Incremental Connected Components
This section describes a family of functions and classes that work
together to calculate the connected components of an undirected graph.
The algorithm used here is based on the disjoint-sets (fast
union-find) data structure [8,27]
which is a good method to use for situations where the graph is
growing (edges are being added) and the connected components
information needs to be updated repeatedly. This method does not cover
the situation where edges are both added and removed from the graph,
hence it is called incremental[42] (and not
fully dynamic). The disjoint-sets class is described in Section Disjoint Sets.
The following five operations are the primary functions that
is
used to calculate and maintain the connected components. The objects
used here are a graph g, a disjoint-sets structure ds,
and vertices u
and v.
- initialize_incremental_components(g, ds)
Basic initialization of the disjoint-sets structure. Each
vertex in the graph g is in its own set.
- incremental_components(g, ds)
The connected components are calculated based on the edges in the graph
g and the information is embedded in ds.
- ds.find_set(v)
Extracts the component information for vertex v
from the
disjoint-sets.
- ds.union_set(u, v)
Update the disjoint-sets structure when edge
(u,v) is added to the graph.
Complexity
The time complexity for the whole process is O(V + E
alpha(E,V)) where E is the total number of edges in the
graph (by the end of the process) and V is the number of
vertices. alpha is the inverse of Ackermann's function which
has explosive recursively exponential growth. Therefore its inverse
function grows very slowly. For all practical purposes
alpha(m,n) <= 4 which means the time complexity is only
slightly larger than O(V + E).
Example
Maintain the connected components of a graph while adding edges using
the disjoint-sets data structure. The full source code for this
example can be found in examples/incremental_components.cpp.
typedef adjacency_list <vecS, vecS, undirectedS> Graph;
typedef graph_traits<Graph>::vertex_descriptor Vertex;
typedef graph_traits<Graph>::vertices_size_type size_type;
const int N = 6;
Graph G(N);
std::vector<size_type> rank(num_vertices(G));
std::vector<Vertex> parent(num_vertices(G));
typedef size_type* Rank;
typedef Vertex* Parent;
disjoint_sets<Rank, Parent> ds(&rank[0], &parent[0]);
initialize_incremental_components(G, ds);
incremental_components(G, ds);
graph_traits<Graph>::edge_descriptor e;
bool flag;
boost::tie(e,flag) = add_edge(0, 1, G);
ds.union_set(0,1);
boost::tie(e,flag) = add_edge(1, 4, G);
ds.union_set(1,4);
boost::tie(e,flag) = add_edge(4, 0, G);
ds.union_set(4,0);
boost::tie(e,flag) = add_edge(2, 5, G);
ds.union_set(2,5);
cout << "An undirected graph:" << endl;
print_graph(G, get(vertex_index, G));
cout << endl;
graph_traits<Graph>::vertex_iterator i,end;
for (boost::tie(i, end) = vertices(G); i != end; ++i)
cout << "representative[" << *i << "] = " <<
ds.find_set(*i) << endl;;
cout << endl;
typedef component_index<unsigned int> Components;
Components components(&parent[0], &parent[0] + parent.size());
for (Components::size_type c = 0; c < components.size(); ++c) {
cout << "component " << c << " contains: ";
Components::value_type::iterator
j = components[c].begin(),
jend = components[c].end();
for ( ; j != jend; ++j)
cout << *j << " ";
cout << endl;
}
initialize_incremental_components
| Graphs: |
undirected |
|---|
| Properties: |
rank, parent (in disjoint-sets) |
|---|
| Complexity: |
|
|---|
template <class VertexListGraph, class DisjointSets>
void initialize_incremental_components(VertexListGraph& G, DisjointSets& ds)
This prepares the disjoint-sets data structure for the incremental
connected components algorithm by making each vertex in the graph a
member of its own component (or set).
Where Defined
boost/graph/incremental_components.hpp
incremental_components
| Graphs: |
undirected |
|---|
| Properties: |
rank, parent (in disjoint-sets) |
|---|
| Complexity: |
O(E) |
|---|
template <class EdgeListGraph, class DisjointSets>
void incremental_components(EdgeListGraph& g, DisjointSets& ds)
This function calculates the connected components of the graph,
embedding the results in the disjoint-sets data structure.
Where Defined
boost/graph/incremental_components.hpp
Requirements on Types
| Properties: |
rank, parent (in disjoint-sets) |
|---|
| Complexity: |
O(alpha(E,V)) |
|---|
template <class Vertex, class DisjointSet>
bool same_component(Vertex u, Vertex v, DisjointSet& ds)
This function determines whether/font> uu
and v are in the same
component.
Where Defined
boost/graph/incremental_components.hpp
Requirements on Types
- Vertex must be compatible with the rank and parent
property maps of the DisjointSets
data structure.
component_index
component_index<Index>
The is a class that provides an STL container-like view for the
components of the graph. Each component is a container-like object,
and the component_index object provides access to the
component objects via operator[]. Acomponent_index
object is initialized with the parents property in the disjoint-sets
calculated from the incremental_components() function.
Where Defined
boost/graph/incremental_components.hpp
Members
| Memberr | Description |
| size_type |
The type used for representing the number of components. |
| value_type |
The type for a component object. The component type has the following members.
|
| value_type::value_type |
The value type of a component object is a vertex ID.
|
| value_type::iterator |
This iterator can be used to traverse all of the vertices
in the component. This iterator dereferences to give a vertex ID.
|
| value_type::const_iterator |
The const iterator.
|
| value_type::iterator value_type::begin() const |
Return an iterator pointing to the first vertex in the component.
|
| value_type::iterator value_type::end() const |
Return an iterator pointing past the end of the last vertex in the
component.
|
|
template <class ComponentsContainer>
component_index(const ComponentsContainer& c)
|
Construct the
component_index
using the information
from the components container c which was the result
of executing connected_components_on_edgelist.
|
| value_type operator[](size_type i) |
Returns the ith component in the graph.
|
| size_type component_index::size() |
Returns the number of components in the graph.
|
