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Diffstat (limited to 'src/Collapse/doc/intro_edge_collapse.h')
| -rw-r--r-- | src/Collapse/doc/intro_edge_collapse.h | 77 |
1 files changed, 40 insertions, 37 deletions
diff --git a/src/Collapse/doc/intro_edge_collapse.h b/src/Collapse/doc/intro_edge_collapse.h index 70c816f4..0691ccf6 100644 --- a/src/Collapse/doc/intro_edge_collapse.h +++ b/src/Collapse/doc/intro_edge_collapse.h @@ -24,9 +24,9 @@ namespace collapse { * \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$v^{\prime}\f$. + * \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 @@ -35,59 +35,62 @@ namespace collapse { * 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$K^1\f$. + * 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 \f$1\f$- minimal complex and the core \f$K^1\f$ of simplicial + * complex 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$v^{\prime}\f$ + * <i>if and only if</i> all the maximal simplices of \f$K\f$ that contain \f$e\f$ 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$v^{\prime}\f$. + * 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. + * This module implements edge collapse of a filtered flag complex, in particular it reduces a filtration of + * Vietoris-Rips 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 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 \f$i\f$ 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. + * 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 + * \subsection edgecollapseexample Basic edge collapse * - * 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. + * This example builds the `Flag_complex_sparse_matrix` from a proximity graph represented as a list of + * `Flag_complex_sparse_matrix::Filtered_edge`. + * Then it collapses edges and displays a new list of `Flag_complex_sparse_matrix::Filtered_edge` (with less edges) + * that will preserve the persistence homology computation. * - * Then, it is asked to display the distance matrix after the collapse operation. + * \include Collapse/edge_collapse_basic_example.cpp * - * \include Strong_collapse/strong_collapse_from_points.cpp + * When launching the example: * - * \code $> ./strong_collapse_from_points + * \code $> ./Edge_collapse_example_basic * \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 . - * + * \include Collapse/edge_collapse_example_basic.txt */ /** @} */ // end defgroup strong_collapse |