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#ifndef WITNESS_COMPLEX_DOC_H_
#define WITNESS_COMPLEX_DOC_H_
/**
\defgroup witness_complex Witness complex
\author Siargey Kachanovich
\image html "Witness_complex_representation.png" "Witness complex representation"
\section Definitions
Witness complex \f$ Wit(W,L) \f$ is a simplicial complex defined on two sets of points in \f$\mathbb{R}^D\f$:
\li \f$W\f$ set of **witnesses** and
\li \f$L \subseteq W\f$ set of **landmarks**.
The simplices are based on landmarks
and a simplex belongs to the witness complex if and only if it is witnessed, that is:
\f$ \sigma \subset L \f$ is witnessed if there exists a point \f$w \in W\f$ such that
w is closer to the vertices of \f$ \sigma \f$ than other points in \f$ L \f$ and all of its faces are witnessed as well.
The data structure is described in \cite boissonnatmariasimplextreealgorithmica .
\section Implementation
The principal class of this module is Gudhi::Witness_complex.
In both cases, the constructor for this class takes a {witness}x{closest_landmarks} table, where each row represents a witness and consists of landmarks sorted by distance to this witness.
This table can be constructed by two additional classes Landmark_choice_by_furthest_point and Landmark_choice_by_random_point also included in the module.
*\image html "bench_Cy8.png" "Running time as function on number of landmarks" width=10cm
*\image html "bench_sphere.png" "Running time as function on number of witnesses for |L|=300" width=10cm
\copyright GNU General Public License v3.
*/
#endif // WITNESS_COMPLEX_DOC_H_
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