Filtered Complexes

A simplicial complex \(\mathbf{K}\) on a set of vertices \(V = \{1, \cdots ,|V|\}\) is a collection of simplices \(\{\sigma\}\), \(\sigma \subseteq V\) such that \(\tau \subseteq \sigma \in \mathbf{K} \rightarrow \tau \in \mathbf{K}\). The dimension \(n=|\sigma|-1\) of \(\sigma\) is its number of elements minus \(1\).

A filtration of a simplicial complex is a function \(f:\mathbf{K} \rightarrow \mathbb{R}\) satisfying \(f(\tau)\leq f(\sigma)\) whenever \(\tau \subseteq \sigma\). 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.

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 [7]

Here is a list of simplex tree examples :

- Simplex_tree/simple_simplex_tree.cpp - Simple simplex tree construction and basic function use.

- Simplex_tree/simplex_tree_from_cliques_of_graph.cpp - Simplex tree construction from cliques of graph read in a file.

Simplex tree construction with \(\mathbb{Z}/3\mathbb{Z}\) coefficients on weighted graph Klein bottle file:

$> ./simplex_tree_from_cliques_of_graph ../../data/points/Klein_bottle_complex.txt 3

Insert the 1-skeleton in the simplex tree in 0.000404 s.

max_dim = 3

Expand the simplex tree in 3.8e-05 s.

Information of the Simplex Tree:

Number of vertices = 10 Number of simplices = 98

- Simplex_tree/example_alpha_shapes_3_simplex_tree_from_off_file.cpp - Simplex tree is computed and displayed from a 3D alpha complex (Requires CGAL, GMP and GMPXX to be installed).

- Simplex_tree/graph_expansion_with_blocker.cpp - Simple simplex tree construction from a one-skeleton graph with a simple blocker expansion method.

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.

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