# Connected Components

    template<typename Graph, typename ComponentMap>
    inline typename property_traits<ComponentMap>::value_type
    strong_components( const Graph& g, ComponentMap c);

    namespace graph {
      template<typename Graph, typename VertexComponentMap>
      void
      fleischer_hendrickson_pinar_strong_components(const Graph& g, VertexComponentMap r);

      template<typename Graph, typename ReverseGraph,
               typename ComponentMap, typename IsoMapFR, typename IsoMapRF>
      inline typename property_traits<ComponentMap>::value_type
      fleischer_hendrickson_pinar_strong_components(const Graph& g,
                                                    ComponentMap c,
                                                    const ReverseGraph& gr,
                                                    IsoMapFR fr, IsoMapRF rf);
    }

The `strong_components()` function computes the strongly connected components of a directed graph. The distributed strong components algorithm uses the sequential strong components algorithm to identify components local to a processor. The distributed portion of the algorithm is built on the distributed breadth first search algorithm and is based on the work of Fleischer, Hendrickson, and Pinar [FHP00]. The interface is a superset of the interface to the BGL sequential strong components algorithm. The number of strongly-connected components in the graph is returned to all processes.

The distributed strong components algorithm works on both `directed` and `bidirectional` graphs. In the bidirectional case, a reverse graph adapter is used to produce the required reverse graph. In the directed case, a separate graph is constructed in which all the edges are reversed.

Contents

- Where Defined
- Parameters
- Complexity
- Algorithm Description
- Bibliography

# Where Defined

<`boost/graph/distributed/strong_components.hpp`>

also accessible from

<`boost/graph/strong_components.hpp`>

# Parameters

IN: `const Graph& g`
  The graph type must be a model of Distributed Graph. The graph type must also model the Incidence Graph and be directed.
OUT: `ComponentMap c`
  The algorithm computes how many strongly connected components are in the graph, and assigns each component an integer label. The algorithm then records to which component each vertex in the graph belongs by recording the component number in the component property map. The `ComponentMap` type must be a Distributed Property Map. The value type must be the `vertices_size_type` of the graph. The key type must be the graph's vertex descriptor type.
UTIL: `VertexComponentMap r`
  The algorithm computes a mapping from each vertex to the representative of the strong component, stored in this property map. The `VertexComponentMap` type must be a Distributed Property Map. The value and key types must be the vertex descriptor of the graph.
IN: `const ReverseGraph& gr`

  The reverse (or transpose) graph of `g`, such that for each directed edge (u, v) in `g` there exists a directed edge (fr(v), fr(u)) in `gr` and for each edge (v', u') in gr there exists an edge (rf(u'), rf(v')) in `g`. The functions fr and rf map from vertices in the graph to the reverse graph and vice-verse, and are represented as property map arguments. The concept requirements on this graph type are equivalent to those on the `Graph` type, but the types need not be the same.

  Default: Either a `reverse_graph` adaptor over the original graph (if the graph type is bidirectional, i.e., models the Bidirectional Graph concept) or a distributed adjacency list constructed from the input graph.

IN: `IsoMapFR fr`

  A property map that maps from vertices in the input graph `g` to vertices in the reversed graph `gr`. The type `IsoMapFR` must model the Readable Property Map concept and have the graph's vertex descriptor as its key type and the reverse graph's vertex descriptor as its value type.

  Default: An identity property map (if the graph type is bidirectional) or a distributed `iterator_property_map` (if the graph type is directed).

IN: `IsoMapRF rf`

  A property map that maps from vertices in the reversed graph `gr` to vertices in the input graph `g`. The type `IsoMapRF` must model the Readable Property Map concept and have the reverse graph's vertex descriptor as its key type and the graph's vertex descriptor as its value type.

  Default: An identity property map (if the graph type is bidirectional) or a distributed `iterator_property_map` (if the graph type is directed).

# Complexity

The local phase of the algorithm is O(V + E). The parallel phase of the algorithm requires at most O(V) supersteps each containing two breadth first searches which are O(V + E) each.

# Algorithm Description

Prior to executing the sequential phase of the algorithm, each process identifies any completely local strong components which it labels and removes from the vertex set considered in the parallel phase of the algorithm.

The parallel phase of the distributed strong components algorithm consists of series of supersteps. Each superstep starts with one or more vertex sets which are guaranteed to completely contain any remaining strong components. A distributed breadth first search is performed starting from the first vertex in each vertex set. All of these breadth first searches are performed in parallel by having each processor call `breadth_first_search()` with a different starting vertex, and if necessary inserting additional vertices into the `distributed queue` used for breadth first search before invoking the algorithm. A second distributed breadth first search is performed on the reverse graph in the same fashion. For each initial vertex set, the successor set (the vertices reached in the forward breadth first search), and the predecessor set (the vertices reached in the backward breadth first search) is computed. The intersection of the predecessor and successor sets form a strongly connected component which is labeled as such. The remaining vertices in the initial vertex set are partitioned into three subsets each guaranteed to completely contain any remaining strong components. These three sets are the vertices in the predecessor set not contained in the identified strongly connected component, the vertices in the successor set not in the strongly connected component, and the remaing vertices in the initial vertex set not in the predecessor or successor sets. Once new vertex sets are identified, the algorithm begins a new superstep. The algorithm halts when no vertices remain.

To boost performance in sparse graphs when identifying small components, when less than a given portion of the initial number of vertices remain in active vertex sets, a filtered graph adapter is used to limit the graph seen by the breadth first search algorithm to the active vertices.

# Bibliography

[FHP00] | Lisa Fleischer, Bruce Hendrickson, and Ali Pinar. On Identifying Strongly Connected Components in Parallel. In Parallel and Distributed Processing (IPDPS), volume 1800 of Lecture Notes in Computer Science, pages 505--511, 2000. Springer.

Copyright (C) 2004, 2005 The Trustees of Indiana University.

Authors: Nick Edmonds, Douglas Gregor, and Andrew Lumsdaine