Classes
class	Gudhi::Simplex_tree< SimplexTreeOptions >
	Simplex Tree data structure for representing simplicial complexes. More...

struct	Gudhi::simplex_tree::empty_filtration_value_t
	Returns `Filtration_value(0)` when converted to `Filtration_value`. More...

class	Gudhi::Simplex_tree_simplex_vertex_iterator< SimplexTree >
	Iterator over the vertices of a simplex in a SimplexTree. More...

class	Gudhi::Simplex_tree_boundary_simplex_iterator< SimplexTree >
	Iterator over the simplices of the boundary of a simplex. More...

class	Gudhi::Simplex_tree_boundary_opposite_vertex_simplex_iterator< SimplexTree >
	Iterator over the simplices of the boundary of a simplex and their opposite vertices. More...

class	Gudhi::Simplex_tree_complex_simplex_iterator< SimplexTree >
	Iterator over the simplices of a simplicial complex. More...

class	Gudhi::Simplex_tree_skeleton_simplex_iterator< SimplexTree >
	Iterator over the simplices of the skeleton of a given dimension of the simplicial complex. More...

class	Gudhi::Simplex_tree_dimension_simplex_iterator< SimplexTree >
	Iterator over the simplices of the simplicial complex that match the dimension specified by the parameter. More...

struct	Gudhi::Simplex_tree_node_explicit_storage< SimplexTree >
	Node of a simplex tree with filtration value and simplex key. More...

struct	Gudhi::Simplex_tree_options_default

struct	Gudhi::Simplex_tree_options_full_featured

struct	Gudhi::Simplex_tree_options_minimal

class	Gudhi::Simplex_tree_siblings< SimplexTree, MapContainer >
	Data structure to store a set of nodes in a SimplexTree sharing the same parent node. More...

class	Extended_simplex_type
	Extended simplex type data structure for representing the type of simplices in an extended filtration. More...

Functions
template<typename Arithmetic_filtration_value>
bool	Gudhi::is_positive_infinity (const Arithmetic_filtration_value &f)
	Returns true if and only if the given filtration value is at infinity. This is the overload for when FiltrationValue is an arithmetic type, like double, int etc. It simply tests equality with `std::numeric_limits<FiltrationValue>::infinity()` if defined or with `std::numeric_limits<FiltrationValue>::max()` otherwise. Can therefore be also used with other classes as long as infinity is defined that way.

template<typename Arithmetic_filtration_value>
bool	Gudhi::unify_lifetimes (Arithmetic_filtration_value &f1, const Arithmetic_filtration_value &f2)
	Given two filtration values at which a simplex exists, stores in the first value the minimal union of births generating a lifetime including those two values. This is the overload for when FiltrationValue is an arithmetic type, like double, int etc. Because the filtration values are totally ordered then, the union is simply the minimum of the two values.

template<typename Arithmetic_filtration_value>
bool	Gudhi::intersect_lifetimes (Arithmetic_filtration_value &f1, const Arithmetic_filtration_value &f2)
	Given two filtration values, stores in the first value the lowest common upper bound of the two values. If a filtration value has value `NaN`, it should be considered as the lowest value possible. This is the overload for when FiltrationValue is an arithmetic type, like double, float, int etc. Because the filtration values are totally ordered then, the upper bound is always the maximum of the two values.

Detailed Description

Author: Clément Maria

A simplicial complex $\mathbf{K}$ on a set of vertices $V = \{1, \cdots ,|V|\}$ is a collection of simplices $\{\sigma\}$, $\sigma \subseteq V$ such that $\tau \subseteq \sigma \in \mathbf{K} \rightarrow \tau \in \mathbf{K}$. The dimension $n=|\sigma|-1$ of $\sigma$ is its number of elements minus $1$.

A filtration of a simplicial complex is a function $f:\mathbf{K} \rightarrow \mathbb{R}$ satisfying $f(\tau)\leq f(\sigma)$ whenever $\tau \subseteq \sigma$. Ordering the simplices by increasing filtration values (breaking ties so as a simplex appears after its subsimplices of same filtration value) provides an indexing scheme.

Implementations

Simplex tree

There are two implementation of complexes. The first on is the Simplex_tree data structure. The simplex tree is an efficient and flexible data structure for representing general (filtered) simplicial complexes. The data structure is described in [7]

Simplex tree representation

Examples

Here is a list of simplex tree examples :

Simplex_tree/simple_simplex_tree.cpp - Simple simplex tree construction and basic function use.

Simplex_tree/simplex_tree_from_cliques_of_graph.cpp - Simplex tree construction from cliques of graph read in a file.

Simplex tree construction with $\mathbb{Z}/3\mathbb{Z}$ coefficients on weighted graph Klein bottle file:

$> ./simplex_tree_from_cliques_of_graph ../../data/points/Klein_bottle_complex.txt 3

 Insert the 1-skeleton in the simplex tree in 0.000404 s. 
max_dim = 3
Expand the simplex tree in 3.8e-05 s. 
Information of the Simplex Tree: 
Number of vertices = 10   Number of simplices = 98 

Simplex_tree/example_alpha_shapes_3_simplex_tree_from_off_file.cpp - Simplex tree is computed and displayed from a 3D alpha complex (Requires CGAL, GMP and GMPXX to be installed).

Simplex_tree/graph_expansion_with_blocker.cpp - Simple simplex tree construction from a one-skeleton graph with a simple blocker expansion method.

Hasse complex

The second one is the Hasse_complex. The Hasse complex is a data structure representing explicitly all co-dimension 1 incidence relations in a complex. It is consequently faster when accessing the boundary of a simplex, but is less compact and harder to construct from scratch.

Iterators and range types for the Simplex_tree.

Represents a node of a Simplex_tree.

Pre-defined options for the Simplex_tree.

Represents a set of node of a Simplex_tree that share the same parent.

Function Documentation

◆ unify_lifetimes()

template<typename Arithmetic_filtration_value>

bool Gudhi::unify_lifetimes	(	Arithmetic_filtration_value &	f1,
		const Arithmetic_filtration_value &	f2 )

Given two filtration values at which a simplex exists, stores in the first value the minimal union of births generating a lifetime including those two values. This is the overload for when FiltrationValue is an arithmetic type, like double, int etc. Because the filtration values are totally ordered then, the union is simply the minimum of the two values.

NaN values are not supported.

Classes