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Incremental Connected Components

A family of functions and classes for computing connected components of an undirected graph incrementally using disjoint-sets.

Complexity: O(V + E * alpha(E, V))
Defined in: <boost/graph/incremental_components.hpp>

Example

#include <iostream>
#include <vector>
#include <boost/graph/adjacency_list.hpp>
#include <boost/graph/incremental_components.hpp>
#include <boost/pending/disjoint_sets.hpp>

int main() {
    using Graph = boost::adjacency_list<boost::vecS, boost::vecS, boost::undirectedS>;
    Graph g(5);
    std::vector<std::size_t> rank(5, 0);
    std::vector<std::size_t> parent(5);
    boost::disjoint_sets<std::size_t*, std::size_t*> ds(rank.data(), parent.data());
    for (std::size_t i = 0; i < 5; ++i) ds.make_set(i);

    auto add_and_report = [&](int u, int v) {
        boost::add_edge(u, v, g);
        ds.union_set(static_cast<std::size_t>(u), static_cast<std::size_t>(v));
        // Count components
        std::size_t count = 0;
        for (std::size_t i = 0; i < 5; ++i)
            if (ds.find_set(i) == i) ++count;
        std::cout << "  After adding " << u << "-" << v << ": " << count << " components\n";
    };

    std::cout << "Incremental connected components:\n";
    add_and_report(0, 1);
    add_and_report(2, 3);
    add_and_report(1, 2);
    add_and_report(3, 4);
}
Incremental connected components:
  After adding 0-1: 4 components
  After adding 2-3: 3 components
  After adding 1-2: 2 components
  After adding 3-4: 1 components

initialize_incremental_components

template <class VertexListGraph,
          class DisjointSets>
void initialize_incremental_components(
    VertexListGraph& G,
    DisjointSets& ds);
Direction Parameter Description

IN

VertexListGraph& G

An undirected graph. The graph type must be a model of Vertex List Graph.

IN/OUT

DisjointSets& ds

A disjoint-sets data structure. Must provide a make_set operation. The rank and parent properties are embedded in this structure.

incremental_components

Complexity: O(E)

template <class EdgeListGraph,
          class DisjointSets>
void incremental_components(
    EdgeListGraph& g,
    DisjointSets& ds);
Direction Parameter Description

IN

EdgeListGraph& g

An undirected graph. The graph type must be a model of Edge List Graph.

IN/OUT

DisjointSets& ds

A disjoint-sets data structure. Must provide a union_set operation. The rank and parent properties are embedded in this structure.

same_component

Complexity: O(alpha(E,V))

template <class Vertex,
          class DisjointSet>
bool same_component(
    Vertex u,
    Vertex v,
    DisjointSet& ds);
Direction Parameter Description

IN

Vertex u

The first vertex. Vertex must be compatible with the rank and parent property maps of the DisjointSet data structure.

IN

Vertex v

The second vertex.

IN

DisjointSet& ds

A disjoint-sets data structure. Must provide a find_set operation.

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 to index property map is passed in (identity_property_map is used by default).

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()).

Description

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 [5,22] 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.

initialize_incremental_components 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).

incremental_components calculates the connected components of the graph, embedding the results in the disjoint-sets data structure.

same_component determines whether u and v are in the same component.

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