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/* This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
* See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
* Author(s): Clément Maria
*
* Copyright (C) 2014 Inria
*
* Modification(s):
* - YYYY/MM Author: Description of the modification
*/
#ifndef DOC_SIMPLEX_TREE_INTRO_SIMPLEX_TREE_H_
#define DOC_SIMPLEX_TREE_INTRO_SIMPLEX_TREE_H_
// needs namespace for Doxygen to link on classes
namespace Gudhi {
/** \defgroup simplex_tree Filtered Complexes
* @{
* \author Clément Maria
*
* A simplicial complex \f$\mathbf{K}\f$ on a set of vertices \f$V = \{1, \cdots ,|V|\}\f$ is a collection of
* simplices \f$\{\sigma\}\f$, \f$\sigma \subseteq V\f$ such that
* \f$\tau \subseteq \sigma \in \mathbf{K} \rightarrow \tau \in \mathbf{K}\f$. The dimension \f$n=|\sigma|-1\f$ of
* \f$\sigma\f$ is its number of elements minus \f$1\f$.
*
* A filtration of a simplicial complex is a function \f$f:\mathbf{K} \rightarrow \mathbb{R}\f$ satisfying
* \f$f(\tau)\leq f(\sigma)\f$ whenever \f$\tau \subseteq \sigma\f$. 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.
*
* \section filteredcomplexesimplementation Implementations
* \subsection filteredcomplexessimplextree 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 \cite boissonnatmariasimplextreealgorithmica
* \image html "Simplex_tree_representation.png" "Simplex tree representation"
*
* \subsubsection filteredcomplexessimplextreeexamples Examples
*
* Here is a list of simplex tree examples :
* \li <a href="_simplex_tree_2simple_simplex_tree_8cpp-example.html">
* Simplex_tree/simple_simplex_tree.cpp</a> - Simple simplex tree construction and basic function use.
*
* \li <a href="_simplex_tree_2simplex_tree_from_cliques_of_graph_8cpp-example.html">
* Simplex_tree/simplex_tree_from_cliques_of_graph.cpp</a> - Simplex tree construction from cliques of graph read in
* a file.
*
* Simplex tree construction with \f$\mathbb{Z}/3\mathbb{Z}\f$ coefficients on weighted graph Klein bottle file:
* \code $> ./simplex_tree_from_cliques_of_graph ../../data/points/Klein_bottle_complex.txt 3 \endcode
* \code 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 \endcode
*
* \li <a href="_simplex_tree_2example_alpha_shapes_3_simplex_tree_from_off_file_8cpp-example.html">
* Simplex_tree/example_alpha_shapes_3_simplex_tree_from_off_file.cpp</a> - Simplex tree is computed and displayed
* from a 3D alpha complex (Requires CGAL, GMP and GMPXX to be installed).
*
* \li <a href="_simplex_tree_2graph_expansion_with_blocker_8cpp-example.html">
* Simplex_tree/graph_expansion_with_blocker.cpp</a> - Simple simplex tree construction from a one-skeleton graph with
* a simple blocker expansion method.
*
* \subsection filteredcomplexeshassecomplex 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.
*
* @}
*/
} // namespace Gudhi
#endif // DOC_SIMPLEX_TREE_INTRO_SIMPLEX_TREE_H_
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