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diff --git a/src/cython/doc/simplex_tree_user.rst b/src/cython/doc/simplex_tree_user.rst new file mode 100644 index 00000000..6b55c4e7 --- /dev/null +++ b/src/cython/doc/simplex_tree_user.rst @@ -0,0 +1,67 @@ +======================== +Simplex tree user manual +======================== +Definition +---------- + +.. include:: simplex_tree_sum.rst + +A simplicial complex :math:`\mathbf{K}` on a set of vertices :math:`V = \{1, \cdots ,|V|\}` is a collection of +simplices :math:`\{\sigma\}`, :math:`\sigma \subseteq V` such that +:math:`\tau \subseteq \sigma \in \mathbf{K} \rightarrow \tau \in \mathbf{K}`. The dimension :math:`n=|\sigma|-1` of +:math:`\sigma` is its number of elements minus `1`. + +A filtration of a simplicial complex is a function :math:`f:\mathbf{K} \rightarrow \mathbb{R}` satisfying +:math:`f(\tau)\leq f(\sigma)` whenever :math:`\tau \subseteq \sigma`. Ordering the simplices by increasing filtration +values (breaking ties so as a simplex appears after its subsimplices of same filtration value) provides an indexing +scheme. + + +Implementation +-------------- + +There are two implementation of complexes. The first on is the Simplex_tree data structure. +The simplex tree is an efficient and flexible data structure for representing general (filtered) simplicial complexes. +The data structure is described in :cite`boissonnatmariasimplextreealgorithmica`. + +The second one is the Hasse_complex. The Hasse complex is a data structure representing explicitly all co-dimension 1 +incidence relations in a complex. It is consequently faster when accessing the boundary of a simplex, but is less +compact and harder to construct from scratch. + +Example +------- + +.. testcode:: + + import gudhi + st = gudhi.SimplexTree() + if st.insert_simplex([0, 1]): + print("[0, 1] inserted") + if st.insert_simplex_and_subfaces([0, 1, 2], filtration=4.0): + print("[0, 1, 2] inserted") + if st.find([0, 1]): + print("[0, 1] found") + print("num_vertices=", st.num_vertices()) + print("num_simplices=", st.num_simplices()) + print("skeleton_tree(2) =") + for sk_value in st.get_skeleton_tree(2): + print(sk_value) + + +The output is: + +.. testoutput:: + + [0, 1] inserted + [0, 1, 2] inserted + [0, 1] found + ('num_vertices=', 3) + ('num_simplices=', 7) + skeleton_tree(2) = + ([0, 1, 2], 4.0) + ([0, 1], 0.0) + ([0, 2], 4.0) + ([0], 0.0) + ([1, 2], 4.0) + ([1], 4.0) + ([2], 4.0) |