# 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`.

      // Create a graph
      typedef adjacency_list <vecS, vecS, undirectedS> Graph;
      typedef Graph::vertex_descriptor Vertex;
      const int N = 6;
      Graph G(N);
      add_edge(0, 1, G);
      add_edge(1, 4, G);

      // create the disjoint-sets structure, which requires
      // rank and parent vertex properties
      std::vector<Vertex> rank(num_vertices(G));
      std::vector<Vertex> parent(num_vertices(G));
      typedef std::vector<Graph::size_type>::iterator Rank;
      typedef std::vector<Vertex>::iterator Parent;
      disjoint_sets<Rank, Parent> ds(rank.begin(), parent.begin());

      // determine the connected components
      // the results are embedded in the disjoint-sets structure
      initialize_incremental_components(G, ds);
      incremental_components(G, ds);

      Graph::edge_descriptor e;
      bool flag;

      // Add a couple more edges and update the disjoint-sets
      tie(e,flag) = add_edge(4, 0, G);
      ds.union_set(4,0);

      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::vertex_iterator i,end;
      for (tie(i,end) = vertices(G); i != end; ++i)
        cout << "representative[" << *i << "] = " <<
          ds.find_set(*i) << endl;;
      cout << endl;

      typedef component_index<int> Components;
      Components components(parent.begin(), parent.end());

      for (Components::size_type i = 0; i < components.size(); ++i) {
        cout << "component " << i << " contains: ";
        Components::value_type::iterator
          j = components[i].begin(),
          jend = components[i].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

- The graph type must be a model of EdgeListGraph.

## `same_component`

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 `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 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[]`. A `component_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

Member | 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 `i`th component in the graph.
`size_type component_index::size()` | Returns the number of components in the graph.

Copyright © 2000 | Jeremy Siek, Univ.of Notre Dame (jsiek@lsc.nd.edu) Lie-Quan Lee, Univ.of Notre Dame (llee1@lsc.nd.edu) Andrew Lumsdaine, Univ.of Notre Dame (lums@lsc.nd.edu)