Classes
class	Gudhi::Hasse_complex< FiltrationValue, SimplexKey, VertexHandle >
	Data structure representing a Hasse diagram, i.e. a complex where all codimension 1 incidence relations are explicitly encoded. More...

class	Gudhi::Simplex_tree< IndexingTag, FiltrationValue, SimplexKey, VertexHandle >
	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

Copyright: GNU General Public License v3.

Classes

Detailed Description