Projective max-plus matrices

Overview

Defined in element.hpp.

This page contains the documentation for the class template libsemigroups::ProjectiveMaxPlusMatrix.

Full API

class ProjectiveMaxPlusMatrix : public libsemigroups::detail::MatrixOverSemiringBase<int64_t, ProjectiveMaxPlusMatrix>

Class for projective max-plus matrices.

These matrices belong to the quotient of the monoid of all max-plus matrices by the congruence where two matrices are related if they differ by a scalar multiple.

Matrices in this class are modified when constructed to be in a normal form which is obtained by subtracting the maximum finite entry in the matrix from the every finite entry.

See

MaxPlusSemiring and MatrixOverSemiring.

Public Functions

ProjectiveMaxPlusMatrix(std::vector<int64_t> const&, Semiring<int64_t> const *)

A constructor.

See MatrixOverSemiring::MatrixOverSemiring for details about this member function.

The parameter matrix is converted into its normal form when when the object is constructed.

ProjectiveMaxPlusMatrix(std::vector<std::vector<int64_t>> const&, Semiring<int64_t> const *)

A constructor.

See MatrixOverSemiring::MatrixOverSemiring for details about this member function.

The copy of the parameter matrix in the object constructed is converted into its normal form when the object is constructed.

ProjectiveMaxPlusMatrix(ProjectiveMaxPlusMatrix const&)

A copy constructor.

ProjectiveMaxPlusMatrix operator*(ElementWithVectorData const&) const

Returns the product of this and y.

This returns the product of this and y, as defined by ProjectiveMaxPlusMatrix::redefine, without altering this or y. This is specialised to avoid trying to call ProjectiveMaxPlusMatrix::ProjectiveMaxPlusMatrix(degree) in Element::redefine (as for MatrixOverSemiring).

virtual bool operator==(Element const&) const = 0

Returns true if this equals that.

This member function checks the mathematical equality of two Element objects in the same subclass of Element.

virtual bool operator<(Element const&) const = 0

Returns true if this is less than that.

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 if this is greater than that.

This member function returns true if this is greater than that, under the ordering defined by the operator <.

bool operator!=(Element const &that) const

Returns true if this is not equal to that.

This member function returns true if this is mathematically not equal to that.

bool operator<=(Element const &that) const

Returns true if this is less than or equal to that.

This member function returns true if this is less than (under the order defined by the operator <) or mathematically equal to that.

bool operator>=(Element const &that) const

Returns true if this is less than or equal to that.

This member function returns true if this is greater than (under the order defined by the operator <) or mathematically equal to that.

virtual size_t complexity() const = 0

Returns the approximate time complexity of multiplying two Element objects in a given subclass.

This member function returns an integer which represents the approximate time complexity of multiplying two objects in the same subclass of Element which have the same Element::degree. For example, the approximate time complexity of multiplying two \(3\times 3\) matrices over a common semiring is \(O(3 ^ 3)\), and 27 is returned by MatrixOverSemiring::complexity.

The returned value is used in, for example, FroidurePin::fast_product and FroidurePin::nr_idempotents to decide if it is better to multiply elements or follow a path in the Cayley graph.

virtual size_t degree() const = 0

Returns the degree of an Element.

This member function returns an integer which represents the size of the element, and is used to determine whether or not two elements are compatible for multiplication. For example, two Transformation objects of different degrees cannot be multiplied, and a Bipartition of degree 10 cannot be an element of a monoid of bipartitions of degree 3.

See the relevant subclass for the particular meaning of the return value of this member function for each subclass.

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.

virtual void swap(Element&) = 0

Swap another Element with this.

This member function swaps the defining data of x and this.

virtual void redefine(Element const &x, Element const &y)

Multiplies x and y and stores the result in this.

Redefine this to be the product of x and y. 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 and y and stores the result in this.

This version of the member function takes const pointers rather than const references, but otherwise behaves like the other Element::redefine.

virtual void redefine(Element const &x, Element const &y, size_t)

Multiplies x and y and stores the result in this.

Redefine this to be the product of x and y. 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 of x and y.

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 and y and stores the result in this.

This member function differs from the the previous only in taking pointers instead of references.

virtual void increase_degree_by(size_t)

Increases the degree of this by deg.

This does not make sense for all subclasses of Element.

virtual 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.

virtual 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.