#include <CGAL/HDVF/Hdvf_duality.h>

Inherits from

CGAL::Homological_discrete_vector_field::Hdvf_core< ChainComplex >.

Definition

template<typename ChainComplex>
class CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >

The class Hdvf_duality is the implementation of homological discrete vector fields (HDVF for short) for Alexander duality computation.

Warning: The ring of coefficients provided should be a field.

In dimension \(n\), given a complex \(L\) homeomorphic to \(\mathcal S^n\) and a sub-complex \(K\subseteq L\), Alexander duality states that for all \(q\leqslant n\):

\[\tilde H_q(K) \simeq \tilde H^{n-q-1}(L-K)\]

where \(\tilde H_q\) and \(\tilde H^q\) denote reduced homology and cohomology groups.

In [Gonzalez and al. 2025], the authors prove that, even if \(L-K\) is not a sub-complex, it produces a valid chain complex (we call "co-complex" such a complementary of sub-complex). Hence, its homology/cohomology can be computed and for all \(q\leqslant n\):

\[\tilde H_q(K) \simeq \tilde H^{q+1}(L-K)\]

HDVFs provide a fast and convenient mean to compute this isomorphism. In order to work with convenient finite complexes, the complex \(L\) must be homeomorphic to a ball of dimension \(n\) (thus \(\mathcal S^n\) is actually homeomorphic to \(L\) plus an infinite \(n\)-cell closing its boundary).

Perfect HDVFs are first computed over \(K\) and \(L-K\) (providing corresponding relative homology) respectively and Alexander isomorphism gives rise to a pairing between critical cells in \(K\) and \(L-K\), that is a pairing between homology/cohomology generators in \(K\) and \(L-K\).

The class provides HDVF constuction operations: compute_perfect_hdvf() and compute_rand_perfect_hdvf(), which build perfect HDVFs over \(K\) and \(L-K\) respectively. Then, compute_alexander_pairing() computes Alexander isomorphism (and provides a pairing between homology/cohomology generators in \(K\) and \(L-K\)).

Example of Alexander duality isomorphism. The twirl mesh is a subcomplex K of a larger complex L depicted in yellow, homeomorphic to the ball of dimension 3 (right - sectional view).

Figure 98.1 Example of "homological quartet for the twirl model". 1: Homology generators of the twirl \(H_1(K)\), 2: Cohomology generators of the twirl \(H^1(K)\), 3: Homology generators of the complementary of the twirl \(H_1(L-K)\), 4: Cohomology generators of the complementary of the twirl \(H^1(L-K)\). Alexander isomorphism is represented through colours (paired generators have similar colours).

Hence, each hole in \(K\) gives rise to four generators (called its "homological quarted": its homology and cohomology generators in \(K_q\) and the homology and cohomology generators paired with them in \((L-K)_{q+1}\)).

In order to compute relative homology, a sub chain complex mask is used to partially screen the complex L and thus restrict HDVF computation. This mask is called "current mask" (and can be set over K or L-K).

Is model of: HDVF

Template Parameters

ChainComplex a model of the AbstractChainComplex concept, providing the type of abstract chain complex used. The sparse matrix provided by ChainComplex must be an instance of the Sub_sparse_matrix template.

[Gonzalez and al. 2025] Gonzalez-Lorenzo, A., Bac, A. & Gazull, YS. A constructive approach of Alexander duality. J Appl. and Comput. Topology 9, 2 (2025).

Public Types
typedef ChainComplex	Chain_complex
	Chain complex type.

typedef Chain_complex::Coefficient_ring	Coefficient_ring
	Type of coefficients used to compute homology.

typedef Hdvf_core< ChainComplex >	Base
	Type of parent Hdvf_core class.

using	Column_chain = typename Base::Column_chain

using	Row_chain = typename Base::Row_chain

Public Types inherited from CGAL::Homological_discrete_vector_field::Hdvf_core< ChainComplex >
typedef ChainComplex	Chain_complex
	Type of the underlying chain complex.

typedef ChainComplex::Coefficient_ring	Coefficient_ring
	Type of coefficients used to compute homology.

typedef ChainComplex::Sparse_matrix_struct	Sparse_matrix_struct
	Type of sparse matrix structure used to compute homology.

template<typename CR , int SF>
using	Sparse_chain_base = typename Sparse_matrix_struct::template Sparse_chain_type< CR, SF >
	Template type of underlying sparse chains.

template<typename CR , int SF>
using	Sparse_matrix_base = typename Sparse_matrix_struct::template Sparse_matrix_type< CR, SF >
	Template type of underlying sparse matrices.

typedef Sparse_chain_base< Coefficient_ring, CGAL::OSM::COLUMN >	Column_chain
	Type of column-major chains.

typedef Sparse_chain_base< Coefficient_ring, CGAL::OSM::ROW >	Row_chain
	Type of row-major chains.

typedef Sparse_matrix_base< Coefficient_ring, CGAL::OSM::COLUMN >	Column_matrix
	Type of column-major sparse matrices.

typedef Sparse_matrix_base< Coefficient_ring, CGAL::OSM::ROW >	Row_matrix
	Type of row-major sparse matrices.

Public Member Functions
	Hdvf_duality (const Chain_complex &L, Sub_chain_complex_mask< Chain_complex > &K, int hdvf_opt=OPT_FULL)
	Hdvf_duality constructor ( from a complex `L` and a sub-complex `K`)

Cell_pair	find_pair_A (int q, bool &found) const
	Finds a valid cell pair of dimension q / q+1 for A in the current sub chain complex.

Cell_pair	find_pair_A (int q, bool &found, size_t gamma) const
	Finds a valid cell pair for A containing `gamma` in the current sub chain complex (a cell of dimension `q`)

std::vector< Cell_pair >	find_pairs_A (int q, bool &found) const
	Finds all valid cell pairs of dimension q / q+1 in the current sub chain complex for A.

std::vector< Cell_pair >	find_pairs_A (int q, bool &found, size_t gamma) const
	Finds all valid cell pairs for `A`containing `gamma` in the current sub chain complex (a cell of dimension `q`)

void	set_mask_K ()
	Sets the current sub chain complex masks over `K`.

void	set_mask_L_K ()
	Sets the current sub chain complex masks over `L-K`.

Sub_chain_complex_mask< ChainComplex >	get_current_mask () const
	Returns the value of the current sub chain complex mask.

std::vector< Cell_pair >	compute_perfect_hdvf (bool verbose=false)
	Computes a perfect HDVF over the current sub chain complex.

std::vector< Cell_pair >	compute_rand_perfect_hdvf (bool verbose=false)
	Computes a random perfect HDVF over the current sub chain complex.

bool	is_perfect_hdvf (int dimension_restriction=-2)
	Tests if HDVFs over `K` and `L-K` are perfect.

std::vector< Cell_pair >	compute_pairing_hdvf ()
	Computes a "pairing" HDVF between K and L-K.

std::vector< Cell_pair >	compute_rand_pairing_hdvf ()
	Computes a random "pairing" HDVF between K and L-K.

std::vector< std::vector< size_t > >	psc_flags (PSC_flag flag) const
	Gets cells with a given `PSC_flag` in any dimension in the current sub chain complex.

std::vector< size_t >	psc_flags (PSC_flag flag, int q) const
	Gets cells with a given `PSC_flag` in dimension `q` in the current sub chain complex.

std::ostream &	insert_reduction (std::ostream &out=std::cout)
	Prints the homology and cohomology reduction information for `K` and `L-K`.

std::ostream &	write_hdvf_reduction (std::ostream &out) const
	Writes a `Hdvf_duality` together with the associated reduction (f, g, h, d matrices)

void	write_hdvf_reduction (std::string filename) const
	Writes a `Hdvf_duality` together with the associated reduction to a file (f, g, h, d matrices).

std::istream &	read_hdvf_reduction (std::istream &in)
	Reads a `Hdvf_duality` together with the associated reduction (f, g, h, d matrices)

void	read_hdvf_reduction (std::string filename)
	Loads a `Hdvf_duality` together with the associated reduction from a file (f, g, h, d matrices)

std::ostream &	print_reduction_sub (std::ostream &out=std::cout)
	Prints the homology and cohomology reduction information for the current sub chain complex.

std::ostream &	print_bnd_pairing (std::ostream &out=std::cout)
	Prints the reduced boundary over critical cells of `K` and `L-K`.

std::vector< std::vector< int > >	psc_labels () const
	Exports primary/secondary/critical labels of the current sub chain complex for vtk export.

Column_chain	homology_chain (size_t cell_index, int dim) const
	Exports homology generators of the current sub chain complex associated to `cell_index` (critical cell) of dimension `q` (used by vtk export).

Column_chain	cohomology_chain (size_t cell_index, int dim) const
	Exports cohomology generators of the current sub chain complex associated to `cell_index` (critical cell) of dimension `q` (used by vtk export).

bool	compare (const Hdvf_duality &other, bool full_compare=false) const
	Compares the `HDVF_duality` with another `HDVF_duality` over the same underlying complex and sub-complex.

Public Member Functions inherited from CGAL::Homological_discrete_vector_field::Hdvf_core< ChainComplex >
int	dimension_restriction () const
	Returns the dimension of Hdvf computation.

void	dimension_restriction (int dimension)
	Changes the dimension of Hdvf computation.

	Hdvf_core (const ChainComplex &K, int hdvf_opt=OPT_FULL, int dimension_restriction=-1)
	Constructor from a chain complex.

	Hdvf_core (const Hdvf_core &hdvf)

Hdvf_core &	operator= (const Hdvf_core &hdvf)
	Affectation operator.

	Hdvf_core (const ChainComplex &K, const std::vector< std::vector< PSC_flag > > &flags, bool with_build_reduction=false, int hdvf_opt=OPT_FULL, int dimension_restriction=-1)
	Constructor from the PRIMARY/SECONDARY/CRITICAL labels.

	~Hdvf_core ()

bool	combinatorially_coherent ()
	Check the "combinatorial" coherence of a HDVF \(X(P,S,C)\).

bool	is_valid_pair_for_A (size_t gamma, size_t gamma_prime, int q)
	Checks if the pair of cells \((\gamma, \gamma')\), of dimensions q / q+1, is valid for A.

virtual Cell_pair	find_pair_A (int q, bool &found) const
	Finds a valid Cell_pair of dimension q / q+1 for A.

virtual Cell_pair	find_pair_A (int q, bool &found, size_t gamma) const
	Finds a valid Cell_pair for A containing `gamma` (a cell of dimension `q`)

virtual std::vector< Cell_pair >	find_pairs_A (int q, bool &found) const
	Finds all valid Cell_pair of dimension q / q+1 for A.

virtual std::vector< Cell_pair >	find_pairs_A (int q, bool &found, size_t gamma) const
	Finds all valid Cell_pair for A containing `gamma` (a cell of dimension `q`)

void	A (size_t gamma1, size_t gamma2, int q)
	A operation: pairs critical cells.

void	A (const Cell_pair &p)
	A operation: pairs critical cells.

std::vector< Cell_pair >	compute_perfect_hdvf (bool verbose=false)
	Computes a perfect HDVF.

std::vector< Cell_pair >	compute_rand_perfect_hdvf (bool verbose=false)
	Computes a random perfect HDVF.

bool	is_perfect_hdvf (int dimension_restriction=-2)
	Tests if a HDVF is perfect.

size_t	number_of_cells_by_flag (PSC_flag flag, int q) const
	Gets the number of cells with a given flag in dimension `q`.

virtual std::vector< std::vector< size_t > >	psc_flags (PSC_flag flag) const
	Gets cells with a given `PSC_flag` in any dimension.

virtual std::vector< size_t >	psc_flags (PSC_flag flag, int q) const
	Gets cells with a given `PSC_flag` in dimension `q`.

virtual const std::vector< PSC_flag > &	psc_flags (int q) const
	Gets the`PSC_flag` of all cells in dimension `q`.

virtual const std::vector< std::vector< PSC_flag > > &	psc_flags () const
	Gets the`PSC_flag` of all cells.

PSC_flag	psc_flag (size_t tau, int q) const
	Gets the PSC_flag of the cell `tau` in dimension `q`.

int	hdvf_opts () const
	Gets HDVF computation option.

const Row_matrix &	matrix_f (int q) const
	Gets the row-major matrix of \(f\) (from the reduction associated to the HDVF).

const Column_matrix &	matrix_g (int q) const
	Gets the column-major matrix of \(g\) (from the reduction associated to the HDVF).

const Column_matrix &	matrix_h (int q) const
	Gets the column-major matrix of \(h\) (from the reduction associated to the HDVF).

const Column_matrix &	matrix_dd (int q) const
	Gets the column-major matrix of \(\partial'\), reduced boundary operator (from the reduction associated to the HDVF).

const Chain_complex &	complex ()
	Gets a constant reference over the underlying chain complex.

std::ostream &	write_flags (std::ostream &out=std::cout, int dimension_restriction=-2) const
	Writes the PSC flags of cells along each dimension.

std::ostream &	write_matrices (std::ostream &out=std::cout, int dimension_restriction=-2) const
	Writes the matrices of the reduction.

std::ostream &	write_reduction (std::ostream &out=std::cout, int dimension_restriction=-2) const
	Writes the homology and cohomology reduction information.

virtual std::vector< std::vector< int > >	psc_labels (int dimension_restriction=-2) const
	Exports primary/secondary/critical integers encoding labels (in particular for vtk export)

virtual Column_chain	homology_chain (size_t cell_index, int q) const
	Gets homology generators associated to `cell` (critical cell) of dimension `q` (used by vtk export).

virtual Column_chain	cohomology_chain (size_t cell_index, int dim) const
	Gets cohomology generators associated to `cell_index` (critical cell) of dimension `q` (used by vtk export).

std::ostream &	write_hdvf_reduction (std::ostream &out) const
	Writes a HDVF together with the associated reduction (f, g, h, d matrices)

void	write_hdvf_reduction (std::string filename) const
	Writes a HDVF together with the associated reduction to a file (f, g, h, d matrices).

std::istream &	read_hdvf_reduction (std::istream &in_stream)
	Loads a HDVF together with the associated reduction (f, g, h, d matrices)

void	read_hdvf_reduction (std::string filename)
	Loads a HDVF together with the associated reduction from a file (f, g, h, d matrices)

bool	compare (const Hdvf_core &other, bool full_compare=false) const
	Compares the HDVF with another HDVF over the same underlying complex.

bool	operator== (const Hdvf_core &other)
	Comparison operator.

Protected Member Functions
	Hdvf_duality (const Hdvf_duality &other)

Protected Member Functions inherited from CGAL::Homological_discrete_vector_field::Hdvf_core< ChainComplex >
Sub_matrix_data< Sparse_matrix_base< Coefficient_ring, OSM::ROW > >	extract_sub (const Sparse_matrix_base< Coefficient_ring, OSM::ROW > &M, int q_rows, int q_cols, PSC_flag flag_rows, PSC_flag flag_cols)

Sub_matrix_data< Sparse_matrix_base< Coefficient_ring, OSM::COLUMN > >	extract_sub (const Sparse_matrix_base< Coefficient_ring, OSM::COLUMN > &M, int q_rows, int q_cols, PSC_flag flag_rows, PSC_flag flag_cols)

template<typename MatrixType >
void	fill_sub (MatrixType &M, const Sub_matrix_data< MatrixType > &sub)

template<typename MatrixType >
void	restrict_matrix (MatrixType &M, int q_rows, int q_cols, PSC_flag flag_rows, PSC_flag flag_cols)

void	build_reduction ()

std::ostream &	write_hdvf_reduction_main (std::ostream &out) const

std::istream &	read_hdvf_reduction_main (std::istream &in_stream)

template<int StorageFormat>
Sparse_chain_base< Coefficient_ring, StorageFormat >	projection (const Sparse_chain_base< Coefficient_ring, StorageFormat > &chain, PSC_flag flag, int q) const

void	progress_bar (size_t i, size_t n)

Additional Inherited Members
Protected Attributes inherited from CGAL::Homological_discrete_vector_field::Hdvf_core< ChainComplex >
std::vector< std::vector< PSC_flag > >	_flag

std::vector< size_t >	_nb_P

std::vector< size_t >	_nb_S

std::vector< size_t >	_nb_C

std::vector< Row_matrix >	_F_row

std::vector< Column_matrix >	_G_col

std::vector< Column_matrix >	_H_col

std::vector< Column_matrix >	_DD_col

const ChainComplex &	_K

int	_hdvf_opt

int	_dimension_restriction

int	_min_dimension

int	_max_dimension

Member Typedef Documentation

◆ Chain_complex

template<typename ChainComplex >

typedef ChainComplex CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::Chain_complex

Chain complex type.

Constructor & Destructor Documentation

◆ Hdvf_duality()

template<typename ChainComplex >

CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::Hdvf_duality	(	const Chain_complex &	L,
		Sub_chain_complex_mask< Chain_complex > &	K,
		int	hdvf_opt = `OPT_FULL`
	)

Hdvf_duality constructor ( from a complex L and a sub-complex K)

L is a complex of a given dimension \(n\) homeomorphic to \(\mathcal B^n\) and K is a sub-complex of L described by a bitboard (cells of K have a bit set to 1, cells of K have a bit set to 0).

Initially, the sub chain complex mask is set to K.

Parameters

L	A complex of a given dimension \(n\) homeomorphic to \(\mathcal B^n\).
K	A sub complex of `L` encoded through a bitboard.
hdvf_opt	Option for HDVF computation (`OPT_BND`, `OPT_F`, `OPT_G` or `OPT_FULL`).

Exceptions

Empty_complex If the complex `L` is empty, raises a `std::runtime_error`.

Member Function Documentation

◆ cohomology_chain()

template<typename ChainComplex >

Column_chain CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::cohomology_chain	(	size_t	cell_index,
		int	dim
	)		const

virtual

Exports cohomology generators of the current sub chain complex associated to cell_index (critical cell) of dimension q (used by vtk export).

The method exports the chain \(f^\star(\sigma)\) for \(\sigma\) the cell of index cell_index and dimension q.

Returns: A column-major chain.

Reimplemented from CGAL::Homological_discrete_vector_field::Hdvf_core< ChainComplex >.

◆ compare()

template<typename ChainComplex >

bool CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::compare	(	const Hdvf_duality< ChainComplex > &	other,
		bool	full_compare = `false`
	)		const

Compares the HDVF_duality with another HDVF_duality over the same underlying complex and sub-complex.

Parameters

other	Other `HDVF_duality` to compare.
full_compare	Turns on "in depth" HDVF comparison (reduction matrices).

◆ compute_pairing_hdvf()

template<typename ChainComplex >

std::vector< Cell_pair > CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::compute_pairing_hdvf

Computes a "pairing" HDVF between K and L-K.

Warning: Run compute_perfect_hdvf() first (to build perfect HDVFs over K and L-K respectively).

The function computes a perfect HDVF over remaining critical cells. Each pair of cells inserted with the A() operation maps corresponding homology/cohomology generators in the Alexander isomorphism.

Returns: The vector of paired critical cells (encoding Alexander isomorphism).

Exceptions

Non_perfect_hdvfs If the HDVFs over `K` and `L-K` are not perfect, raises a `std::runtime_error`.

◆ compute_perfect_hdvf()

template<typename ChainComplex >

std::vector< Cell_pair > CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::compute_perfect_hdvf ( bool verbose = false )

Computes a perfect HDVF over the current sub chain complex.

As long as valid pairs for A exist in the current sub chain complex, the function selects the first available pair (returned by find_pair_A()) and applies the corresponding A() operation. If the coefficient ring of coefficients is a field, this operation always produces a perfect HDVF (ie. the reduced boundary is null and the reduction provides homology and cohomology information). Otherwise the operation produces a maximal HDVF with a residual boundary matrix over critical cells.

If the HDVF is initially not trivial (some cells have already been paired), the function completes it into a perfect HDVF.

Parameters

verbose If this parameter is true, all intermediate reductions are printed out.

Returns: The vector of all Cell_pair paired with A.

◆ compute_rand_pairing_hdvf()

template<typename ChainComplex >

std::vector< Cell_pair > CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::compute_rand_pairing_hdvf

Computes a random "pairing" HDVF between K and L-K.

Warning: Run compute_perfect_hdvf() first (to build perfect HDVFs over K and L-K respectively).

The function computes a random perfect HDVF over remaining critical cells. Each pair of cells inserted with the A() operation maps corresponding homology/cohomology generators in the Alexander isomorphism.

Returns: The vector of paired critical cells (encoding Alexander isomorphism).

Exceptions

Non_perfect_hdvfs If the HDVFs over `K` and `L-K` are not perfect, raises a `std::runtime_error`.

◆ compute_rand_perfect_hdvf()

template<typename ChainComplex >

std::vector< Cell_pair > CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::compute_rand_perfect_hdvf ( bool verbose = false )

Computes a random perfect HDVF over the current sub chain complex.

As long as valid pairs for A exist in the current sub chain complex, the function selects a random pair (among pairs returned by find_pairs_A()) and applies the corresponding A() operation. If the coefficient ring is a field, this operation always produces a perfect HDVF (that is the reduced boundary is null and the reduction provides homology and cohomology information).

If the HDVF is initially not trivial (some cells have already been paired), the function randomly completes it into a perfect HDVF.

Warning: This method is slower that compute_perfect_hdvf() (finding out all possible valid pairs requires additional time).

Parameters

verbose If this parameter is true, all intermediate reductions are printed out.

Returns: The vector of all pairs of cells used for apply A.

◆ find_pair_A() [1/2]

template<typename ChainComplex >

Cell_pair CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::find_pair_A	(	int	q,
		bool &	found
	)		const

virtual

Finds a valid cell pair of dimension q / q+1 for A in the current sub chain complex.

The function searches a pair of critical cells, in the current sub chain complex, \((\gamma_1, \gamma2)\) of dimension q / q+1, valid for A (ie. such that \(\langle \partial_{q+1}(\gamma_2), \gamma_1 \rangle\) invertible). It returns the first valid pair found by iterators.

Parameters

q	Lower dimension of the pair.
found	Reference to a Boolean variable. The method sets `found` to `true` if a valid pair is found, `false` otherwise.

Reimplemented from CGAL::Homological_discrete_vector_field::Hdvf_core< ChainComplex >.

◆ find_pair_A() [2/2]

template<typename ChainComplex >

Cell_pair CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::find_pair_A	(	int	q,
		bool &	found,
		size_t	gamma
	)		const

virtual

Finds a valid cell pair for A containing gamma in the current sub chain complex (a cell of dimension q)

The function searches a cell \(\gamma'\) in the current sub chain complex such that one of the following conditions holds:

\(\gamma'\) has dimension q+1 and \((\gamma, \gamma')\) is valid for A (ie. such that \(\langle \partial_{q+1}(\gamma'), \gamma \rangle\) invertible),
\(\gamma'\) has dimension q-1 and \((\gamma', \gamma)\) is valid for A (ie. such that \(\langle \partial_{q}(\gamma), \gamma' \rangle\) invertible).

Parameters

q	Dimension of the cell `gamma`.
found	Reference to a Boolean variable. The method sets `found` to `true` if a valid pair is found, `false` otherwise.
gamma	Index of a cell to pair.

Reimplemented from CGAL::Homological_discrete_vector_field::Hdvf_core< ChainComplex >.

◆ find_pairs_A() [1/2]

template<typename ChainComplex >

std::vector< Cell_pair > CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::find_pairs_A	(	int	q,
		bool &	found
	)		const

virtual

Finds all valid cell pairs of dimension q / q+1 in the current sub chain complex for A.

The function searches all pairs of critical cells \((\gamma_1, \gamma2)\) in the current sub chain complex of dimension q / q+1, valid for A (ie. such that \(\langle \partial_{q+1}(\gamma_2), \gamma_1 \rangle\) invertible). It returns a vector of such pairs.

Parameters

q	Lower dimension of the pairs.
found	Reference to a Boolean variable. The method sets `found` to `true` if a valid pair is found, `false` otherwise.

Reimplemented from CGAL::Homological_discrete_vector_field::Hdvf_core< ChainComplex >.

◆ find_pairs_A() [2/2]

template<typename ChainComplex >

std::vector< Cell_pair > CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::find_pairs_A	(	int	q,
		bool &	found,
		size_t	gamma
	)		const

virtual

Finds all valid cell pairs for Acontaining gamma in the current sub chain complex (a cell of dimension q)

The function searches all CRITICAL cells \(\gamma'\) in the current sub chain complex such that one of the following conditions holds:

\(\gamma'\) has dimension q+1 and \((\gamma, \gamma')\) is valid for A (ie. such that \(\langle \partial_{q+1}(\gamma'), \gamma \rangle\) invertible),
\(\gamma'\) has dimension q-1 and \((\gamma', \gamma)\) is valid for A (ie. such that \(\langle \partial_{q}(\gamma), \gamma' \rangle\) invertible). It returns a vector of such pairs.

Parameters

q	Dimension of the cell `gamma`.
found	Reference to a Boolean variable. The method sets `found` to `true` if a valid pair is found, `false` otherwise.
gamma	Index of a cell to pair.

Reimplemented from CGAL::Homological_discrete_vector_field::Hdvf_core< ChainComplex >.

◆ homology_chain()

template<typename ChainComplex >

Column_chain CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::homology_chain	(	size_t	cell_index,
		int	dim
	)		const

virtual

Exports homology generators of the current sub chain complex associated to cell_index (critical cell) of dimension q (used by vtk export).

The method exports the chain \(g(\sigma)\) for \(\sigma\) the cell of index cell_index and dimension q.

Returns: A column-major chain.

Reimplemented from CGAL::Homological_discrete_vector_field::Hdvf_core< ChainComplex >.

◆ insert_reduction()

template<typename ChainComplex >

std::ostream & CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::insert_reduction ( std::ostream & out = std::cout )

Prints the homology and cohomology reduction information for K and L-K.

Prints \(f^*\), \(g\) \(\partial'\) the reduced boundary over each critical cell.

By default, outputs the complex to std::cout.

◆ is_perfect_hdvf()

template<typename ChainComplex >

bool CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::is_perfect_hdvf ( int dimension_restriction = -2 )

Tests if HDVFs over K and L-K are perfect.

The function returns true if the HDVFs over K and L-Kare perfect, that is, if the reduced boundary matrix is null and false otherwise. The functions tests all dimensions if dimension_restriction is -1 (default value) and tests only dimension dimension_restriction otherwise.

Parameters

dimension_restriction If positive, restricts the test to a single dimension , if -1 is provided, all dimensions are tested, if -2 is provided (default) test is carried out according to the HDVF dimension restriction value.

◆ print_bnd_pairing()

template<typename ChainComplex >

std::ostream & CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::print_bnd_pairing ( std::ostream & out = std::cout )

Prints the reduced boundary over critical cells of K and L-K.

The method prints out the reduced boundary matrix in each dimension, restricted to critical cells of K and L-K (ie. the matrix used to compute Alexander pairing).

Warning: Call this method after compute_perfect_hdvf().

By default, outputs the complex to std::cout.

◆ print_reduction_sub()

template<typename ChainComplex >

std::ostream & CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::print_reduction_sub ( std::ostream & out = std::cout )

Prints the homology and cohomology reduction information for the current sub chain complex.

Prints \(f^*\), \(g\) \(\partial'\) the reduced boundary over each critical cell.

By default, outputs the complex to std::cout.

◆ psc_flags() [1/2]

template<typename ChainComplex >

std::vector< std::vector< size_t > > CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::psc_flags ( PSC_flag flag ) const

virtual

Gets cells with a given PSC_flag in any dimension in the current sub chain complex.

The function returns a vector containing, for each dimension, the vector of cells with a given PSC_flag.

Parameters

flag	PSC_flag to select.

Reimplemented from CGAL::Homological_discrete_vector_field::Hdvf_core< ChainComplex >.

◆ psc_flags() [2/2]

template<typename ChainComplex >

std::vector< size_t > CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::psc_flags	(	PSC_flag	flag,
		int	q
	)		const

virtual

Gets cells with a given PSC_flag in dimension q in the current sub chain complex.

The function returns the vector of cells of dimension q with a given PSC_flag.

Parameters

flag	PSC_flag to select.
q	Dimension visited.

Reimplemented from CGAL::Homological_discrete_vector_field::Hdvf_core< ChainComplex >.

◆ psc_labels()

template<typename ChainComplex >

std::vector< std::vector< int > > CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::psc_labels ( ) const

Exports primary/secondary/critical labels of the current sub chain complex for vtk export.

The method exports the labels of every cells in each dimension. Exports the label 2 (ie. NONE) for cells out of the current sub chain complex.

Returns: A vector containing, for each dimension, the vector of labels by cell index.

◆ read_hdvf_reduction() [1/2]

template<typename ChainComplex >

std::istream & CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::read_hdvf_reduction ( std::istream & in )

Reads a Hdvf_duality together with the associated reduction (f, g, h, d matrices)

Reads a Hdvf_duality from a stream in hdvf file format (a simple text file format, see for a specification).

Parameters

in	Input stream.

◆ read_hdvf_reduction() [2/2]

template<typename ChainComplex >

void CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::read_hdvf_reduction ( std::string filename )

Loads a Hdvf_duality together with the associated reduction from a file (f, g, h, d matrices)

Load a Hdvf_duality and its reduction from a file in hdvf file format, a simple text file format (see for a specification).

Warning: The underlying complex is not stored in the file!

Parameters

filename Input file name.

Exceptions

File_not_found If the file `filename` cannot be opened, raises a `std::runtime_error`.

◆ set_mask_K()

template<typename ChainComplex >

void CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::set_mask_K ( )

Sets the current sub chain complex masks over K.

Further HDVF computations will be restricted to K (ie. computation of reduced homology).

◆ set_mask_L_K()

template<typename ChainComplex >

void CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::set_mask_L_K ( )

Sets the current sub chain complex masks over L-K.

Further HDVF computations will be restricted to L-K (ie. computation of reduced homology).

◆ write_hdvf_reduction() [1/2]

template<typename ChainComplex >

std::ostream & CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::write_hdvf_reduction ( std::ostream & out ) const

Writes a Hdvf_duality together with the associated reduction (f, g, h, d matrices)

Writes a HDVF_duality to a stream in hdvf file format (a simple text file format, see for a specification).

Parameters

out	Output stream.

◆ write_hdvf_reduction() [2/2]

template<typename ChainComplex >

void CGAL::Homological_discrete_vector_field::Hdvf_duality< ChainComplex >::write_hdvf_reduction ( std::string filename ) const

Writes a Hdvf_duality together with the associated reduction to a file (f, g, h, d matrices).

Writes a Hdvf_duality to a file in hdvf file format (a simple text file format, see for a specification).

Parameters

filename Output file name.

Exceptions

File_not_found If the file `filename` cannot be opened, raises a `std::runtime_error`.

Inherits from

Definition

Public Types

Public Member Functions

Protected Member Functions

Additional Inherited Members

Member Typedef Documentation

◆ Chain_complex

Constructor & Destructor Documentation

◆ Hdvf_duality()

Member Function Documentation

◆ cohomology_chain()

◆ compare()

◆ compute_pairing_hdvf()

◆ compute_perfect_hdvf()

◆ compute_rand_pairing_hdvf()

◆ compute_rand_perfect_hdvf()

◆ find_pair_A() [1/2]

◆ find_pair_A() [2/2]

◆ find_pairs_A() [1/2]

◆ find_pairs_A() [2/2]

◆ homology_chain()

◆ insert_reduction()

◆ is_perfect_hdvf()

◆ print_bnd_pairing()

◆ print_reduction_sub()

◆ psc_flags() [1/2]

◆ psc_flags() [2/2]

◆ psc_labels()

◆ read_hdvf_reduction() [1/2]

◆ read_hdvf_reduction() [2/2]

◆ set_mask_K()

◆ set_mask_L_K()

◆ write_hdvf_reduction() [1/2]

◆ write_hdvf_reduction() [2/2]