SimplexTree

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.

Attributes

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

Properties

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.

Methods

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.

adjacent

SimplexTree.adjacent(self, simplices)

Checks for adjacencies between simplices.

card

SimplexTree.card(self, p=None)

Returns the cardinality of various skeleta of the complex.

Parameters

p : int, optional (default=None)

dimension parameter. Defaults to None.

Returns

cardinalities : Union[int, tuple],

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

coface_roots

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.

Parameters

sigma : Collection, optional (default=[])

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

Returns

coface_roots : list[Collection],

the coface roots of sigma.

cofaces

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

Returns the cofaces of sigma.

Note, by definition, sigma itself is considered as a coface.

Parameters

sigma : Collection, optional (default=[])

the simplex to obtain cofaces of.

Returns

cofaces : list[Collection],

the cofaces of sigma.

collapse

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).

Parameters

sigma : Collection, required

maximal simplex to collapse

tau : Collection, required

face of sigma to collapse

Returns

collapsed : bool,

whether the pair was collapsed

Examples

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

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

print(st)

degree

SimplexTree.degree(self, vertices=None)

Computes the degree of select vertices in the trie.

Parameters

vertices : ArrayLike, optional (default=None)

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

Returns

degrees : Union[ArrayLike, int],

degree of each vertex id given in ‘vertices’.

expand

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.

Parameters

k : int, required

maximum dimension to expand to.

f : Callable[[Collection], bool], optional (default=None)

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

Examples

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)

faces

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

Wrapper for simplices function.

find

SimplexTree.find(self, simplices)

Finds whether simplices exist in Simplex Tree.

Parameters

simplices : Iterable[Collection], required

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

Returns

found : np.ndarray,

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.

insert

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.

Parameters

simplices : Iterable[Collection], required

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.

Examples

from simplextree import SimplexTree
st = SimplexTree([range(3)])
print(st)
Simplex Tree with (3, 3, 1) (0, 1, 2)-simplices
st.insert([[0,1]])
print(st)
Simplex Tree with (3, 3, 1) (0, 1, 2)-simplices

print(st)

maximal

SimplexTree.maximal(self)

Returns the maximal simplices in the complex.

reindex

SimplexTree.reindex(self, labels=None)

Reindexes the vertex labels of the complex.

remove

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.

Parameters

simplices : Iterable[Collection], required

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.

Examples

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

simplices

SimplexTree.simplices(self, p=None)

Returns the p-simplices in the complex.

skeleton

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.

Parameters

p : int, optional (default=None)

the dimension of the skeleton.

sigma : Collection, optional (default=[])

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

Returns

list : Iterable[Collection],

the simplices in the p-skeleton of sigma.

traverse

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).

Parameters

order : str, optional (default=‘preorder’)

the type of traversal of the simplex tree to execute.

f : Callable, optional (default=print)

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

sigma : Collection, optional (default=[])

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

p : int, optional (default=0)

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

vertex_collapse

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

Maps a pair of vertices into a single vertex.

Parameters

u : int, required

the first vertex in the free pair.

v : int, required

the second vertex in the free pair.

w : int, required

the target vertex to collapse to.

Returns

collapsed : bool,

whether the collapse was performed.