.. Copyright (C) 2004-2008 The Trustees of Indiana University.
   Use, modification and distribution is subject to the Boost Software
   License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
   http://www.boost.org/LICENSE_1_0.txt)

============================
|Logo| Minimum Spanning Tree
============================

The Parallel BGL contains four `minimum spanning tree`_ (MST)
algorithms [DG98]_ for use on undirected, weighted, distributed
graphs. The graphs need not be connected: each algorithm will compute
a minimum spanning forest (MSF) when provided with a disconnected
graph.

The interface to each of the four algorithms is similar to the
implementation of 'Kruskal's algorithm'_ in the sequential BGL. Each
accepts, at a minimum, a graph, a weight map, and an output
iterator. The edges of the MST (or MSF) will be output via the output
iterator on process 0: other processes may receive edges on their
output iterators, but the set may not be complete, depending on the
algorithm. The algorithm parameters are documented together, because
they do not vary greatly. See the section `Selecting an MST
algorithm`_ for advice on algorithm selection.

The graph itself must model the `Vertex List Graph`_ concept and the
Distributed Edge List Graph concept. Since the most common
distributed graph structure, the `distributed adjacency list`_, only
models the Distributed Vertex List Graph concept, it may only be used
with these algorithms when wrapped in a suitable adaptor, such as the
`vertex_list_adaptor`_. 

.. contents::

Where Defined
-------------
<``boost/graph/distributed/dehne_gotz_min_spanning_tree.hpp``>

Parameters
----------

IN: ``Graph& g``
  The graph type must be a model of `Vertex List Graph`_ and
  `Distributed Edge List Graph`_. 


IN/OUT: ``WeightMap weight``
  The weight map must be a `Distributed Property Map`_ and a `Readable
  Property Map`_ whose key type is the edge descriptor of the graph
  and whose value type is numerical.


IN/OUT: ``OutputIterator out``
  The output iterator through which the edges of the MSF will be
  written. Must be capable of accepting edge descriptors for output. 


IN: ``VertexIndexMap index``
  A mapping from vertex descriptors to indices in the range *[0,
  num_vertices(g))*. This must be a `Readable Property Map`_ whose
  key type is a vertex descriptor and whose value type is an integral
  type, typically the ``vertices_size_type`` of the graph.

  **Default:** ``get(vertex_index, g)``


IN/UTIL: ``RankMap rank_map``
  Stores the rank of each vertex, which is used for maintaining
  union-find data structures. This must be a `Read/Write Property Map`_
  whose key type is a vertex descriptor and whose value type is an
  integral type. 

  **Default:** An ``iterator_property_map`` built from an STL vector
  of the ``vertices_size_type`` of the graph and the vertex index map.


IN/UTIL: ``ParentMap parent_map``
  Stores the parent (representative) of each vertex, which is used for
  maintaining union-find data structures. This must be a `Read/Write
  Property Map`_ whose key type is a vertex descriptor and whose value
  type is also a vertex descriptor.

  **Default:** An ``iterator_property_map`` built from an STL vector
  of the ``vertex_descriptor`` of the graph and the vertex index map.


IN/UTIL: ``SupervertexMap supervertex_map``
  Stores the supervertex index of each vertex, which is used for
  maintaining the supervertex list data structures. This must be a
  `Read/Write Property Map`_ whose key type is a vertex descriptor and
  whose value type is an integral type.

  **Default:** An ``iterator_property_map`` built from an STL vector
  of the ``vertices_size_type`` of the graph and the vertex index map.


``dense_boruvka_minimum_spanning_tree``
---------------------------------------

:: 

  namespace graph {
    template<typename Graph, typename WeightMap, typename OutputIterator, 
             typename VertexIndexMap, typename RankMap, typename ParentMap, 
             typename SupervertexMap>
    OutputIterator
    dense_boruvka_minimum_spanning_tree(const Graph& g, WeightMap weight_map,
                                      OutputIterator out, 
                                      VertexIndexMap index,
                                      RankMap rank_map, ParentMap parent_map,
                                      SupervertexMap supervertex_map);

    template<typename Graph, typename WeightMap, typename OutputIterator, 
             typename VertexIndex>
    OutputIterator
    dense_boruvka_minimum_spanning_tree(const Graph& g, WeightMap weight_map,
                                      OutputIterator out, VertexIndex index);

    template<typename Graph, typename WeightMap, typename OutputIterator>
    OutputIterator
    dense_boruvka_minimum_spanning_tree(const Graph& g, WeightMap weight_map,
                                      OutputIterator out);
  }

Description
~~~~~~~~~~~

The dense Boruvka distributed minimum spanning tree algorithm is a
direct parallelization of the sequential MST algorithm by
Boruvka. The algorithm first creates a *supervertex* out of each
vertex. Then, in each iteration, it finds the smallest-weight edge
incident to each vertex, collapses supervertices along these edges,
and removals all self loops. The only difference between the
sequential and parallel algorithms is that the parallel algorithm
performs an all-reduce operation so that all processes have the
global minimum set of edges.

Unlike the other three algorithms, this algorithm emits the complete
list of edges in the minimum spanning forest via the output iterator
on all processes. It may therefore be more useful than the others
when parallelizing sequential BGL programs.

Complexity
~~~~~~~~~~

The distributed algorithm requires *O(log n)* BSP supersteps, each of
which requires *O(m/p + n)* time and *O(n)* communication per
process. 

Performance
~~~~~~~~~~~

The following charts illustrate the performance of this algorithm on
various random graphs. We see that the algorithm scales well up to 64
or 128 processors, depending on the type of graph, for dense
graphs. However, for sparse graphs performance tapers off as the
number of processors surpases *m/n*, i.e., the average degree (which
is 30 for this graph). This behavior is expected from the algorithm.

.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeSparse&columns=5
  :align: left
.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeSparse&columns=5&speedup=1

.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeDense&columns=5
  :align: left
.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeDense&columns=5&speedup=1

``merge_local_minimum_spanning_trees``
--------------------------------------

::

  namespace graph {
    template<typename Graph, typename WeightMap, typename OutputIterator,
             typename VertexIndexMap>
    OutputIterator
    merge_local_minimum_spanning_trees(const Graph& g, WeightMap weight,
                                       OutputIterator out, 
                                       VertexIndexMap index);

    template<typename Graph, typename WeightMap, typename OutputIterator>
    inline OutputIterator
    merge_local_minimum_spanning_trees(const Graph& g, WeightMap weight,
                                       OutputIterator out);
  }

Description
~~~~~~~~~~~

The merging local MSTs algorithm operates by computing minimum
spanning forests from the edges stored on each process. Then the
processes merge their edge lists along a tree. The child nodes cease
participating in the computation, but the parent nodes recompute MSFs
from the newly acquired edges. In the final round, the root of the
tree computes the global MSFs, having received candidate edges from
every other process via the tree.

Complexity
~~~~~~~~~~

This algorithm requires *O(log_D p)* BSP supersteps (where *D* is the
number of children in the tree, and is currently fixed at 3). Each
superstep requires *O((m/p) log (m/p) + n)* time and *O(m/p)*
communication per process.

Performance
~~~~~~~~~~~

The following charts illustrate the performance of this algorithm on
various random graphs. The algorithm only scales well for very dense
graphs, where most of the work is performed in the initial stage and
there is very little work in the later stages. 

.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeSparse&columns=6
  :align: left
.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeSparse&columns=6&speedup=1

.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeDense&columns=6
  :align: left
.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeDense&columns=6&speedup=1


``boruvka_then_merge``
----------------------

::

  namespace graph {
    template<typename Graph, typename WeightMap, typename OutputIterator,
             typename VertexIndexMap, typename RankMap, typename ParentMap,
             typename SupervertexMap>
    OutputIterator
    boruvka_then_merge(const Graph& g, WeightMap weight, OutputIterator out,
                       VertexIndexMap index, RankMap rank_map, 
                       ParentMap parent_map, SupervertexMap
                       supervertex_map);

    template<typename Graph, typename WeightMap, typename OutputIterator,
             typename VertexIndexMap>
    inline OutputIterator
    boruvka_then_merge(const Graph& g, WeightMap weight, OutputIterator out,
                        VertexIndexMap index);

    template<typename Graph, typename WeightMap, typename OutputIterator>
    inline OutputIterator
    boruvka_then_merge(const Graph& g, WeightMap weight, OutputIterator out);
  }

Description
~~~~~~~~~~~

This algorithm applies both Boruvka steps and local MSF merging steps
together to achieve better asymptotic performance than either
algorithm alone. It first executes Boruvka steps until only *n/(log_d
p)^2* supervertices remain, then completes the MSF computation by
performing local MSF merging on the remaining edges and
supervertices. 

Complexity
~~~~~~~~~~

This algorithm requires *log_D p* + *log log_D p* BSP supersteps. The
time required by each superstep depends on the type of superstep
being performed; see the distributed Boruvka or merging local MSFS
algorithms for details.

Performance
~~~~~~~~~~~

The following charts illustrate the performance of this algorithm on
various random graphs. We see that the algorithm scales well up to 64
or 128 processors, depending on the type of graph, for dense
graphs. However, for sparse graphs performance tapers off as the
number of processors surpases *m/n*, i.e., the average degree (which
is 30 for this graph). This behavior is expected from the algorithm.

.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeSparse&columns=7
  :align: left
.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeSparse&columns=7&speedup=1

.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeDense&columns=7
  :align: left
.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeDense&columns=7&speedup=1

``boruvka_mixed_merge``
-----------------------

::

  namespace {
    template<typename Graph, typename WeightMap, typename OutputIterator,
             typename VertexIndexMap, typename RankMap, typename ParentMap,
             typename SupervertexMap>
    OutputIterator
    boruvka_mixed_merge(const Graph& g, WeightMap weight, OutputIterator out,
                        VertexIndexMap index, RankMap rank_map, 
                        ParentMap parent_map, SupervertexMap
                        supervertex_map);

    template<typename Graph, typename WeightMap, typename OutputIterator,
             typename VertexIndexMap>
    inline OutputIterator
    boruvka_mixed_merge(const Graph& g, WeightMap weight, OutputIterator out,
                        VertexIndexMap index);

    template<typename Graph, typename WeightMap, typename OutputIterator>
    inline OutputIterator
    boruvka_mixed_merge(const Graph& g, WeightMap weight, OutputIterator out);
  }

Description
~~~~~~~~~~~

This algorithm applies both Boruvka steps and local MSF merging steps
together to achieve better asymptotic performance than either method
alone. In each iteration, the algorithm first performs a Boruvka step
and then merges the local MSFs computed based on the supervertex
graph. 

Complexity
~~~~~~~~~~

This algorithm requires *log_D p* BSP supersteps. The
time required by each superstep depends on the type of superstep
being performed; see the distributed Boruvka or merging local MSFS
algorithms for details. However, the algorithm is
communication-optional (requiring *O(n)* communication overall) when
the graph is sufficiently dense, i.e., *m/n >= p*.

Performance
~~~~~~~~~~~

The following charts illustrate the performance of this algorithm on
various random graphs. We see that the algorithm scales well up to 64
or 128 processors, depending on the type of graph, for dense
graphs. However, for sparse graphs performance tapers off as the
number of processors surpases *m/n*, i.e., the average degree (which
is 30 for this graph). This behavior is expected from the algorithm.

.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeSparse&columns=8
  :align: left
.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeSparse&columns=8&speedup=1

.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeDense&columns=8
  :align: left
.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER,SF,SW&dataset=TimeDense&columns=8&speedup=1


Selecting an MST algorithm
--------------------------

Dehne and Gotz reported [DG98]_ mixed results when evaluating these
four algorithms. No particular algorithm was clearly better than the
others in all cases. However, the asymptotically best algorithm
(``boruvka_mixed_merge``) seemed to perform more poorly in their tests
than the other merging-based algorithms. The following performance
charts illustrate the performance of these four minimum spanning tree
implementations. 

Overall, ``dense_boruvka_minimum_spanning_tree`` gives the most
consistent performance and scalability for the graphs we
tested. Additionally, it may be more suitable for sequential programs
that are being parallelized, because it emits complete MSF edge lists
via the output iterators in every process.

Performance on Sparse Graphs
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER&dataset=TimeSparse&columns=5,6,7,8
  :align: left
.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER&dataset=TimeSparse&columns=5,6,7,8&speedup=1

.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=SF&dataset=TimeSparse&columns=5,6,7,8
  :align: left
.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=SF&dataset=TimeSparse&columns=5,6,7,8&speedup=1

.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=SW&dataset=TimeSparse&columns=5,6,7,8
  :align: left
.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=SW&dataset=TimeSparse&columns=5,6,7,8&speedup=1

Performance on Dense Graphs
~~~~~~~~~~~~~~~~~~~~~~~~~~~
.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER&dataset=TimeDense&columns=5,6,7,8
  :align: left
.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=ER&dataset=TimeDense&columns=5,6,7,8&speedup=1

.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=SF&dataset=TimeDense&columns=5,6,7,8
  :align: left
.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=SF&dataset=TimeDense&columns=5,6,7,8&speedup=1

.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=SW&dataset=TimeDense&columns=5,6,7,8
  :align: left
.. image:: http://www.osl.iu.edu/research/pbgl/performance/chart.php?generator=SW&dataset=TimeDense&columns=5,6,7,8&speedup=1

-----------------------------------------------------------------------------

Copyright (C) 2004 The Trustees of Indiana University.

Authors: Douglas Gregor and Andrew Lumsdaine

.. |Logo| image:: pbgl-logo.png
            :align: middle
            :alt: Parallel BGL
            :target: http://www.osl.iu.edu/research/pbgl

.. _minimum spanning tree: http://www.boost.org/libs/graph/doc/graph_theory_review.html#sec:minimum-spanning-tree
.. _Kruskal's algorithm: http://www.boost.org/libs/graph/doc/kruskal_min_spanning_tree.html
.. _Vertex list graph: http://www.boost.org/libs/graph/doc/VertexListGraph.html
.. _distributed adjacency list: distributed_adjacency_list.html
.. _vertex_list_adaptor: vertex_list_adaptor.html
.. _Distributed Edge List Graph: DistributedEdgeListGraph.html
.. _Distributed property map: distributed_property_map.html
.. _Readable Property Map: http://www.boost.org/libs/property_map/ReadablePropertyMap.html
.. _Read/Write Property Map: http://www.boost.org/libs/property_map/ReadWritePropertyMap.html

.. [DG98] Frank Dehne and Silvia Gotz. *Practical Parallel Algorithms
    for Minimum Spanning Trees*. In Symposium on Reliable Distributed Systems, 
    pages 366--371, 1998.