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diff --git a/src/Skeleton_blocker/doc/SkeletonBlockerMainPage.h b/src/Skeleton_blocker/doc/SkeletonBlockerMainPage.h deleted file mode 100644 index 5a80bc8d..00000000 --- a/src/Skeleton_blocker/doc/SkeletonBlockerMainPage.h +++ /dev/null @@ -1,156 +0,0 @@ - -/*! \mainpage Skeleton blockers data-structure - -\author David Salinas - -\section Introduction -The Skeleton-blocker data-structure had been introduced in the two papers -\cite skbl_socg2011 \cite skbl_ijcga2012. -It proposes a light encoding for simplicial complexes by storing only an *implicit* representation of its -simplices. -Intuitively, it just stores the 1-skeleton of a simplicial complex with a graph and the set of its "missing faces" that -is very small in practice (see next section for a formal definition). -This data-structure handles every classical operations used for simplicial complexes such as - as simplex enumeration or simplex removal but operations that are particularly efficient - are operations that do not require simplex enumeration such as edge iteration, link computation or simplex contraction. - - -\todo{image wont include} - -\section Definitions - -We recall briefly classical definitions of simplicial complexes \cite Munkres. -An abstract simplex is a finite non-empty set and its dimension is its number of elements minus 1. -Whenever \f$\tau \subset \sigma\f$ and \f$\tau \neq \emptyset \f$, \f$ \tau \f$ is called a face of -\f$ \sigma\f$ and \f$ \sigma\f$ is called a coface of \f$ \tau \f$ . Furthermore, -when \f$ \tau \neq \sigma\f$ we say that \f$ \tau\f$ is a proper-face of \f$ \sigma\f$. -An abstract simplicial complex is a set of simplices that contains all the faces of its simplices. -The 1-skeleton of a simplicial complex (or its graph) consists of its elements of dimension lower than 2. - - -\image latex "images/ds_representation.eps" "My application" width=10cm - - -To encode, a simplicial complex, one can encodes all its simplices. -In case when this number gets too large, -a lighter and implicit version consists of encoding only its graph plus some elements called missing faces or blockers. -A blocker is a simplex of dimension greater than 1 -that does not belong to the complex but whose all proper faces does. - - -Remark that for a clique complex (i.e. a simplicial complex whose simplices are cliques of its graph), the set of blockers -is empty and the data-structure is then particularly sparse. -One famous example of clique-complex is the Rips complex which is intensively used -in topological data-analysis. -In practice, the set of blockers of a simplicial complex -remains also small when simplifying a Rips complex with edge contractions -but also for most of the simplicial complexes used in topological data-analysis such as Delaunay, Cech or Witness complexes. -For instance, the numbers of blockers is depicted for a random 3 dimensional sphere embedded into \f$R^4\f$ -in figure X. - - -\image latex "images/blockers_curve.eps" "My application" width=10cm - - - - -\section API - -\subsection Overview - -Four classes define a simplicial complex namely : - -\li <Code>Skeleton_blocker_complex</Code> : a simplicial complex with basic operations such as vertex/edge/simplex enumeration and construction -\li <Code>Skeleton_blocker_link_complex</Code> : the link of a simplex in a parent complex. It is represented as a sub complex -of the parent complex -\li <Code>Skeleton_blocker_simplifiable_complex</Code> : a simplicial complex with simplification operations such as edge contraction or simplex collapse -\li <Code>Skeleton_blocker_geometric_complex</Code> : a simplicial complex who has access to geometric points in \f$R^d\f$ - -The two last classes are derived classes from the <Code>Skeleton_blocker_complex</Code> class. The class <Code>Skeleton_blocker_link_complex</Code> inheritates from a template passed parameter -that may be either <Code>Skeleton_blocker_complex</Code> or <Code>Skeleton_blocker_geometric_complex</Code> (a link may store points coordinates or not). - -\todo{include links} - -\subsection Visitor - -The class <Code>Skeleton_blocker_complex</Code> has a visitor that is called when usual operations such as adding an edge or remove a vertex are called. -You may want to use this visitor to compute statistics or to update another data-structure (for instance this visitor is heavily used in the -<Code>Contraction</Code> package). - - - -\section Example - - -\subsection s Iterating through vertices, edges, blockers and simplices - -Iteration through vertices, edges, simplices or blockers is straightforward with c++11 for range loops. -Note that simplex iteration with this implicit data-structure just takes -a few more time compared to iteration via an explicit representation -such as the Simplex Tree. The following example computes the Euler Characteristic -of a simplicial complex. - - \code{.cpp} - typedef Skeleton_blocker_complex<Skeleton_blocker_simple_traits> Complex; - typedef Complex::Vertex_handle Vertex_handle; - typedef Complex::Simplex_handle Simplex; - - const int n = 15; - - // build a full complex with 10 vertices and 2^n-1 simplices - Complex complex; - for(int i=0;i<n;i++) - complex.add_vertex(); - for(int i=0;i<n;i++) - for(int j=0;j<i;j++) - //note that add_edge adds the edge and all its cofaces - complex.add_edge(Vertex_handle(i),Vertex_handle(j)); - - // this is just to illustrate iterators, to count number of vertices - // or edges, complex.num_vertices() and complex.num_edges() are - // more appropriated! - unsigned num_vertices = 0; - for(auto v : complex.vertex_range()){ - ++num_vertices; - } - - unsigned num_edges = 0; - for(auto e : complex.edge_range()) - ++num_edges; - - unsigned euler = 0; - unsigned num_simplices = 0; - // we use a reference to a simplex instead of a copy - // value here because a simplex is a set of integers - // and copying it cost time - for(const Simplex & s : complex.simplex_range()){ - ++num_simplices; - if(s.dimension()%2 == 0) - euler += 1; - else - euler -= 1; - } - std::cout << "Saw "<<num_vertices<<" vertices, "<<num_edges<<" edges and "<<num_simplices<<" simplices"<<std::endl; - std::cout << "The Euler Characteristic is "<<euler<<std::endl; - \endcode - - -\verbatim -./SkeletonBlockerIteration -Saw 15 vertices, 105 edges and 32767 simplices -The Euler Characteristic is 1 - 0.537302s wall, 0.530000s user + 0.000000s system = 0.530000s CPU (98.6%) -\endverbatim - - - -\subsection Acknowledgements -The author wishes to thank Dominique Attali and André Lieutier for -their collaboration to write the two initial papers about this data-structure - and also Dominique for leaving him use a prototype. - - -\copyright GNU General Public License v3. -\verbatim Contact: David Salinas, david.salinas@inria.fr \endverbatim - -*/ |