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<Title>Floyd-Warshall All Pairs Shortest Paths</Title>
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<H1><A NAME="sec:floyd-warshall">
<TT>floyd_warshall_all_pairs_shortest_paths</TT>
</H1>

<PRE><em>// Named parameters version</em>
template &lt;class VertexListGraph, class DistanceMatrix,
    class P, class T, class R&gt;
bool floyd_warshall_initialized_all_pairs_shortest_paths(
    const VertexListGraph&amp; g, DistanceMatrix&amp; d,
    const bgl_named_params&lt;P, T, R&gt;&amp; params)

template &lt;class VertexAndEdgeListGraph, class DistanceMatrix,
    class P, class T, class R&gt;
bool floyd_warshall_all_pairs_shortest_paths(
    const VertexAndEdgeListGraph&amp; g, DistanceMatrix&amp; d,
    const bgl_named_params&lt;P, T, R&gt;&amp; params)

<em>// Positional parameter versions</em>
\begin{verbatim}
template &lt;typename VertexListGraph, typename DistanceMatrix,
    typename BinaryPredicate, typename BinaryFunction,
    typename Infinity, typename Zero&gt;
bool floyd_warshall_initialized_all_pairs_shortest_paths(
    const VertexListGraph&amp; g, DistanceMatrix&amp; d,
    const BinaryPredicate&amp; compare, const BinaryFunction&amp; combine,
    const Infinity&amp; inf, const Zero&amp; zero)

template &lt;typename VertexAndEdgeListGraph, typename DistanceMatrix,
    typename WeightMap, typename BinaryPredicate,
    typename BinaryFunction, typename Infinity, typename Zero&gt;
bool floyd_warshall_all_pairs_shortest_paths(
    const VertexAndEdgeListGraph&amp; g, DistanceMatrix&amp; d,
    const WeightMap&amp; w, const BinaryPredicate&amp; compare,
    const BinaryFunction&amp; combine,
    const Infinity&amp; inf, const Zero&amp; zero)</PRE>

<P>
These algorithms find the shortest distance between every pair of
vertices in the graph.  The algorithms return false if there is a
negative weight cycle in the graph, true otherwise.  The shortest
distance between each pair of vertices is stored in the distance
matrix <code>d</code>.  The difference between the two algorithms is in
whether the distance matrix is assumed to be initialized or not, as
discussed below under the OUT parameter description.

<P>This algorithm should be used to compute shortest paths between
every pair of vertices for dense graphs. For sparse graphs, use <a
href="johnson_all_pairs_shortest.html"><code>johnson_all_pairs_shortest_paths</code></a>. 

<H3>Where Defined</H3>

<P>
<a href="../../../boost/graph/floyd_warshall_shortest.hpp"><TT>boost/graph/floyd_warshall_shortest.hpp</TT></a>

<h3>Parameters</h3>
IN: <code>Graph&amp; g</code>
<blockquote>
A directed or undirected graph.  The graph must be a model of <a href="VertexListGraph.html">Vertex List Graph</a> for calls to
<code>floyd_warshall_initialized_all_pairs_shortest_paths</code>, and
<a href="VertexAndEdgeListGraph.html">Vertex And Edge List Graph</a> for calls to
<code>floyd_warshall_all_pairs_shortest_paths</code>.<br>
</blockquote>

OUT: <code>DistanceMatrix&amp; d</code>
<blockquote>
The length of the shortest path between each pair of vertices
<code>u</code>,<code>v</code> are  
stored in the matrix at location <code>D[u][v]</code>.  The
DistanceMatrix must be 
of type <code>{M, I, V}</code> where <code>I</code> is of type
<code>vertex_descriptor</code> and <code>V</code> is the
type of the <code>weight_map</code>.  The set of types must be a model of
<a href="BasicMatrix.html">BasicMatrix</a>, with the exceptions that
it isn't required to 
run in constant time, and it must be mutable.  The matrix must be
properly initialized when it is passed to the function
<code>floyd_warshall_initialized_all_pairs_shortest_paths</code>.  If the
function <code>floyd_warshall_all_pairs_shortest_paths</code> is used then the
matrix will be initialized for the user.
</blockquote>

<h3>Named Parameters</h3>

IN: <code>weight_map(WeightMap w)</code>
<blockquote>
The weight of length of each edge in the graph.  The <code>WeightMap</code>
must be a model of <a href="../../property_map/doc/ReadablePropertyMap.html">Readable Property
Map</a>.  The edge descriptor 
type of the graph needs to be usable as the key type for the weight
map.  The <code>value_type</code> of the weight map must be the type of the
<code>DistanceMatrix</code>, and must always either be part of the
graph passed to the function, or passed in as a parameter.<br>
<b>Default:</b> <code>get(edge_weight, g)</code>
</blockquote>

IN: <code>distance_compare(CompareFunction cmp)</code>
<blockquote>
The function used to compare distances to determine which target
vertex is closer to the source vertex.  The CompareFunction must be a
model of <code>BinaryPredicate</code>.  The argument types must match the
value type of the <code>WeightMap</code>.<br>

<b>Default:</b> <code>std::less&lt;WM&gt;</code>with <code>WM = typename property_traits&lt;WeightMap&gt;::value_type</code>
</blockquote>

IN: <code>distance_combine(CombineFunction cmb)</code>
<blockquote>
The function used to combine distance to compute the distance of a
path.  The CombineFunction must be a model of <code>BinaryFunction</code>.
The argument types must match the value type of the <code>WeightMap</code>.
The result type must be the same as the distance value type.<br>

<b>Default:</b> <code>boost::closed_plus&lt;WM&gt;</code> with <code>WM = typename property_traits&lt;WeightMap&gt;::value_type</code>
</blockquote>

IN: <code>distance_inf(WM inf)</code>
<blockquote>
The value used to initialize the distance for each vertex before
starting the algorithm, and to represent the distance between vertices
for which there is not path.  Should be larger than any possible valid
path length.  The argument type must match the value type of the <code>
WeightMap</code>.<br>

<b>Default:</b> <code>std::numeric_limits&lt;WM&gt;::max()</code> with <code>WM = typename property_traits&lt;WeightMap&gt;::value_type</code>
</blockquote>

IN: <code>distance_zero(WM zero)</code>
<blockquote>
The value used to represent the distance from a vertex to itself, and
to determine if a value is negative.  The argument type must match the
value type of the <code>WeightMap</code>.<br>
<b>Default:</b> <code>0</code>
</blockquote>

<h3>Complexity</h3>

The time complexity is <i>O(V<sup>3</sup>)</i>.

<br>
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<TABLE>
<TR valign=top>
<TD nowrap>Copyright &copy; 2002-2004</TD><TD>
  Lauren Foutz, Rensselaer Polytechnic Institute</td>
  </tr><tr valign="top"><td></td>
  <td>Scott Hill, Rensselaer Polytechnic Institute
</TD></TR></TABLE>

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