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 you will use 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.

using namespace boost;

int main(int argc, char* argv[]) 
{
  typedef adjacency_list  Graph;
  typedef graph_traits::vertex_descriptor Vertex;
  typedef graph_traits::vertices_size_type VertexIndex;

  const int VERTEX_COUNT = 6;
  Graph graph(VERTEX_COUNT);

  std::vector rank(num_vertices(graph));
  std::vector parent(num_vertices(graph));

  typedef VertexIndex* Rank;
  typedef Vertex* Parent;

  disjoint_sets ds(&rank[0], &parent[0]);

  initialize_incremental_components(graph, ds);
  incremental_components(graph, ds);

  graph_traits::edge_descriptor edge;
  bool flag;

  boost::tie(edge, flag) = add_edge(0, 1, graph);
  ds.union_set(0,1);

  boost::tie(edge, flag) = add_edge(1, 4, graph);
  ds.union_set(1,4);

  boost::tie(edge, flag) = add_edge(4, 0, graph);
  ds.union_set(4,0);

  boost::tie(edge, flag) = add_edge(2, 5, graph);
  ds.union_set(2,5);
    
  std::cout << "An undirected graph:" << std::endl;
  print_graph(graph, get(boost::vertex_index, graph));
  std::cout << std::endl;
    
  BOOST_FOREACH(Vertex current_vertex, vertices(graph)) {
    std::cout << "representative[" << current_vertex << "] = " <<
      ds.find_set(current_vertex) << std::endl;
  }

  std::cout << std::endl;

  typedef component_index Components;

  // NOTE: Because we're using vecS for the graph type, we're
  // effectively using identity_property_map for a vertex index map.
  // If we were to use listS instead, the index map would need to be
  // explicity passed to the component_index constructor.
  Components components(parent.begin(), parent.end());

  // Iterate through the component indices
  BOOST_FOREACH(VertexIndex current_index, components) {
    std::cout << "component " << current_index << " contains: ";

    // Iterate through the child vertex indices for [current_index]
    BOOST_FOREACH(VertexIndex child_index,
                  components[current_index]) {
      std::cout << child_index << " ";
    }

    std::cout << std::endl;
  }

  return (0);
}

initialize_incremental_components

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

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

• The graph type must be a model of EdgeListGraph.

same_component

template <class Vertex, class DisjointSet>
bool same_component(Vertex u, Vertex v, DisjointSet& ds)

This function determines whether u 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 component_index class provides an STL container-like view for the components of the graph. Each component is a container-like object, and access is provided via the operator[]. A component_index object is initialized with the parents property in the disjoint-sets calculated from the incremental_components() function. Optionally, a vertex -> index property map is passed in (identity_property_map is used by default).

Where Defined

boost/graph/incremental_components.hpp

Members

Member	Description
value_type/size_type	The type for a component index (same as Index).
size_type size()	Returns the number of components in the graph.
iterator/const_iterator	Iterators used to traverse the available component indices [0 to size()).
iterator begin() const	Returns an iterator at the start of the component indices (0).
iterator end() const	Returns an iterator past the end of the component indices (size()).
std::pair<component_iterator, component_iterator> operator[size_type index] const	Returns a pair of iterators for the component at index where index is in [0, size()).