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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([0, 1]):
print("[0, 1] inserted")
if st.insert([0, 1, 2], filtration=4.0):
print("[0, 1, 2] inserted")
if st.find([0, 1]):
print("[0, 1] found")
result_str = 'num_vertices=' + repr(st.num_vertices())
print(result_str)
result_str = 'num_simplices=' + repr(st.num_simplices())
print(result_str)
print("skeleton(2) =")
for sk_value in st.get_skeleton(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(2) =
([0, 1, 2], 4.0)
([0, 1], 0.0)
([0, 2], 4.0)
([0], 0.0)
([1, 2], 4.0)
([1], 0.0)
([2], 4.0)
|
