this (self
) shell.An upper cover of \(x\) is an element \(y\) of the poset such that \(x < y\) and there is no element \(z\) of the poset so that \(x < z < y\).
INPUT:
shell
– the shell for which to find the covering shells. There is no restriction ofshell
being contained in the poset. Ifshell
is contained in the poset, then use the more efficient methodspredecessors()
andsuccessors()
.reverse
– (default:False
) if set, then find the lower covers (see alsolower_covers()
) instead of the upper covers.
OUTPUT:
A set of
shells
.Note
Suppose
reverse
isFalse
. This method starts at the calling shell (self
) and searches towards'null'
. Thus, only shells which are (not necessarily direct) predecessors of this shell are considered.If
reverse
isTrue
, then the reverse direction is taken.EXAMPLES:
sage: from sage.data_structures.mutable_poset import MutablePoset as MP sage: class T(tuple): ....: def __le__(left, right): ....: return all(l <= r for l, r in zip(left, right)) sage: P = MP([T((1, 1)), T((1, 3)), T((2, 1)), ....: T((4, 4)), T((1, 2)), T((2, 2))]) sage: e = P.shell(T((2, 2))); e (2, 2) sage: sorted(P.null.lower_covers(e), ....: key=lambda c: repr(c.element)) [(1, 2), (2, 1)] sage: set(_) == e.predecessors() True sage: sorted(P.oo.upper_covers(e), ....: key=lambda c: repr(c.element)) [(4, 4)] sage: set(_) == e.successors() True
sage: Q = MP([T((3, 2))]) sage: f = next(Q.shells()) sage: sorted(P.null.lower_covers(f), ....: key=lambda c: repr(c.element)) [(2, 2)] sage: sorted(P.oo.upper_covers(f), ....: key=lambda c: repr(c.element)) [(4, 4)]
See also
- sage.data_structures.mutable_poset.is_MutablePoset(P)¶
Test whether
P
inherits fromMutablePoset
.See also