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-
-/*! \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
-
-*/