```
from simplextree import SimplexTree
= SimplexTree([range(3)])
st print(st)
```

`Simplex Tree with (3, 3, 1) (0, 1, 2)-simplices`

`SimplexTree(self, simplices=None)`

SimplexTree provides lightweight wrapper around a Simplex Tree data structure.

This class exposes a native extension module wrapping a simplex tree implemented with modern C++.

The Simplex Tree was originally introduced in the paper: > Boissonnat, Jean-Daniel, and Clément Maria. “The simplex tree: An efficient data structure for general simplicial complexes.” Algorithmica 70.3 (2014): 406-427.

Name | Type | Description |
---|---|---|

n_simplices | ndarray | number of simplices |

dimension | int | maximal dimension of the complex |

id_policy | str | policy for generating new vertex ids |

Name | Type | Description |
---|---|---|

vertices | ndarray | 0-simplices in the complex. |

edges | ndarray | 1-simplices in the complex. |

triangles | ndarray | 2-simplices in the complex. |

quads | ndarray | 3-simplices in the complex. |

connected_components | ndarray | connected component ids. |

Name | Description |
---|---|

adjacent | Checks for adjacencies between simplices. |

card | Returns the cardinality of various skeleta of the complex. |

coface_roots | Returns the roots whose subtrees span the cofaces of `sigma` . |

cofaces | Returns the cofaces of `sigma` . |

collapse | Performs an elementary collapse on two given simplices. |

degree | Computes the degree of select vertices in the trie. |

expand | Performs a k-expansion of the complex. |

faces | Wrapper for simplices function. |

find | Finds whether simplices exist in Simplex Tree. |

insert | Inserts simplices into the Simplex Tree. |

link | Returns the simplices in the link of `sigma` . |

maximal | Returns the maximal simplices in the complex. |

reindex | Reindexes the vertex labels of the complex. |

remove | Removes simplices into the Simplex Tree. |

simplices | Returns the p-simplices in the complex. |

skeleton | Returns the simplices in the p-skeleton of `sigma` . |

traverse | Traverses the simplex tree in the specified order, calling `f` on each simplex encountered. |

vertex_collapse | Maps a pair of vertices into a single vertex. |

`SimplexTree.adjacent(self, simplices)`

Checks for adjacencies between simplices.

`SimplexTree.card(self, p=None)`

Returns the cardinality of various skeleta of the complex.

dimension parameter. Defaults to None.

if p is an integer, the number of p-simplices in the complex. Otherwise a tuple indicating the number of simplices of all dimensions.

`SimplexTree.coface_roots(self, sigma=\[\])`

Returns the roots whose subtrees span the cofaces of `sigma`

.

Note that `sigma`

itself is included in the set of its cofaces.

the simplex to obtain cofaces of. Defaults to the empty set (root node).

the coface roots of `sigma`

.

`SimplexTree.cofaces(self, sigma=\[\])`

Returns the cofaces of `sigma`

.

Note, by definition, `sigma`

itself is considered as a coface.

the simplex to obtain cofaces of.

the cofaces of `sigma`

.

`SimplexTree.collapse(self, tau, sigma)`

Performs an elementary collapse on two given simplices.

Checks whether its possible to collapse \sigma through \tau, and if so, both simplices are removed. A simplex \sigma is said to be collapsible through one of its faces \tau if \sigma is the only coface of \tau (excluding \tau itself).

maximal simplex to collapse

face of sigma to collapse

whether the pair was collapsed

from splex import SimplexTree st = SimplexTree([[0,1,2]]) print(st)

st.collapse([1,2], [0,1,2])

print(st)

`SimplexTree.degree(self, vertices=None)`

Computes the degree of select vertices in the trie.

Retrieves vertex degrees If no vertices are specified, all degrees are computed. Non-existing vertices by default have degree 0.

degree of each vertex id given in ‘vertices’.

`SimplexTree.expand(self, k, f=None)`

Performs a k-expansion of the complex.

This function is particularly useful for expanding clique complexes beyond their 1-skeleton.

maximum dimension to expand to.

boolean predicate which returns whether a simplex should added to the complex (and further expanded).

from simplextree import SimplexTree from itertools import combinations st = SimplexTree(combinations(range(8), 2)) print(st)

st.expand(k=2, lambda s: 2 in s) # Expand only triangles containing 2 as a vertex print(st)

st.expand(k=2) # Expand all 2-cliques print(st)

`SimplexTree.faces(self, p=None, sigma=\[\], **kwargs)`

Wrapper for simplices function.

`SimplexTree.find(self, simplices)`

Finds whether simplices exist in Simplex Tree.

Iterable of simplices to insert (each of which are SimplexLike)

boolean array indicating whether each simplex was found in the complex

Note

```
If the iterable is an 2-dim np.ndarray, then the p-simplex to find is given by each contiguous p+1 stride.
Otherwise, each element of the iterable to casted to a Simplex and then searched for in the tree.
```

`SimplexTree.insert(self, simplices)`

Inserts simplices into the Simplex Tree.

By definition, inserting a simplex also inserts all of its faces. If the simplex already exists in the complex, the tree is not modified.

Iterable of simplices to insert (each of which are SimplexLike)

Note

```
If the iterable is an 2-dim np.ndarray, then a p-simplex is inserted along each contiguous p+1 stride.
Otherwise, each element of the iterable to casted to a Simplex and then inserted into the tree.
```

`Simplex Tree with (3, 3, 1) (0, 1, 2)-simplices`

print(st)

`SimplexTree.link(self, sigma=\[\])`

Returns the simplices in the link of `sigma`

.

`SimplexTree.maximal(self)`

Returns the maximal simplices in the complex.

`SimplexTree.reindex(self, labels=None)`

Reindexes the vertex labels of the complex.

`SimplexTree.remove(self, simplices)`

Removes simplices into the Simplex Tree.

By definition, removing a face also removes all of its cofaces. If the simplex does not exist in the complex, the tree is not modified.

Iterable of simplices to insert (each of which are SimplexLike).

Note

```
If the iterable is an 2-dim np.ndarray, then a p-simplex is removed along each contiguous p+1 stride.
Otherwise, each element of the iterable to casted to a Simplex and then removed from the tree.
```

st = SimplexTree([range(3)]) print(st) st.remove([[0,1]]) print(st)

`SimplexTree.simplices(self, p=None)`

Returns the p-simplices in the complex.

`SimplexTree.skeleton(self, p=None, sigma=\[\])`

Returns the simplices in the p-skeleton of `sigma`

.

Note that, when dim(`sigma`

) <= `p`

, `sigma`

is included in the skeleton.

the dimension of the skeleton.

the simplex to obtain cofaces of. Defaults to the empty set (root node).

the simplices in the p-skeleton of `sigma`

.

`SimplexTree.traverse(self, order='preorder', f=print, sigma=\[\], p=0)`

Traverses the simplex tree in the specified order, calling `f`

on each simplex encountered.

Supported traversals include breadth-first / level order (“bfs”, “levelorder”), depth-first / prefix (“dfs”, “preorder”). faces, cofaces, coface roots (“coface_roots”), p-skeleton, p-simplices, maximal simplices (“maximal”), and link.

Where applicable, each traversal begins its traversal `sigma`

, which defaults to the empty set (root node).

the type of traversal of the simplex tree to execute.

a function to evaluate on every simplex in the traversal. Defaults to print.

simplex to start the traversal at, where applicable. Defaults to the root node (empty set).

dimension of simplices to restrict to, where applicable. Defaults to 0.

`SimplexTree.vertex_collapse(self, u, v, w)`

Maps a pair of vertices into a single vertex.

the first vertex in the free pair.

the second vertex in the free pair.

the target vertex to collapse to.

whether the collapse was performed.