Skeleton-Blocker

Classes
struct	Gudhi::skeleton_blocker::Skeleton_blocker_simple_geometric_traits< GeometryTrait >
	Simple traits that is a model of SkeletonBlockerGeometricDS and can be passed as a template argument to Skeleton_blocker_geometric_complex. More...
struct	Gudhi::skeleton_blocker::Skeleton_blocker_simple_traits
	Simple traits that is a model of SkeletonBlockerDS and can be passed as a template argument to Skeleton_blocker_complex. More...
class	Gudhi::skeleton_blocker::Skeleton_blocker_geometric_complex< SkeletonBlockerGeometricDS >
	Class that represents a geometric complex that can be simplified. The class allows access to points of vertices. More...
class	Gudhi::skeleton_blocker::Skeleton_blocker_link_complex< ComplexType >
	Class representing the link of a simplicial complex encoded by a skeleton/blockers pair. It inherits from Skeleton_blocker_sub_complex because such complex is a sub complex of a root complex. More...
class	Gudhi::skeleton_blocker::Skeleton_blocker_complex< SkeletonBlockerDS >
	Abstract Simplicial Complex represented with a skeleton/blockers pair. More...

Detailed Description

Author

David Salinas

Introduction

The Skeleton-Blocker data-structure proposes a light encoding for simplicial complexes by storing only an implicit representation of its simplices [2],[3]. Intuitively, it just stores the 1-skeleton of a simplicial complex with a graph and the set of its "missing faces" that is very small in practice (see next section for a formal definition). This data-structure handles all simplicial complexes operations such as as simplex enumeration or simplex removal but operations that are particularly efficient are operations that do not require simplex enumeration such as edge iteration, link computation or simplex contraction.

Definitions

We recall briefly classical definitions of simplicial complexes [37]. An abstract simplex is a finite non-empty set and its dimension is its number of elements minus 1. Whenever \(\tau \subset \sigma\) and \(\tau \neq \emptyset \), \( \tau \) is called a face of \( \sigma\) and \( \sigma\) is called a coface of \( \tau \) . Furthermore, when \( \tau \neq \sigma\) we say that \( \tau\) is a proper-face of \( \sigma\). An abstract simplicial complex is a set of simplices that contains all the faces of its simplices. The 1-skeleton of a simplicial complex (or its graph) consists of its elements of dimension lower than 2.

Skeleton-blocker representation

To encode, a simplicial complex, one can encodes all its simplices. In case when this number gets too large, a lighter and implicit version consists of encoding only its graph plus some elements called missing faces or blockers. A blocker is a simplex of dimension greater than 1 that does not belong to the complex but whose all proper faces does.

Remark that for a clique complex (i.e. a simplicial complex whose simplices are cliques of its graph), the set of blockers is empty and the data-structure is then particularly sparse. One famous example of clique-complex is the Rips complex which is intensively used in topological data-analysis. In practice, the set of blockers of a simplicial complex remains also small when simplifying a Rips complex with edge contractions but also for most of the simplicial complexes used in topological data-analysis such as Delaunay, Cech or Witness complexes. For instance, the numbers of blockers is depicted for random 3-dimensional spheres embedded into \(R^4\) in next figure. Storing the graph and blockers of such simplicial complexes is much compact in this case than storing their simplices.

Number of blockers of random triangulations of 3-spheres

API

Overview

Two main classes of this package are Skeleton_blocker_complex and Skeleton_blocker_geometric_complex. The first one can be used to represent an abstract simplicial complex and supports most used operations in a simplicial complex such as :

• vertex/edge/simplex enumeration
• simplifications operations such as remove star, add star (e.g. general form of collapse), edge contractions

The class Skeleton_blocker_geometric_complex supports the same methods as Skeleton_blocker_complex and point access in addition.

Visitor

The class Skeleton_blocker_complex has a visitor that is called when usual operations such as adding an edge or remove a vertex are called. You may want to use this visitor to compute statistics or to update another data-structure (for instance this visitor is heavily used in the Edge contraction package).

Example

Iterating through vertices, edges, blockers and simplices

Iteration through vertices, edges, simplices or blockers is straightforward with c++11 for range loops. Note that simplex iteration with this implicit data-structure just takes a few more time compared to iteration via an explicit representation such as the Simplex Tree. The following example computes the Euler Characteristic of a simplicial complex.

typedef Skeleton_blocker_complex<Skeleton_blocker_simple_traits> Complex;
typedef Complex::Vertex_handle Vertex_handle;
typedef Complex::Simplex Simplex;

const int n = 15;

// build a full complex with 10 vertices and 2^n-1 simplices
Complex complex;
for(int i=0;i<n;i++)
        complex.add_vertex();
for(int i=0;i<n;i++)
        for(int j=0;j<i;j++)
                complex.add_edge_without_blockers(Vertex_handle(i),Vertex_handle(j));

// this is just to illustrate iterators, to count number of vertices
// or edges, complex.num_vertices() and complex.num_edges() are
// more appropriated!
unsigned num_vertices = 0;
for(auto v : complex.vertex_range()){
        ++num_vertices;
}

unsigned num_edges = 0;
for(auto e : complex.edge_range())
        ++num_edges;

unsigned euler = 0;
unsigned num_simplices = 0;
// we use a reference to a simplex instead of a copy
// value here because a simplex is a set of integers
// and copying it cost time
for(const Simplex & s : complex.star_simplex_range()){
        ++num_simplices;
        if(s.dimension()%2 == 0) 
                euler += 1;
        else 
                euler -= 1;
}
std::clog << "Saw "<<num_vertices<<" vertices, "<<num_edges<<" edges and "<<num_simplices<<" simplices"<<std::endl;
std::clog << "The Euler Characteristic is "<<euler<<std::endl;

./SkeletonBlockerIteration
Saw 15 vertices, 105 edges and 32767 simplices
The Euler Characteristic is 1
 0.537302s wall, 0.530000s user + 0.000000s system = 0.530000s CPU (98.6%)

Constructing a skeleton-blockers from a list of maximal faces or from a list of faces

std::vector<Simplex> simplices;

//add 4 triangles of a tetrahedron 0123
simplices.push_back(Simplex(Vertex_handle(0),Vertex_handle(1),Vertex_handle(2)));
simplices.push_back(Simplex(Vertex_handle(1),Vertex_handle(2),Vertex_handle(3)));
simplices.push_back(Simplex(Vertex_handle(3),Vertex_handle(0),Vertex_handle(2)));
simplices.push_back(Simplex(Vertex_handle(3),Vertex_handle(0),Vertex_handle(1)));

Complex complex;
//get complex from top faces
make_complex_from_top_faces(complex,simplices.begin(),simplices.end());

std::clog << "Simplices:"<<std::endl;
for(const Simplex & s : complex.star_simplex_range())
        std::clog << s << " ";
std::clog << std::endl;

//One blocker as simplex 0123 is not in the complex but all its proper faces are.
std::clog << "Blockers: "<<complex.blockers_to_string()<<std::endl;

//now build a complex from its full list of simplices
simplices.clear();
simplices.push_back(Simplex(Vertex_handle(0)));
simplices.push_back(Simplex(Vertex_handle(1)));
simplices.push_back(Simplex(Vertex_handle(2)));
simplices.push_back(Simplex(Vertex_handle(0),Vertex_handle(1)));
simplices.push_back(Simplex(Vertex_handle(1),Vertex_handle(2)));
simplices.push_back(Simplex(Vertex_handle(2),Vertex_handle(0)));
complex = Complex(simplices.begin(),simplices.end());

std::clog << "Simplices:"<<std::endl;
for(const Simplex & s : complex.star_simplex_range())
        std::clog << s << " ";
std::clog << std::endl;

//One blocker as simplex 012 is not in the complex but all its proper faces are.
std::clog << "Blockers: "<<complex.blockers_to_string()<<std::endl;

./SkeletonBlockerFromSimplices
Simplices:
{0} {0,1} {0,2} {0,3} {0,1,2} {0,1,3} {0,2,3} {1} {1,2} {1,3} {1,2,3} {2} {2,3} {3} 
Blockers: {0,1,2,3}

Simplices:
{0} {0,1} {0,2} {1} {1,2} {2} 
Blockers: {0,1,2}

Acknowledgements

The author wishes to thank Dominique Attali and André Lieutier for their collaboration to write the two initial papers [2],[3] about this data-structure and also Dominique for leaving him use a prototype.