Data structures for maps between finite sets¶
This module implements several fast Cython data structures for maps
between two finite set. Those classes are not intended to be used
directly. Instead, such a map should be constructed via its parent,
using the class FiniteSetMaps.
EXAMPLES:
To create a map between two sets, one first creates the set of such maps:
sage: M = FiniteSetMaps(["a", "b"], [3, 4, 5])
The map can then be constructed either from a function:
sage: f1 = M(lambda c: ord(c)-94); f1
map: a -> 3, b -> 4
or from a dictionary:
sage: f2 = M.from_dict({'a':3, 'b':4}); f2
map: a -> 3, b -> 4
The two created maps are equal:
sage: f1 == f2
True
Internally, maps are represented as the list of the ranks of the
images f(x) in the co-domain, in the order of the domain:
sage: list(f2)
[0, 1]
A third fast way to create a map it to use such a list. it should be kept for internal use:
sage: f3 = M._from_list_([0, 1]); f3
map: a -> 3, b -> 4
sage: f1 == f3
True
AUTHORS:
Florent Hivert
- class sage.sets.finite_set_map_cy.FiniteSetEndoMap_N¶
Bases:
sage.sets.finite_set_map_cy.FiniteSetMap_MNMaps from
range(n)to itself.See also
FiniteSetMap_MNfor assumptions on the parent
- class sage.sets.finite_set_map_cy.FiniteSetEndoMap_Set¶
Bases:
sage.sets.finite_set_map_cy.FiniteSetMap_SetMaps from a set to itself
See also
FiniteSetMap_Setfor assumptions on the parent
- class sage.sets.finite_set_map_cy.FiniteSetMap_MN¶
Bases:
sage.structure.list_clone.ClonableIntArrayData structure for maps from
range(m)torange(n).We assume that the parent given as argument is such that:
mis stored inself.parent()._mnis stored inself.parent()._nthe domain is in
self.parent().domain()the codomain is in
self.parent().codomain()
- check()¶
Performs checks on
selfCheck that
selfis a proper function and then callsparent.check_element(self)whereparentis the parent ofself.
- codomain()¶
Returns the codomain of
selfEXAMPLES:
sage: FiniteSetMaps(4, 3)([1, 0, 2, 1]).codomain() {0, 1, 2}
- domain()¶
Returns the domain of
selfEXAMPLES:
sage: FiniteSetMaps(4, 3)([1, 0, 2, 1]).domain() {0, 1, 2, 3}
- fibers()¶
Returns the fibers of
selfOUTPUT:
a dictionary
dsuch thatd[y]is the set of allxindomainsuch thatf(x) = yEXAMPLES:
sage: FiniteSetMaps(4, 3)([1, 0, 2, 1]).fibers() {0: {1}, 1: {0, 3}, 2: {2}} sage: F = FiniteSetMaps(["a", "b", "c"]) sage: F.from_dict({"a": "b", "b": "a", "c": "b"}).fibers() == {'a': {'b'}, 'b': {'a', 'c'}} True
- getimage(i)¶
Returns the image of
ibyselfINPUT:
i– any object.
Note
if you need speed, please use instead
_getimage()EXAMPLES:
sage: fs = FiniteSetMaps(4, 3)([1, 0, 2, 1]) sage: fs.getimage(0), fs.getimage(1), fs.getimage(2), fs.getimage(3) (1, 0, 2, 1)
- image_set()¶
Returns the image set of
selfEXAMPLES:
sage: FiniteSetMaps(4, 3)([1, 0, 2, 1]).image_set() {0, 1, 2} sage: FiniteSetMaps(4, 3)([1, 0, 0, 1]).image_set() {0, 1}
- items()¶
The items of
selfReturn the list of the ordered pairs
(x, self(x))EXAMPLES:
sage: FiniteSetMaps(4, 3)([1, 0, 2, 1]).items() [(0, 1), (1, 0), (2, 2), (3, 1)]
- setimage(i, j)¶
Set the image of
iasjinselfWarning
selfmust be mutable; otherwise an exception is raised.INPUT:
i,j– twoobject’s
OUTPUT:
NoneNote
if you need speed, please use instead
_setimage()EXAMPLES:
sage: fs = FiniteSetMaps(4, 3)([1, 0, 2, 1]) sage: fs2 = copy(fs) sage: fs2.setimage(2, 1) sage: fs2 [1, 0, 1, 1] sage: with fs.clone() as fs3: ....: fs3.setimage(0, 2) ....: fs3.setimage(1, 2) sage: fs3 [2, 2, 2, 1]
- class sage.sets.finite_set_map_cy.FiniteSetMap_Set¶
Bases:
sage.sets.finite_set_map_cy.FiniteSetMap_MNData structure for maps
We assume that the parent given as argument is such that:
the domain is in
parent.domain()the codomain is in
parent.codomain()parent._mcontains the cardinality of the domainparent._ncontains the cardinality of the codomainparent._unrank_domainandparent._rank_domainis a pair of reciprocal rank and unrank functions between the domain andrange(parent._m).parent._unrank_codomainandparent._rank_codomainis a pair of reciprocal rank and unrank functions between the codomain andrange(parent._n).
- classmethod from_dict(t, parent, d)¶
Creates a
FiniteSetMapfrom a dictionaryWarning
no check is performed !
- classmethod from_list(t, parent, lst)¶
Creates a
FiniteSetMapfrom a listWarning
no check is performed !
- getimage(i)¶
Returns the image of
ibyselfINPUT:
i– anint
EXAMPLES:
sage: F = FiniteSetMaps(["a", "b", "c", "d"], ["u", "v", "w"]) sage: fs = F._from_list_([1, 0, 2, 1]) sage: list(map(fs.getimage, ["a", "b", "c", "d"])) ['v', 'u', 'w', 'v']
- image_set()¶
Returns the image set of
selfEXAMPLES:
sage: F = FiniteSetMaps(["a", "b", "c"]) sage: sorted(F.from_dict({"a": "b", "b": "a", "c": "b"}).image_set()) ['a', 'b'] sage: F = FiniteSetMaps(["a", "b", "c"]) sage: F(lambda x: "c").image_set() {'c'}
- items()¶
The items of
selfReturn the list of the couple
(x, self(x))EXAMPLES:
sage: F = FiniteSetMaps(["a", "b", "c"]) sage: F.from_dict({"a": "b", "b": "a", "c": "b"}).items() [('a', 'b'), ('b', 'a'), ('c', 'b')]
- setimage(i, j)¶
Set the image of
iasjinselfWarning
selfmust be mutable otherwise an exception is raised.INPUT:
i,j– twoobject’s
OUTPUT:
NoneEXAMPLES:
sage: F = FiniteSetMaps(["a", "b", "c", "d"], ["u", "v", "w"]) sage: fs = F(lambda x: "v") sage: fs2 = copy(fs) sage: fs2.setimage("a", "w") sage: fs2 map: a -> w, b -> v, c -> v, d -> v sage: with fs.clone() as fs3: ....: fs3.setimage("a", "u") ....: fs3.setimage("c", "w") sage: fs3 map: a -> u, b -> v, c -> w, d -> v
- sage.sets.finite_set_map_cy.FiniteSetMap_Set_from_dict(t, parent, d)¶
Creates a
FiniteSetMapfrom a dictionaryWarning
no check is performed !
- sage.sets.finite_set_map_cy.FiniteSetMap_Set_from_list(t, parent, lst)¶
Creates a
FiniteSetMapfrom a listWarning
no check is performed !
- sage.sets.finite_set_map_cy.fibers(f, domain)¶
Returns the fibers of the function
fon the finite setdomainINPUT:
f– a function or callabledomain– a finite iterable
OUTPUT:
a dictionary
dsuch thatd[y]is the set of allxindomainsuch thatf(x) = y
EXAMPLES:
sage: from sage.sets.finite_set_map_cy import fibers, fibers_args sage: fibers(lambda x: 1, []) {} sage: fibers(lambda x: x^2, [-1, 2, -3, 1, 3, 4]) {1: {1, -1}, 4: {2}, 9: {3, -3}, 16: {4}} sage: fibers(lambda x: 1, [-1, 2, -3, 1, 3, 4]) {1: {1, 2, 3, 4, -3, -1}} sage: fibers(lambda x: 1, [1,1,1]) {1: {1}}
See also
fibers_args()if one needs to pass extra arguments tof.
- sage.sets.finite_set_map_cy.fibers_args(f, domain, *args, **opts)¶
Returns the fibers of the function
fon the finite setdomainIt is the same as
fibers()except that one can pass extra argument forf(with a small overhead)EXAMPLES:
sage: from sage.sets.finite_set_map_cy import fibers_args sage: fibers_args(operator.pow, [-1, 2, -3, 1, 3, 4], 2) {1: {1, -1}, 4: {2}, 9: {3, -3}, 16: {4}}