Partial Transformations¶
Overview¶
Defined in element.hpp
.
This page contains the documentation for the class template
libsemigroups::PartialTransformation
.
Full API¶
-
template<typename
T
, typenameTSubclass
>
classlibsemigroups
::
PartialTransformation
: public libsemigroups::detail::ElementWithVectorDataDefaultHash<T, TSubclass>¶ Abstract class for partial transformations.
This is a template class for partial transformations.
The template parameter
T
is the type of image values, i.e. uint16_t, and so on. The value of the template parameterT
can be used to reduce (or increase) the amount of memory required by instances of this class.The template parameter
TSubclass
is the subclass of PartialTransformation used by the PartialTransformation::identity member function to construct an identity, so that the return value of the member function PartialTransformation::identity is of typeTSubclass
instead of type PartialTransformation.This class is abstract but this will likely change in a future version of libsemigroups.
A partial transformation \(f\) is just a function defined on a subset of \(\{0, 1, \ldots, n - 1\}\) for some integer \(n\) called the degree of f. A partial transformation is stored as a vector of the images of \(\{0, 1, \ldots, n -1\}\), i.e. \(\{(0)f, (1)f, \ldots, (n - 1)f\}\) where the value libsemigroups::UNDEFINED is used to indicate that \((i)f\) is, you guessed it, undefined (i.e. not among the points where \(f\) is defined).
Public Functions
-
PartialTransformation
()¶ A constructor.
Constructs an uninitialised PartialTransformation.
-
PartialTransformation
(std::vector<T> const &vec)¶ A constructor.
Constructs an PartialTransformation f from
vec
and validates it. The vectorvec
(of length some non-negative integer n) should contain in position i the image \((i)f\), or libsemigroups::UNDEFINED if f is not defined on i, for all \(0 < i < n\).
-
PartialTransformation
(std::initializer_list<T> imgs)¶ A constructor.
Constructs a vector from
imgs
and calls the corresponding constructor.
-
void
validate
() const¶ Validates the data defining
this
.This member function throws a libsemigroups::LibsemigroupsException if any value of
this
is out of bounds (i.e. less than 0, greater than or equal todegree()
, and not libsemigroups::UNDEFINED).
-
size_t
complexity
() const override¶ Returns the approximate time complexity of multiplying two partial transformations.
The approximate time complexity of multiplying partial transformations is just their degree.
-
size_t
degree
() const final¶ Returns the degree of a partial transformation.
The degree of a partial transformation is the number of points used in its definition.
-
size_t
crank
() const¶ Returns the rank of a partial transformation.
The rank of a partial transformation is the number of its distinct image values, not including libsemigroups::UNDEFINED. This function recomputes the return value every time it is called.
-
TSubclass
identity
() const override¶ Returns the identity transformation with same degree as
this
.This member function returns a new partial transformation with degree equal to the degree of
this
that fixes every value from 0 to the degree ofthis
.
-
bool
operator==
(Element const&) const = 0¶ Returns
true
ifthis
equalsthat
.This member function checks the mathematical equality of two Element objects in the same subclass of Element.
-
bool
operator<
(Element const&) const = 0¶ Returns
true
ifthis
is less thanthat
.This member function defines a total order on the set of objects in a given subclass of Element with a given Element::degree. The definition of this total order depends on the member function for the operator < in the subclass.
-
bool
operator>
(Element const &that) const¶ Returns
true
ifthis
is greater thanthat
.This member function returns
true
ifthis
is greater thanthat
, under the ordering defined by the operator <.
-
bool
operator!=
(Element const &that) const¶ Returns
true
ifthis
is not equal tothat
.This member function returns
true
ifthis
is mathematically not equal tothat
.
-
bool
operator<=
(Element const &that) const¶ Returns
true
ifthis
is less than or equal tothat
.This member function returns
true
ifthis
is less than (under the order defined by the operator <) or mathematically equal tothat
.
-
bool
operator>=
(Element const &that) const¶ Returns
true
ifthis
is less than or equal tothat
.This member function returns
true
ifthis
is greater than (under the order defined by the operator <) or mathematically equal tothat
.
-
size_t
hash_value
() const¶ Return the hash value of an Element.
This member function returns a hash value for an object in a subclass of Element. This value is only computed the first time this member function is called.
-
void
swap
(Element&) = 0¶ Swap another Element with
this
.This member function swaps the defining data of
x
andthis
.
-
void
redefine
(Element const &x, Element const &y)¶ Multiplies
x
andy
and stores the result inthis
.Redefine
this
to be the product ofx
andy
. This is in-place multiplication to avoid allocation of memory for products which do not need to be stored for future use.The implementation of this member function in the Element base class simply calls the 3 parameter version with third parameter 0. Any subclass of Element can implement either a two or three parameter version of this member function and the base class member function implements the other member function.
-
void
redefine
(Element const *x, Element const *y)¶ Multiplies
x
andy
and stores the result inthis
.This version of the member function takes const pointers rather than const references, but otherwise behaves like the other Element::redefine.
-
void
redefine
(Element const &x, Element const &y, size_t)¶ Multiplies
x
andy
and stores the result inthis
.Redefine
this
to be the product ofx
andy
. This is in-place multiplication to avoid allocation of memory for products which do not need to be stored for future use.The implementation of this member function in the Element base class simply calls the 2 parameter version and ignores the third parameter
thread_id
. Any subclass of Element can implement either a two or three parameter version of this member function and the base class member function implements the other member function.The parameter
thread_id
is required in some derived classes of Element because some temporary storage is required to find the product ofx
andy
.Note that if different threads call this member function on a derived class of Element where static temporary storage is used in the redefine member function with the same value of
thread_id
, then bad things may happen.
-
void
redefine
(Element const *x, Element const *y, size_t)¶ Multiplies
x
andy
and stores the result inthis
.This member function differs from the the previous only in taking pointers instead of references.
-
void
increase_degree_by
(size_t)¶ Increases the degree of
this
bydeg
.This does not make sense for all subclasses of Element.
-
Element *
heap_copy
() const = 0¶ Returns a new element completely independent of
this
.This member function really copies an Element. To minimise the amount of copying when Element objects are inserted in a std::unordered_map and other containers, an Element behaves somewhat like a pointer, in that the actual data in an Element is only copied when this member function is called. Otherwise, if an Element is copied, then its defining data is only stored once.
-
Element *
heap_identity
() const = 0¶ Returns an independent copy of the identity.
This member function returns a copy of the identity element (in the appropriate semigroup) which is independent from previous copies.
-