Gudhi  1.2.0
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Filtered Complexes

Classes

struct  Gudhi::Simplex_tree_options_full_featured
 Model of SimplexTreeOptions. More...
 
class  Gudhi::Simplex_tree< SimplexTreeOptions >
 Simplex Tree data structure for representing simplicial complexes. More...
 

Detailed Description

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.

Implementations: 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 [4]

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.

Author
Clément Maria
Version
1.0
Date
2014