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<Title>Boost Graph Library: Constructing Graph Algorithms</Title>
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<H1>Constructing graph algorithms with BGL</H1>

<P>
The <I>main</I> goal of  BGL is not to provide a nice graph class, or
to provide a comprehensive set of reusable graph algorithms (though
these are goals). The main goal of BGL is to encourage others to
write reusable graph algorithms. By reusable we mean maximally
reusable.  Generic programming is a methodology for making algorithms
maximally reusable, and in this section we will discuss how to apply
generic programming to constructing graph algorithms.

<P>
To illustrate the generic programming process we will step though the
construction of a graph coloring algorithm. The graph coloring problem
(or more specifically, the vertex coloring problem) is to label each
vertex in a graph <i>G</i> with a color such that no two adjacent
vertices are labeled with the same color and such that the minimum
number of colors are used. In general, the graph coloring problem is
NP-complete, and therefore it is impossible to find an optimal
solution in a reasonable amount of time. However, there are many
algorithms that use heuristics to find colorings that are close to the
minimum.

<P>
The particular algorithm we present here is based on the linear time
<TT>SEQ</TT> subroutine that is used in the estimation of sparse
Jacobian and Hessian matrices&nbsp;[<A
HREF="bibliography.html#curtis74:_jacob">9</A>,<A
HREF="bibliography.html#coleman84:_estim_jacob">7</A>,<A
HREF="bibliography.html#coleman85:_algor">6</A>].  This algorithm
visits all of the vertices in the graph according to the order defined
by the input order. At each vertex the algorithm marks the colors of
the adjacent vertices, and then chooses the smallest unmarked color
for the color of the current vertex. If all of the colors are already
marked, a new color is created.  A color is considered marked if its
mark number is equal to the current vertex number. This saves the
trouble of having to reset the marks for each vertex. The
effectiveness of this algorithm is highly dependent on the input
vertex order. There are several ordering algorithms, including the
<I>largest-first</I>&nbsp;[<A HREF="bibliography.html#welsch67">31</A>],
<I>smallest-last</I>&nbsp;[<a
href="bibliography.html#matula72:_graph_theory_computing">29</a>], and
<I>incidence degree</I>&nbsp;[<a
href="bibliography.html#brelaz79:_new">32</a>] algorithms, which
improve the effectiveness of this coloring algorithm.

<P>
The first decision to make when constructing a generic graph algorithm
is to decide what graph operations are necessary for implementing the
algorithm, and which graph concepts the operations map to.  In this
algorithm we will need to traverse through all of the vertices to
intialize the vertex colors. We also need to access the adjacent
vertices. Therefore, we will choose the <a
href="./VertexListGraph.html">VertexListGraph</a> concept because it
is the minimum concept that includes these operations.  The graph type
will be parameterized in the template function for this algorithm. We
do not restrict the graph type to a particular graph class, such as
the BGL <a href="./adjacency_list.html"><TT>adjacency_list</TT></a>,
for this would drastically limit the reusability of the algorithm (as
most algorithms written to date are). We do restrict the graph type to
those types that model <a
href="./VertexListGraph.html">VertexListGraph</a>. This is enforced by
the use of those graph operations in the algorithm, and furthermore by
our explicit requirement added as a concept check with
<TT>function_requires()</TT> (see Section <A
HREF="../../concept_check/concept_check.htm">Concept
Checking</A> for more details about concept checking).

<P>
Next we need to think about what vertex or edge properties will be
used in the algorithm. In this case, the only property is vertex
color. The most flexible way to specify access to vertex color is to
use the propery map interface. This gives the user of the
algorithm the ability to decide how they want to store the properties.
Since we will need to both read and write the colors we specify the
requirements as <a
href="../../property_map/doc/ReadWritePropertyMap.html">ReadWritePropertyMap</a>. The
<TT>key_type</TT> of the color map must be the
<TT>vertex_descriptor</TT> from the graph, and the <TT>value_type</TT>
must be some kind of integer. We also specify the interface for the
<TT>order</TT> parameter as a property map, in this case a <a
href="../../property_map/doc/ReadablePropertyMap.html">ReadablePropertyMap</a>. For
order, the <TT>key_type</TT> is an integer offset and the
<TT>value_type</TT> is a <TT>vertex_descriptor</TT>. Again we enforce
these requirements with concept checks. The return value of this
algorithm is the number of colors that were needed to color the graph,
hence the return type of the function is the graph's
<TT>vertices_size_type</TT>. The following code shows the interface for our
graph algorithm as a template function, the concept checks, and some
typedefs. The implementation is straightforward, the only step not
discussed above is the color initialization step, where we set the
color of all the vertices to ``uncolored''.

<P>
<PRE>
namespace boost {
  template &lt;class VertexListGraph, class Order, class Color&gt;
  typename graph_traits&lt;VertexListGraph&gt;::vertices_size_type
  sequential_vertex_color_ting(const VertexListGraph&amp; G, 
    Order order, Color color)
  {
    typedef graph_traits&lt;VertexListGraph&gt; GraphTraits;
    typedef typename GraphTraits::vertex_descriptor vertex_descriptor;
    typedef typename GraphTraits::vertices_size_type size_type;
    typedef typename property_traits&lt;Color&gt;::value_type ColorType;
    typedef typename property_traits&lt;Order&gt;::value_type OrderType;

    function_requires&lt; VertexListGraphConcept&lt;VertexListGraph&gt; &gt;();
    function_requires&lt; ReadWritePropertyMapConcept&lt;Color, vertex_descriptor&gt; &gt;();
    function_requires&lt; IntegerConcept&lt;ColorType&gt; &gt;();
    function_requires&lt; size_type, ReadablePropertyMapConcept&lt;Order&gt; &gt;();
    typedef typename same_type&lt;OrderType, vertex_descriptor&gt;::type req_same;
    
    size_type max_color = 0;
    const size_type V = num_vertices(G);
    std::vector&lt;size_type&gt; 
      mark(V, numeric_limits_max(max_color));
    
    typename GraphTraits::vertex_iterator v, vend;
    for (tie(v, vend) = vertices(G); v != vend; ++v)
      color[*v] = V - 1; // which means "not colored"
    
    for (size_type i = 0; i &lt; V; i++) {
      vertex_descriptor current = order[i];

      // mark all the colors of the adjacent vertices
      typename GraphTraits::adjacency_iterator ai, aend;
      for (tie(ai, aend) = adjacent_vertices(current, G); ai != aend; ++ai)
        mark[color[*ai]] = i; 

      // find the smallest color unused by the adjacent vertices
      size_type smallest_color = 0;
      while (smallest_color &lt; max_color &amp;&amp; mark[smallest_color] == i) 
        ++smallest_color;

      // if all the colors are used up, increase the number of colors
      if (smallest_color == max_color)
        ++max_color;

      color[current] = smallest_color;
    }
    return max_color;
  }
} // namespace boost
</PRE>

<P>


<br>
<HR>
<TABLE>
<TR valign=top>
<TD nowrap>Copyright &copy; 2000-2001</TD><TD>
<A HREF="http://www.boost.org/people/jeremy_siek.htm">Jeremy Siek</A>,
Indiana University (<A
HREF="mailto:jsiek@osl.iu.edu">jsiek@osl.iu.edu</A>)<br>
<A HREF="http://www.boost.org/people/liequan_lee.htm">Lie-Quan Lee</A>, Indiana University (<A HREF="mailto:llee@cs.indiana.edu">llee@cs.indiana.edu</A>)<br>
<A HREF="http://www.osl.iu.edu/~lums">Andrew Lumsdaine</A>,
Indiana University (<A
HREF="mailto:lums@osl.iu.edu">lums@osl.iu.edu</A>)
</TD></TR></TABLE>

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