Simplex tree user manual#

Definition#

	The simplex tree is an efficient and flexible data structure for representing general (filtered) simplicial complexes. The data structure is described in [5]	Author: Clément Maria Since: GUDHI 2.0.0 License: MIT
Simplex tree user manual	Simplex tree reference manual
	TensorFlow layer for lower-star persistence on simplex trees

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 $\tau \subseteq \sigma \in \mathbf{K}\to \tau \in \mathbf{K}$. The dimension $n=|\sigma |-1$ of $\sigma$ is its number of elements minus 1.

A filtration of a simplicial complex is a function $f:\mathbf{K}\to \mathbb{R}$ satisfying $f(\tau )\le f(\sigma )$ 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.

Implementation#

The simplex tree is an efficient and flexible data structure for representing general (filtered) simplicial complexes. The data structure is described in [5].

Example#

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:

[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)