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	<TITLE>Boost Graph Library: Maximum (Minimum) cycle ratio</TITLE>
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     Authors: Dmitry Bufistov
              Andrey Parfenov
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<H1><TT>[maximum|minimum]_cycle_ratio</TT></H1>
<P>
<PRE>
template &lt;typename Graph, typename VertexIndexMap,
          typename EdgeWeight1, typename EdgeWeight2&gt;
dobule
maximum_cycle_ratio(const Graph &amp;g, VertexIndexMap vim,
                    EdgeWeight1 ewm, EdgeWeight2 ew2m,
                    std::vector&lt;typename boost::graph_traits&lt;Graph&gt;::edge_descriptor&gt; *pcc = 0);

template &lt;typename FloatTraits, Graph, typename VertexIndexMap,
          typename EdgeWeight1, typename EdgeWeight2&gt;
typename FloatTraits::float_type
maximum_cycle_ratio(const Graph &amp;g, VertexIndexMap vim,
                    EdgeWeight1 ewm, EdgeWeight2 ew2m,
                    std::vector&lt;typename boost::graph_traits&lt;Graph&gt;::edge_descriptor&gt; *pcc = 0,
                    FloatTraits = FloatTraits());

template &lt;typename Graph, typename VertexIndexMap,
          typename EdgeWeight1, typename EdgeWeight2&gt;
dobule
minimum_cycle_ratio(const Graph &amp;g, VertexIndexMap vim,
                    EdgeWeight1 ewm, EdgeWeight2 ew2m,
                    std::vector&lt;typename boost::graph_traits&lt;Graph&gt;::edge_descriptor&gt; *pcc = 0);

template &lt;typename FloatTraits, typename <A HREF="http://boost.org/libs/graph/doc/Graph.html">Graph</A>, typename VertexIndexMap,
          typename EdgeWeight1, typename EdgeWeight2&gt;
typename FloatTraits::float_type
minimum_cycle_ratio(const Graph &amp;g, VertexIndexMap vim,
                    EdgeWeight1 ewm, EdgeWeight2 ew2m,
                    std::vector&lt;typename boost::graph_traits&lt;Graph&gt;::edge_descriptor&gt; *pcc = 0,
                    FloatTraits = FloatTraits());
</PRE>
</P>


The <tt>maximum_cycle_ratio()</tt> function calculates the maximum cycle ratio of a
weighted directed multigraph <I>G=(V,E,W1,W2)</I>, where <i>V</i> is a vertex set,
<i>E</i> is an edge set, W1 and W2 are edge weight functions, W2 is nonnegative.
As a multigraph, <i>G</i> can have multiple edges connecting a pair of vertices.
</P>

<P>Let every edge <I>e</I> has two weights <I>W1(e)</I>  and <I>W2(e)</I>.
Let <I>c</I> be a cycle of the graph<I>g</I>. Then, the <i>cycle ratio</i>, <I>cr(c)</I>, is defined  as:</P>
<P>
<IMG SRC="figs/cr.jpg" ALT="Cycle ratio definition" BORDER=0>
</P>

The <I>maximum (minimum) cycle ratio</I> (mcr) is the maximum (minimum) cycle ratio
of all cycles of the graph. A cycle is called <I>critical</I>  if its ratio is equal
to the <I>mcr</I>. The calculated maximum cycle ratio will be the return value
of the function. The <tt>maximum_cycle_ratio()/minimum_cycle_ratio()</tt> returns
<tt>-FloatTraits::infinity()/FloatTraits::infinity()</tt> if graph has no cycles.
If the <i>pcc</i> parameter is not zero then one critical cycle will be written
to the corresponding <tt>std::vector</tt> of edge descriptors.  The edges in the
vector stored in the way such that <tt>*pcc[0], *ppc[1], ..., *ppc[n]</tt> is a
correct path.
</P>
<P>
The algorithm is due to Howard's iteration policy algorithm, descibed in
<A HREF="./cochet-terrasson98numerical.pdf">[1]</A>.
Ali Dasdan, Sandy S. Irani and Rajesh K.Gupta in their paper
<A HREF="./dasdan-dac99.pdf">Efficient Algorithms for Optimum Cycle Mean and Optimum Cost to Time Ratio
Problems</A>state that this is the most efficient algorithm to the time being (1999).</P>

<p>
For the convenience, functions <tt>maximum_cycle_mean()</tt> and <tt>minimum_cycle_mean()</tt>
are also provided. They have the following signatures:
<pre>
template &lt;typename Graph, typename VertexIndexMap,
          typename EdgeWeightMap, typename EdgeIndexMap&gt;
double
maximum_cycle_mean(const Graph &amp;g, VertexIndexMap vim,
                   EdgeWeightMap ewm, EdgeIndexMap eim,
                   std::vector&lt;typename graph_traits&lt;Graph&gt;::edge_descriptor&gt; *pcc = 0);
template &lt;typename FloatTraits, typename Graph, typename VertexIndexMap,
          typename EdgeWeightMap, typename EdgeIndexMap&gt;

typename FloatTraits::float_type
maximum_cycle_mean(const Graph &amp;g, VertexIndexMap vim,
                   EdgeWeightMap ewm, EdgeIndexMap eim,
                   std::vector&lt;typename graph_traits&lt;Graph&gt;::edge_descriptor&gt; *pcc = 0,
                   FloatTraits = FloatTraits());

template &lt;typename Graph, typename VertexIndexMap,
          typename EdgeWeightMap, typename EdgeIndexMap&gt;
double
minimum_cycle_mean(const Graph &amp;g, VertexIndexMap vim,
                   EdgeWeightMap ewm, EdgeIndexMap eim,
                   std::vector&lt;typename graph_traits&lt;Graph&gt;::edge_descriptor&gt; *pcc = 0);
template &lt;typename FloatTraits, typename Graph, typename VertexIndexMap,
          typename EdgeWeightMap, typename EdgeIndexMap&gt;

typename FloatTraits::float_type
minimum_cycle_mean(const Graph &amp;g, VertexIndexMap vim,
                   EdgeWeightMap ewm, EdgeIndexMap eim,
                   std::vector&lt;typename graph_traits&lt;Graph&gt;::edge_descriptor&gt; *pcc = 0,
                   FloatTraits = FloatTraits());
</pre>
</p>

<H3>Where Defined</H3>
<P STYLE="background: transparent"><TT><A HREF="../../../boost/graph/howard_cycle_ratio.hpp">boost/graph/howard_cycle_ratio.hpp</A></TT>
</P>
<H3>Parameters</H3>
<P>IN: <tt>FloatTraits</tt> </P>
<blockquote>
The <tt>FloatTrats</tt> encapsulates customizable limits-like information for
floating point types. This type <i>must</i> provide an associated type,
<tt>value_type</tt> for the floating point type.

The default value is <tt>boost::mcr_float&lt;&gt;</tt>which has the following
definition:<br/>
<pre>
 template &lt;typename Float = double&gt;
 struct mcr_float {
    typedef Float value_type;

    static Float infinity()
    { return (std::numeric_limits&lt;value_type&gt;::max)(); }

    static Float epsilon()
    { return Float(-0.005); }
  };
</pre>
The value <tt>FloatTraits::epsilon()</tt> controls the "tolerance" of the
algorithm. By increasing the absolute value of epsilon you may slightly decrease
the execution time  but there is a risk to skip a global optima. By decreasing
the absolute value you may fall to the infinite loop. The best option is to
leave this parameter unchanged.
</blockquote>
<P>IN: <tt>const Graph&amp; g </tt>
</P>
<BLOCKQUOTE>A weighted directed multigraph.
The graph's type must be a model of <A HREF="http://boost.org/libs/graph/doc/VertexListGraph.html">VertexListGraph</A>
and <A HREF="http://boost.org/libs/graph/doc/IncidenceGraph.html">IncidenceGraph</A>
</BLOCKQUOTE>
<P>IN: <tt>VertexIndexMap vim</tt>
</P>
<BLOCKQUOTE>Maps each vertex of the graph to a unique integer in the
range [0, num_vertices(g)).
</BLOCKQUOTE>
<P>IN: <tt>EdgeWeight1  ew1m</t>
</P>
<BLOCKQUOTE>
The W1 edge weight function.
</BLOCKQUOTE>
<P>IN: <tt>EdgeWeight2 ew2m</tt>
</P>
<BLOCKQUOTE>
The W2 edge weight function. Should be nonnegative. The actual limitation of the
algorithm is the positivity of the total weight of each directed cycle of the graph.
</BLOCKQUOTE>
<P>
OUT: <tt>std::vector&lt;typename boost::graph_traits&lt;Graph&gt;::edge_descriptor&gt;* pcc</tt>
</P>
<BLOCKQUOTE>
If non zero then one critical cycle will be stored in the std::vector. Default
value is 0.
</BLOCKQUOTE>
<P>
IN (only for maximum/minimal_cycle_mean()): <tt>EdgeIndexMap eim</tt>
</P>
<BLOCKQUOTE>
Maps each edge of the graph to a unique integer in the range [0, num_edges(g)).
</BLOCKQUOTE>
<BLOCKQUOTE STYLE="margin-left: 0cm">
All property maps must be models of <A HREF="http://boost.org/libs/property_map/ReadablePropertyMap.html">Readable
Property Map</A>
</BLOCKQUOTE>

<H3>Complexity</H3>
<P>There is no known precise upper bound for the time complexity of the
algorithm. Imperical time complexity is <I>O(|E|)</I>, where the constant tends to
be pretty small (about 20-30). Space complexity is equal to <i>7*|V|</i> plus the
space required to store a graph.
</P>

<H3>Example</H3>
<P>The program in <A HREF="../example/cycle_ratio_example.cpp">libs/graph/example/cycle_ratio_example.cpp</A>
generates a random graph and computes its maximum cycle ratio.
</P>

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        <TD>
            <P>Copyright &copy; 2006-2009</P>
        </TD>
        <TD>
            <P><A HREF="http://www.lsi.upc.edu/~dmitry">Dmitry
            Bufistov</A>, Andrey Parfenov</P>
        </TD>
    </TR>
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