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Diffstat (limited to 'src/Collapse/doc/intro_edge_collapse.h')
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1 files changed, 28 insertions, 48 deletions
diff --git a/src/Collapse/doc/intro_edge_collapse.h b/src/Collapse/doc/intro_edge_collapse.h index 81edd79f..12e909c8 100644 --- a/src/Collapse/doc/intro_edge_collapse.h +++ b/src/Collapse/doc/intro_edge_collapse.h @@ -17,80 +17,60 @@ namespace collapse { /** \defgroup edge_collapse Edge collapse * - * \author Siddharth Pritam + * \author Siddharth Pritam and Marc Glisse * * @{ * - * 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. + * This module implements edge collapse of a filtered flag complex as described in \cite edgecollapsearxiv, in + * particular it reduces a filtration of Vietoris-Rips complex represented by a graph to a smaller flag filtration with + * the same persistent homology. * * \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 \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: + * subcomplex \f$L\f$ in \f$K\f$, such that \f$lk_K(e) = v^{\prime}L\f$. We say that the vertex \f$v^{\prime}\f$ + * \e dominates \f$e\f$ and \f$e\f$ is \e dominated by \f$v^{\prime}\f$. + * An <b> elementary edge collapse </b> is the removal of a dominated edge \f$e\f$ from \f$K\f$ (the cofaces of \f$e\f$ + * are implicitly removed as well). + * Domination is used as a simple sufficient condition that ensures that this removal is a homotopy preserving + * operation. + * + * 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 \f$e\f$ also contain \f$v^{\prime}\f$ + * -- For a general simplicial complex: an edge \f$e \in K\f$ is dominated by another vertex \f$v^{\prime} \in K\f$, + * if and only if 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$v^{\prime}\f$. + * -- For a flag complex: an edge \f$e \in K\f$ is dominated by another vertex \f$v^{\prime} \in K\f$, if and only + * if all the vertices in \f$K\f$ that have an edge with both vertices of \f$e\f$ also have an edge with + * \f$v^{\prime}\f$. Notice that this only depends on the graph. * - * 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 + * In the context of a filtration, an edge collapse may translate into an increase of the filtration value of an edge, + * or its removal if it already had the largest filtration value. + * The algorithm to compute the smaller induced filtration is described in \cite edgecollapsearxiv. + * Edge collapse can be successfully employed to reduce any input 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 \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. - * + * The algorithm implemented here does not produce a minimal filtration. Taking its output and applying the algorithm a + * second time may further simplify the filtration. + * * \subsection edgecollapseexample Basic edge collapse * * This example calls `Gudhi::collapse::flag_complex_collapse_edges()` from a proximity graph represented as a list of * `Filtered_edge`. - * Then it collapses edges and displays a new list of `Filtered_edge` (with less edges) + * Then it collapses edges and displays a new list of `Filtered_edge` (with fewer edges) * that will preserve the persistence homology computation. * - * \include Collapse/edge_collapse_basic_example.cpp + * \include edge_collapse_basic_example.cpp * * When launching the example: * * \code $> ./Edge_collapse_example_basic * \endcode * - * the program output is: + * the program output could be: * - * \include Collapse/edge_collapse_example_basic.txt + * \include edge_collapse_example_basic.txt */ /** @} */ // end defgroup strong_collapse |