Bipartitions

Overview

Defined in element.hpp.

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

Full API

class libsemigroups::Bipartition : public libsemigroups::detail::ElementWithVectorDataDefaultHash<uint32_t, Bipartition>

Class for bipartitions.

A bipartition is a partition of the set \(\{0, ..., 2n - 1\}\) for some integer \(n\); see the Semigroups package for GAP documentation for more details. The Bipartition class is more complex (i.e. has more member functions) than strictly required by the algorithms for a FroidurePin object because the extra member functions are used in the GAP package Semigroups package for GAP.

Public Functions

Bipartition()

A constructor.

Constructs a uninitialised bipartition.

Bipartition(size_t)

A constructor.

Constructs a uninitialised bipartition of degree degree.

Bipartition(std::vector<uint32_t> const&)

A constructor.

The parameter blocks must have length 2n for some positive integer n, consist of non-negative integers, and have the property that if i, i > 0, occurs in blocks, then i - 1 occurs earlier in blocks.

The parameter blocks is copied.

Bipartition(std::vector<uint32_t>&&)

A constructor.

The parameter vector should be a rvalue reference to defining data of the Bipartition.

Returns a Bipartition whose defining data is vec. This constructor moves the data from vec, meaning that vec is changed by this constructor.

Bipartition(std::initializer_list<uint32_t> blocks)

A constructor.

Converts blocks to a vector and uses corresponding constructor.

Bipartition(Bipartition const&)

A copy constructor.

Constructs a Bipartition that is mathematically equal to copy.

Bipartition(std::initializer_list<std::vector<int32_t>> const&)

A constructor.

The argument blocks should be a list of vectors which partition the ranges [-n .. -1] U [1 .. n] for some positive integer n, called the degree of the bipartition. The bipartition constructed has equivalence classes given by the vectors in blocks.

void validate() const

Validates the data defining this.

This member function throws a libsemigroups::LibsemigroupsException if the data defining this is invalid, which could occur if:

  • this->_vector has odd length, or

  • a positive integer i occurs in this->_vector before the integer i - 1

size_t complexity() const override

Returns the approximate time complexity of multiplication.

In the case of a Bipartition of degree n the value 2n ^ 2 is returned.

size_t degree() const override

Returns the degree of the bipartition.

A bipartition is of degree n if it is a partition of \(\{0, \ldots, 2n - 1\}\).

Bipartition identity() const override

Returns an identity bipartition.

The identity bipartition of degree \(n\) has blocks \(\{i, -i\}\) for all \(i\in \{0, \ldots, n - 1\}\). This member function returns a new identity bipartition of degree equal to the degree of this.

void redefine(Element const&, Element const&, size_t) override

Multiply x and y and stores the result in this.

This member function redefines this to be the product (as defined at the top of this page) of the parameters x and y. This member function asserts that the degrees of x, y, and this, are all equal, and that neither x nor y equals this.

The parameter thread_id is required since some temporary storage is required to find the product of x and y. Note that if different threads call this member function with the same value of thread_id then bad things will happen.

size_t rank()

Returns the number of transverse blocks.

The rank of a bipartition is the number of blocks containing both positive and negative values. This value is cached after it is first computed.

uint32_t const_nr_blocks() const

Returns the number of blocks in a bipartition.

This member function differs for Bipartition::nr_blocks in that the number of blocks is not cached if it has not been previously computed.

uint32_t nr_blocks()

Returns the number of blocks in a bipartition.

This value is cached the first time it is computed.

uint32_t nr_left_blocks()

Returns the number of blocks containing a positive integer.

The left blocks of a bipartition is the partition of \(\{0, \ldots, n - 1\}\) induced by the bipartition. This member function returns the number of blocks in this partition.

uint32_t nr_right_blocks()

Returns the number of blocks containing a negative integer.

The right blocks of a bipartition is the partition of \(\{n, \ldots, 2n - 1\}\) induced by the bipartition. This member function returns the number of blocks in this partition.

bool is_transverse_block(size_t)

Returns true if the block with index index is transverse.

A block of a biparition is transverse if it contains integers less than and greater than \(n\), which is the degree of the bipartition. This member function asserts that the parameter index is less than the number of blocks in the bipartition.

Blocks *left_blocks()

Return the left blocks of a bipartition.

The left blocks of a bipartition is the partition of \(\{0, \ldots, n - 1\}\) induced by the bipartition. This member function returns a Blocks object representing this partition.

Blocks *right_blocks()

Return the left blocks of a bipartition.

The right blocks of a bipartition is the partition of \(\{n, \ldots, 2n - 1\}\) induced by the bipartition. This member function returns a Blocks object representing this partition.

void set_nr_blocks(size_t)

Set the cached number of blocks.

This member function sets the cached value of the number of blocks of this to nr_blocks. It asserts that either there is no existing cached value or nr_blocks equals the existing cached value.

void set_nr_left_blocks(size_t)

Set the cached number of left blocks.

This member function sets the cached value of the number of left blocks of this to nr_left_blocks. It asserts that either there is no existing cached value or nr_left_blocks equals the existing cached value.

void set_rank(size_t)

Set the cached rank.

This member function sets the cached value of the rank of this to rank. It asserts that either there is no existing cached value or rank equals the existing cached value.

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.

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.

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 and this.

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.

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.

void increase_degree_by(size_t)

Increases the degree of this by deg.

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.

Public Static Functions

Bipartition identity(size_t n)

Returns an identity bipartition.

The identity bipartition of degree \(n\) has blocks \(\{i, -i\}\) for all \(i\in \{0, \ldots, n - 1\}\). This member function returns a new identity bipartition of degree equal to n.