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diff --git a/src/Skeleton_blocker/doc/SkeletonBlockerMainPage.h b/src/Skeleton_blocker/doc/SkeletonBlockerMainPage.h new file mode 100644 index 00000000..5a80bc8d --- /dev/null +++ b/src/Skeleton_blocker/doc/SkeletonBlockerMainPage.h @@ -0,0 +1,156 @@ + +/*! \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 + +*/ |