isomorphism-impl-v3.w

C++的一个好库。。。现在很流行

but we want the roots of the DFS tree's to be ordered by invariant
multiplicity.  Therefore we implement the outer-loop of the DFS here
and then call \code{depth\_\-first\_\-visit} to handle the recursive
portion of the DFS. The \code{record\_dfs\_order} adapts the DFS to
record the ordering, storing the results in in the
\code{dfs\_vertices} and \code{ordered\_edges} arrays. We then create
the \code{dfs\_num} array which provides a mapping from vertex to DFS
number.

@d Order vertices and edges by DFS
@{
std::vector<default_color_type> color_vec(num_vertices(G1));
safe_iterator_property_map<std::vector<default_color_type>::iterator, IndexMap1>
     color_map(color_vec.begin(), color_vec.size(), index_map1);
record_dfs_order dfs_visitor(dfs_vertices, ordered_edges);
typedef color_traits<default_color_type> Color;
for (vertex_iter u = V_mult.begin(); u != V_mult.end(); ++u) {
    if (color_map[*u] == Color::white()) {
        dfs_visitor.start_vertex(*u, G1);
        depth_first_visit(G1, *u, dfs_visitor, color_map);
    }
}
// Create the dfs_num array and dfs_num_map
dfs_num_vec.resize(num_vertices(G1));
dfs_num = make_safe_iterator_property_map(dfs_num_vec.begin(),
                          dfs_num_vec.size(), index_map1);
size_type n = 0;
for (vertex_iter v = dfs_vertices.begin(); v != dfs_vertices.end(); ++v)
    dfs_num[*v] = n++;
@}

\noindent The definition of the \code{record\_dfs\_order} visitor
class is as follows.

@d DFS visitor to record vertex and edge order
@{
struct record_dfs_order : default_dfs_visitor
{
    record_dfs_order(std::vector<vertex1_t>& v, std::vector<edge1_t>& e) 
        : vertices(v), edges(e) { }

    void discover_vertex(vertex1_t v, const Graph1&) const {
        vertices.push_back(v);
    }
    void examine_edge(edge1_t e, const Graph1& G1) const {
        edges.push_back(e);
    }
    std::vector<vertex1_t>& vertices;
    std::vector<edge1_t>& edges;
};
@}

The final stage of the setup is to reorder the edges so that all edges
belonging to $G_1[k]$ appear before any edges not in $G_1[k]$, for
$k=1,...,n$.

@d Sort edges according to vertex DFS order
@{
sort(ordered_edges, edge_cmp(G1, dfs_num));
@}

\noindent The edge comparison function object is defined as follows.

@d Edge comparison predicate
@{
struct edge_cmp {
    edge_cmp(const Graph1& G1, DFSNumMap dfs_num)
        : G1(G1), dfs_num(dfs_num) { }
    bool operator()(const edge1_t& e1, const edge1_t& e2) const {
        using namespace std;
        vertex1_t u1 = dfs_num[source(e1,G1)], v1 = dfs_num[target(e1,G1)];
        vertex1_t u2 = dfs_num[source(e2,G1)], v2 = dfs_num[target(e2,G1)];
        int m1 = max(u1, v1);
        int m2 = max(u2, v2);
        // lexicographical comparison 
        return make_pair(m1, make_pair(u1, v1))
             < make_pair(m2, make_pair(u2, v2));
    }
    const Graph1& G1;
    DFSNumMap dfs_num;
};
@}


\section{Appendix}


@d Typedefs for commonly used types
@{
typedef typename graph_traits<Graph1>::vertex_descriptor vertex1_t;
typedef typename graph_traits<Graph2>::vertex_descriptor vertex2_t;
typedef typename graph_traits<Graph1>::edge_descriptor edge1_t;
typedef typename graph_traits<Graph2>::edge_descriptor edge2_t;
typedef typename graph_traits<Graph1>::vertices_size_type size_type;
typedef typename Invariant1::result_type invar1_value;
typedef typename Invariant2::result_type invar2_value;
@}

@d Data members for the parameters
@{
const Graph1& G1;
const Graph2& G2;
IsoMapping f;
Invariant1 invariant1;
Invariant2 invariant2;
std::size_t max_invariant;
EdgeCompare edge_compare;
IndexMap1 index_map1;
IndexMap2 index_map2;
@}

@d Internal data structures
@{
std::vector<vertex1_t> dfs_vertices;
typedef typename std::vector<vertex1_t>::iterator vertex_iter;
std::vector<int> dfs_num_vec;
typedef safe_iterator_property_map<typename std::vector<int>::iterator, 
  IndexMap1> DFSNumMap;
DFSNumMap dfs_num;
std::vector<edge1_t> ordered_edges;
typedef typename std::vector<edge1_t>::iterator edge_iter;

std::vector<char> in_S_vec;
typedef safe_iterator_property_map<typename std::vector<char>::iterator,
    IndexMap2> InSMap;
InSMap in_S;

int num_edges_on_k;
@}

@d Isomorphism algorithm constructor
@{
isomorphism_algo(const Graph1& G1, const Graph2& G2, IsoMapping f,
                 Invariant1 invariant1, Invariant2 invariant2, 
                 std::size_t max_invariant,
                 EdgeCompare edge_compare,
                 IndexMap1 index_map1, IndexMap2 index_map2)
    : G1(G1), G2(G2), f(f), invariant1(invariant1), invariant2(invariant2),
      max_invariant(max_invariant), edge_compare(edge_compare),
      index_map1(index_map1), index_map2(index_map2)
{
    in_S_vec.resize(num_vertices(G1));
    in_S = make_safe_iterator_property_map
        (in_S_vec.begin(), in_S_vec.size(), index_map2);
}
@}


@o isomorphism.hpp
@{
// Copyright (C) 2001 Jeremy Siek, Douglas Gregor, Brian Osman
//
// Permission to copy, use, sell and distribute this software is granted
// provided this copyright notice appears in all copies.
// Permission to modify the code and to distribute modified code is granted
// provided this copyright notice appears in all copies, and a notice
// that the code was modified is included with the copyright notice.
//
// This software is provided "as is" without express or implied warranty,
// and with no claim as to its suitability for any purpose.
#ifndef BOOST_GRAPH_ISOMORPHISM_HPP
#define BOOST_GRAPH_ISOMORPHISM_HPP

#include <utility>
#include <vector>
#include <iterator>
#include <algorithm>
#include <boost/graph/iteration_macros.hpp>
#include <boost/graph/depth_first_search.hpp>
#include <boost/utility.hpp>
#include <boost/detail/algorithm.hpp>
#include <boost/pending/indirect_cmp.hpp> // for make_indirect_pmap

namespace boost {

namespace detail {

@<Isomorphism algorithm class@>
    
template <typename Graph, typename InDegreeMap>
void compute_in_degree(const Graph& g, InDegreeMap in_degree_map)
{
    BGL_FORALL_VERTICES_T(v, g, Graph)
        put(in_degree_map, v, 0);

    BGL_FORALL_VERTICES_T(u, g, Graph)
      BGL_FORALL_ADJ_T(u, v, g, Graph)
        put(in_degree_map, v, get(in_degree_map, v) + 1);
}

} // namespace detail


@<Degree vertex invariant functor@>

@<Isomorphism function interface@>
@<Isomorphism function body@>

namespace detail {

struct default_edge_compare {
   template <typename Edge1, typename Edge2>
   bool operator()(Edge1 e1, Edge2 e2) const { return true; }
};
  
template <typename Graph1, typename Graph2, 
          typename IsoMapping, 
          typename IndexMap1, typename IndexMap2,
          typename P, typename T, typename R>
bool isomorphism_impl(const Graph1& G1, const Graph2& G2, 
                      IsoMapping f, IndexMap1 index_map1, IndexMap2 index_map2,
                      const bgl_named_params<P,T,R>& params)
{
  std::vector<std::size_t> in_degree1_vec(num_vertices(G1));
  typedef safe_iterator_property_map<std::vector<std::size_t>::iterator, 
    IndexMap1> InDeg1;
  InDeg1 in_degree1(in_degree1_vec.begin(), in_degree1_vec.size(), index_map1);
  compute_in_degree(G1, in_degree1);

  std::vector<std::size_t> in_degree2_vec(num_vertices(G2));
  typedef safe_iterator_property_map<std::vector<std::size_t>::iterator, 
    IndexMap2> InDeg2;
  InDeg2 in_degree2(in_degree2_vec.begin(), in_degree2_vec.size(), index_map2);
  compute_in_degree(G2, in_degree2);

  degree_vertex_invariant<InDeg1, Graph1> invariant1(in_degree1, G1);
  degree_vertex_invariant<InDeg2, Graph2> invariant2(in_degree2, G2);
  default_edge_compare edge_cmp;

  return isomorphism(G1, G2, f,
        choose_param(get_param(params, vertex_invariant1_t()), invariant1),
        choose_param(get_param(params, vertex_invariant2_t()), invariant2),
        choose_param(get_param(params, vertex_max_invariant_t()), 
                     invariant2.max()),
        choose_param(get_param(params, edge_compare_t()), edge_cmp),
        index_map1, index_map2
        );  
}  
   
} // namespace detail


// Named parameter interface
template <typename Graph1, typename Graph2, class P, class T, class R>
bool isomorphism(const Graph1& g1,
                 const Graph2& g2,
                 const bgl_named_params<P,T,R>& params)
{
  typedef typename graph_traits<Graph2>::vertex_descriptor vertex2_t;
  typename std::vector<vertex2_t>::size_type n = num_vertices(g1);
  std::vector<vertex2_t> f(n);
  return detail::isomorphism_impl
    (g1, g2, 
     choose_param(get_param(params, vertex_isomorphism_t()),
          make_safe_iterator_property_map(f.begin(), f.size(),
                  choose_const_pmap(get_param(params, vertex_index1),
                                    g1, vertex_index), vertex2_t())),
     choose_const_pmap(get_param(params, vertex_index1), g1, vertex_index),
     choose_const_pmap(get_param(params, vertex_index2), g2, vertex_index),
     params
     );
}

// All defaults interface
template <typename Graph1, typename Graph2>
bool isomorphism(const Graph1& g1, const Graph2& g2)
{
  return isomorphism(g1, g2,
    bgl_named_params<int, buffer_param_t>(0));// bogus named param
}


// Verify that the given mapping iso_map from the vertices of g1 to the
// vertices of g2 describes an isomorphism.
// Note: this could be made much faster by specializing based on the graph
// concepts modeled, but since we're verifying an O(n^(lg n)) algorithm,
// O(n^4) won't hurt us.
template<typename Graph1, typename Graph2, typename IsoMap>
inline bool verify_isomorphism(const Graph1& g1, const Graph2& g2, IsoMap iso_map)
{
#if 0
    // problematic for filtered_graph!
  if (num_vertices(g1) != num_vertices(g2) || num_edges(g1) != num_edges(g2))
    return false;
#endif
  
  for (typename graph_traits<Graph1>::edge_iterator e1 = edges(g1).first;
       e1 != edges(g1).second; ++e1) {
    bool found_edge = false;
    for (typename graph_traits<Graph2>::edge_iterator e2 = edges(g2).first;
         e2 != edges(g2).second && !found_edge; ++e2) {
      if (source(*e2, g2) == get(iso_map, source(*e1, g1)) &&
          target(*e2, g2) == get(iso_map, target(*e1, g1))) {
        found_edge = true;
      }
    }
    
    if (!found_edge)
      return false;
  }
  
  return true;
}

} // namespace boost

#include <boost/graph/iteration_macros_undef.hpp>

#endif // BOOST_GRAPH_ISOMORPHISM_HPP
@}

\bibliographystyle{abbrv}
\bibliography{ggcl}

\end{document}
% LocalWords:  Isomorphism Siek isomorphism adjacency subgraph subgraphs OM DFS
% LocalWords:  ISOMORPH Invariants invariants typename IsoMapping bool const
% LocalWords:  VertexInvariant VertexIndexMap iterator typedef VertexG Idx num
% LocalWords:  InvarValue struct invar vec iter tmp_matches mult inserter permute ui
% LocalWords:  dfs cmp isomorph VertexIter edge_iter_t IndexMap desc RPH ATCH pre

% LocalWords:  iterators VertexListGraph EdgeListGraph BidirectionalGraph tmp
% LocalWords:  ReadWritePropertyMap VertexListGraphConcept EdgeListGraphConcept
% LocalWords:  BidirectionalGraphConcept ReadWritePropertyMapConcept indices ei
% LocalWords:  IsoMappingValue ReadablePropertyMapConcept namespace InvarFun
% LocalWords:  MultMap vip inline bitset typedefs fj hpp ifndef adaptor params
% LocalWords:  bgl param pmap endif