/*    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):       Siddharth Pritam
 *
 *    Copyright (C) 2019 Inria
 *
 *    Modification(s):
 *      - YYYY/MM Author: Description of the modification
 */

#ifndef DOC_EDGE_COLLAPSE_INTRO_EDGE_COLLAPSE_H_
#define DOC_EDGE_COLLAPSE_INTRO_EDGE_COLLAPSE_H_

namespace Gudhi {

namespace edge_collapse {

/**  \defgroup edge_collapse Edge collapse
 * 
 * \author    Siddharth Pritam
 * 
 * @{
 * 
 * \section edge_collapse_definition Edge collapse definition
 * 
 * An edge \f$e\f$ in a simplicial complex \f$K\f$ is called a <b>dominated edge</b> if the link of \f$e\f$ in
 * \f$K\f$, \f$lk_K(e)\f$ is a simplicial cone, that is, there exists a vertex \f$v^{\prime} \notin e\f$ and a subcomplex
 * \f$L\f$ in \f$K\f$, such that \f$lk_K(e) = v^{\prime}L\f$. We say that the vertex  \f$v^{\prime}\f$ is {dominating}
 * \f$e\f$ and \f$e\f$ is {dominated} by \f$v^{\prime}\f$. 
 * An <b> elementary egde collapse </b> is the removal of a dominated edge \f$e\f$ from \f$K\f$, 
 * which we denote with \f$K\f$ \f${\searrow\searrow}^1 \f$ \f$K\setminus e\f$. 
 * The symbol \f$\mathbf{K\setminus e}\f$ (deletion of \f$e\f$ from \f$K\f$) refers to the subcomplex of \f$K\f$ which
 * has all simplices of \f$K\f$ except \f$e\f$ and the ones containing \f$e\f$.
 * There is an <b>edge collapse</b> from a simplicial complex \f$K\f$ to its subcomplex \f$L\f$, 
 * if there exists a series of elementary edge collapses from \f$K\f$ to \f$L\f$, denoted as \f$K\f$
 * \f${\searrow\searrow}\f$ \f$L\f$.
 * 
 * An edge collapse is a homotopy preserving operation, and it can be further expressed as sequence of the classical elementary simple collapse. 
 * A complex without any dominated edge is called a $1$- minimal complex and the core \f$K^1\f$ of simplicial comlex is a
 * minimal complex such that \f$K\f$ \f${\searrow\searrow}\f$ \f$K^1\f$.
 * Computation of a core (not unique) involves computation of dominated edges and the dominated edges can be easily
 * characterized as follows:
 * 
 * -- For general simplicial complex: An edge \f$e \in K\f$ is dominated by another vertex \f$v^{\prime} \in K\f$,
 * <i>if and only if</i> all the maximal simplices of \f$K\f$ that contain $e$ also contain \f$v^{\prime}\f$
 * 
 * -- For a flag complex: An edge \f$e \in K\f$ is dominated by another vertex \f$v^{\prime} \in K\f$, <i>if and only
 * if</i> all the vertices in \f$K\f$ that has an edge with both vertices of \f$e\f$  also has an edge with \f$v^{\prime}\f$.

 *  This module implements edge collapse of a filtered flag complex, in particular  it reduces a filtration of Vietoris-Rips (VR) complex from its graph 
 *  to another smaller flag filtration with same persistence. Where a filtration is a sequence of simplicial
 *  (here Rips) complexes connected with inclusions. The algorithm to compute the smaller induced filtration is described in Section 5 \cite edgecollapsesocg2020.
 *  Edge collapse can be successfully employed to reduce any given filtration of flag complexes to a smaller induced
 *  filtration which preserves the persistent homology of the original filtration and is a flag complex as well. 

 * The general idea is that we consider edges in the filtered graph and sort them according to their filtration value giving them a total order.
 * Each edge gets a unique index denoted as \f$i\f$ in this order.  To reduce the filtration, we move forward with increasing filtration value 
 * in the graph and check if the current edge \f$e_i\f$ is dominated in the current graph  \f$G_i := \{e_1, .. e_i\} \f$ or not. 
 * If the edge \f$e_i\f$ is dominated we remove it from the filtration and move forward to the next edge \f$e_{i+1}\f$. 
 * If f$e_i\f$ is non-dominated then we keep it in the reduced filtration and then go backward in the current graph \f$G_i\f$ to look for new non-dominated edges 
 * that was dominated before but might become non-dominated at this point.  
 * If an edge \f$e_j, j < i \f$ during the backward search is found to be non-dominated, we include \f$\e_j\f$ in to the reduced filtration and we set its new filtration value to be $i$ that is the index of \f$e_i\f$.
 * The precise mechanism for this reduction has been described in Section 5 \cite edgecollapsesocg2020. 
 * Here we implement this mechanism for a filtration of Rips complex, 
 * After perfoming the reduction the filtration reduces to a flag-filtration with the same persistence as the original filtration. 
 * 

 * Comment: I think it would be good if you (Vincent) check the later part according to the examples you build.
 * \subsection edge_collapse_from_points_example Example from a point cloud and a distance function
 * 
 * This example builds the edge graph from the given points, threshold value, and distance function.
 * Then it creates a `Flag_complex_edge_collapse` (exact version) with it.
 * 
 * Then, it is asked to display the distance matrix after the collapse operation.
 * 
 * \include Strong_collapse/strong_collapse_from_points.cpp
 * 
 * \code $> ./strong_collapse_from_points
 * \endcode
 *
 * the program output is:
 * 
 * \include Strong_collapse/strong_collapse_from_points_for_doc.txt
 * 
 * A `Gudhi::rips_complex::Rips_complex` can be built from the distance matrix if you want to compute persistence on
 * top of it.

 * For more information about our approach of computing edge collapses and persitent homology via edge collapses,
 * we refer the users to \cite edgecollapsesocg2020 .
 * 
 */
/** @} */  // end defgroup strong_collapse

}  // namespace edge_collapse

}  // namespace Gudhi

#endif  // DOC_EDGE_COLLAPSE_INTRO_EDGE_COLLAPSE_H_