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boost/graph/vf2_sub_graph_iso.hpp

//=======================================================================
// Copyright (C) 2012 Flavio De Lorenzi (fdlorenzi@gmail.com)
// Copyright (C) 2013 Jakob Lykke Andersen, University of Southern Denmark (jlandersen@imada.sdu.dk)
//
// The algorithm implemented here is derived from original ideas by 
// Pasquale Foggia and colaborators. For further information see 
// e.g. Cordella et al. 2001, 2004.
//
// Distributed under 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)
//=======================================================================

// Revision History:
//   8 April 2013: Fixed a typo in vf2_print_callback. (Flavio De Lorenzi) 

#ifndef BOOST_VF2_SUB_GRAPH_ISO_HPP
#define BOOST_VF2_SUB_GRAPH_ISO_HPP

#include <iostream>
#include <iomanip>
#include <iterator>
#include <vector>
#include <utility>

#include <boost/assert.hpp>
#include <boost/concept/assert.hpp>
#include <boost/concept_check.hpp>
#include <boost/graph/graph_utility.hpp>
#include <boost/graph/graph_traits.hpp>
#include <boost/graph/mcgregor_common_subgraphs.hpp> // for always_equivalent
#include <boost/graph/named_function_params.hpp>
#include <boost/type_traits/has_less.hpp>
#include <boost/mpl/int.hpp>
#include <boost/range/algorithm/sort.hpp>
#include <boost/tuple/tuple.hpp>
#include <boost/utility/enable_if.hpp>

#ifndef BOOST_GRAPH_ITERATION_MACROS_HPP
#define BOOST_ISO_INCLUDED_ITER_MACROS // local macro, see bottom of file
#include <boost/graph/iteration_macros.hpp>
#endif

namespace boost {
  
  // Default print_callback
  template <typename Graph1,
            typename Graph2>
  struct vf2_print_callback {
    
    vf2_print_callback(const Graph1& graph1, const Graph2& graph2) 
      : graph1_(graph1), graph2_(graph2) {}
    
    template <typename CorrespondenceMap1To2,
              typename CorrespondenceMap2To1>
    bool operator()(CorrespondenceMap1To2 f, CorrespondenceMap2To1) const {
      
      // Print (sub)graph isomorphism map
      BGL_FORALL_VERTICES_T(v, graph1_, Graph1) 
        std::cout << '(' << get(vertex_index_t(), graph1_, v) << ", " 
                  << get(vertex_index_t(), graph2_, get(f, v)) << ") ";
      
      std::cout << std::endl;
      
      return true;
    }
    
  private:
    const Graph1& graph1_;
    const Graph2& graph2_;
  };
  
  namespace detail {
    
    // State associated with a single graph (graph_this)
    template<typename GraphThis,
             typename GraphOther,
             typename IndexMapThis,
             typename IndexMapOther>
    class base_state {
      
      typedef typename graph_traits<GraphThis>::vertex_descriptor vertex_this_type;
      typedef typename graph_traits<GraphOther>::vertex_descriptor vertex_other_type;
      
      typedef typename graph_traits<GraphThis>::vertices_size_type size_type;
      
      const GraphThis& graph_this_;
      const GraphOther& graph_other_;
      
      IndexMapThis index_map_this_;
      IndexMapOther index_map_other_;
      
      std::vector<vertex_other_type> core_vec_;
      typedef iterator_property_map<typename std::vector<vertex_other_type>::iterator,
                                    IndexMapThis, vertex_other_type, 
                                    vertex_other_type&> core_map_type;
      core_map_type core_;
    
      std::vector<size_type> in_vec_, out_vec_;
      typedef iterator_property_map<typename std::vector<size_type>::iterator,
                                    IndexMapThis, size_type, size_type&> in_out_map_type;
      in_out_map_type in_, out_;

      size_type term_in_count_, term_out_count_, term_both_count_, core_count_;

      // Forbidden 
      base_state(const base_state&);
      base_state& operator=(const base_state&);

    public:

      base_state(const GraphThis& graph_this, const GraphOther& graph_other,
                 IndexMapThis index_map_this, IndexMapOther index_map_other)
        : graph_this_(graph_this), graph_other_(graph_other), 
          index_map_this_(index_map_this), index_map_other_(index_map_other), 
          term_in_count_(0), term_out_count_(0), term_both_count_(0), core_count_(0) {

        core_vec_.resize(num_vertices(graph_this_), graph_traits<GraphOther>::null_vertex());
        core_ = make_iterator_property_map(core_vec_.begin(), index_map_this_);

        in_vec_.resize(num_vertices(graph_this_), 0);
        in_ = make_iterator_property_map(in_vec_.begin(), index_map_this_);

        out_vec_.resize(num_vertices(graph_this_), 0);
        out_ = make_iterator_property_map(out_vec_.begin(), index_map_this_);
      }

      // Adds a vertex pair to the state of graph graph_this
      void push(const vertex_this_type& v_this, const vertex_other_type& v_other) {

        ++core_count_;

        put(core_, v_this, v_other);

        if (!get(in_, v_this)) {   
          put(in_, v_this, core_count_);
          ++term_in_count_;
          if (get(out_, v_this))
            ++term_both_count_;
        }

        if (!get(out_, v_this)) {   
          put(out_, v_this, core_count_);
          ++term_out_count_;
          if (get(in_, v_this))
            ++term_both_count_;
        }

        BGL_FORALL_INEDGES_T(v_this, e, graph_this_, GraphThis) {
          vertex_this_type w = source(e, graph_this_);
          if (!get(in_, w)) {
            put(in_, w, core_count_);
            ++term_in_count_;
            if (get(out_, w))
              ++term_both_count_;
          }
        }
        
        BGL_FORALL_OUTEDGES_T(v_this, e, graph_this_, GraphThis) {
          vertex_this_type w = target(e, graph_this_);
          if (!get(out_, w)) {
            put(out_, w, core_count_);
            ++term_out_count_;
            if (get(in_, w))
              ++term_both_count_;                        
          }
        }
        
      }

      // Removes vertex pair from state of graph_this
      void pop(const vertex_this_type& v_this, const vertex_other_type&) {
        
        if (!core_count_) return;
        
        if (get(in_, v_this) == core_count_) {
          put(in_, v_this, 0);
          --term_in_count_;
          if (get(out_, v_this))
            --term_both_count_;
        }

        BGL_FORALL_INEDGES_T(v_this, e, graph_this_, GraphThis) {
          vertex_this_type w = source(e, graph_this_);
          if (get(in_, w) == core_count_) {
            put(in_, w, 0);
            --term_in_count_;
            if (get(out_, w))
              --term_both_count_;
          }
        }

        if (get(out_, v_this) == core_count_) {
          put(out_, v_this, 0);
          --term_out_count_;
          if (get(in_, v_this))
            --term_both_count_;
        }

        BGL_FORALL_OUTEDGES_T(v_this, e, graph_this_, GraphThis) {
          vertex_this_type w = target(e, graph_this_);
          if (get(out_, w) == core_count_) {
            put(out_, w, 0);
            --term_out_count_;
            if (get(in_, w))
              --term_both_count_;
          }
        }
        put(core_, v_this, graph_traits<GraphOther>::null_vertex());

        --core_count_;
        
      }
            
      // Returns true if the in-terminal set is not empty  
      bool term_in() const {
        return core_count_ < term_in_count_ ;
      }

      // Returns true if vertex belongs to the in-terminal set
      bool term_in(const vertex_this_type& v) const {
        return (get(in_, v) > 0) &&
               (get(core_, v) == graph_traits<GraphOther>::null_vertex());
      }
            
      // Returns true if the out-terminal set is not empty  
      bool term_out() const {
        return core_count_ < term_out_count_;
      }

      // Returns true if vertex belongs to the out-terminal set
      bool term_out(const vertex_this_type& v) const {
        return (get(out_, v) > 0) && 
               (get(core_, v) == graph_traits<GraphOther>::null_vertex());
      }

      // Returns true of both (in- and out-terminal) sets are not empty
      bool term_both() const {
        return core_count_ < term_both_count_;
      }

      // Returns true if vertex belongs to both (in- and out-terminal) sets
      bool term_both(const vertex_this_type& v) const {
        return (get(in_, v) > 0) && (get(out_, v) > 0) && 
               (get(core_, v) == graph_traits<GraphOther>::null_vertex());
      }

      // Returns true if vertex belongs to the core map, i.e. it is in the 
      // present mapping
      bool in_core(const vertex_this_type& v) const {
        return get(core_, v) != graph_traits<GraphOther>::null_vertex();
      }

      // Returns the number of vertices in the mapping
      size_type count() const {
        return core_count_;
      }            

      // Returns the image (in graph_other) of vertex v (in graph_this)
      vertex_other_type core(const vertex_this_type& v) const {
        return get(core_, v);
      }

      // Returns the mapping
      core_map_type get_map() const {
        return core_;
      }

      // Returns the "time" (or depth) when vertex was added to the in-terminal set
      size_type in_depth(const vertex_this_type& v) const {
        return get(in_, v);
      }

      // Returns the "time" (or depth) when vertex was added to the out-terminal set
      size_type out_depth(const vertex_this_type& v) const {
        return get(out_, v);
      }            

      // Returns the terminal set counts
      boost::tuple<size_type, size_type, size_type>
      term_set() const {
        return boost::make_tuple(term_in_count_, term_out_count_, 
                                 term_both_count_);
      }
      
    };


    // Function object that checks whether a valid edge
    // exists. For multi-graphs matched edges are excluded  
    template <typename Graph, typename Enable = void>
    struct equivalent_edge_exists {
      typedef typename boost::graph_traits<Graph>::edge_descriptor edge_type;

      BOOST_CONCEPT_ASSERT(( LessThanComparable<edge_type> ));

      template<typename EdgePredicate>
      bool operator()(typename graph_traits<Graph>::vertex_descriptor s,
                      typename graph_traits<Graph>::vertex_descriptor t, 
                      EdgePredicate is_valid_edge, const Graph& g) {
        
        BGL_FORALL_OUTEDGES_T(s, e, g, Graph) {
          if ((target(e, g) == t) && is_valid_edge(e) && 
              (matched_edges_.find(e) == matched_edges_.end())) {
            matched_edges_.insert(e);
            return true;
          }
        }

        return false;
      }

    private:
      
      std::set<edge_type> matched_edges_;
    };
    
    template <typename Graph>
    struct equivalent_edge_exists<Graph, typename boost::disable_if<is_multigraph<Graph> >::type> {
      template<typename EdgePredicate>
      bool operator()(typename graph_traits<Graph>::vertex_descriptor s,
                      typename graph_traits<Graph>::vertex_descriptor t, 
                      EdgePredicate is_valid_edge, const Graph& g) {
        
        typename graph_traits<Graph>::edge_descriptor e;
        bool found;
        boost::tie(e, found) = edge(s, t, g);
        if (!found)
          return false;
        else if (is_valid_edge(e))
          return true;
        
        return false;
      }
      
    };


    // Generates a predicate for edge e1 given  a binary predicate and a 
    // fixed edge e2
    template <typename Graph1,
              typename Graph2,
              typename EdgeEquivalencePredicate>
    struct edge1_predicate {
      
      edge1_predicate(EdgeEquivalencePredicate edge_comp, 
                      typename graph_traits<Graph2>::edge_descriptor e2)
        : edge_comp_(edge_comp), e2_(e2) {}
      
      bool operator()(typename graph_traits<Graph1>::edge_descriptor e1) {
        return edge_comp_(e1, e2_);
      }

      EdgeEquivalencePredicate edge_comp_;
      typename graph_traits<Graph2>::edge_descriptor e2_;
    };


    // Generates a predicate for edge e2 given given a binary predicate and a
    // fixed edge e1
    template <typename Graph1,
              typename Graph2,
              typename EdgeEquivalencePredicate>
    struct edge2_predicate {
      
      edge2_predicate(EdgeEquivalencePredicate edge_comp, 
                      typename graph_traits<Graph1>::edge_descriptor e1)
        : edge_comp_(edge_comp), e1_(e1) {}

      bool operator()(typename graph_traits<Graph2>::edge_descriptor e2) {
        return edge_comp_(e1_, e2);
      }

      EdgeEquivalencePredicate edge_comp_;
      typename graph_traits<Graph1>::edge_descriptor e1_;
    };


    enum problem_selector {subgraph_mono, subgraph_iso, isomorphism };
    
    // The actual state associated with both graphs
    template<typename Graph1,
             typename Graph2,
             typename IndexMap1,
             typename IndexMap2,
             typename EdgeEquivalencePredicate,
             typename VertexEquivalencePredicate,
             typename SubGraphIsoMapCallback,
             problem_selector problem_selection>
    class state {

      typedef typename graph_traits<Graph1>::vertex_descriptor vertex1_type;
      typedef typename graph_traits<Graph2>::vertex_descriptor vertex2_type;

      typedef typename graph_traits<Graph1>::edge_descriptor edge1_type;
      typedef typename graph_traits<Graph2>::edge_descriptor edge2_type;

      typedef typename graph_traits<Graph1>::vertices_size_type graph1_size_type;
      typedef typename graph_traits<Graph2>::vertices_size_type graph2_size_type;

      const Graph1& graph1_;
      const Graph2& graph2_;
      
      IndexMap1 index_map1_;
      
      EdgeEquivalencePredicate edge_comp_;
      VertexEquivalencePredicate vertex_comp_;
                
      base_state<Graph1, Graph2, IndexMap1, IndexMap2> state1_;
      base_state<Graph2, Graph1, IndexMap2, IndexMap1> state2_;

      // Three helper functions used in Feasibility and Valid functions to test
      // terminal set counts when testing for:
      // - graph sub-graph monomorphism, or
      inline bool comp_term_sets(graph1_size_type a, 
                                 graph2_size_type b,
                                 boost::mpl::int_<subgraph_mono>) const {
        return a <= b;
      }

      // - graph sub-graph isomorphism, or
      inline bool comp_term_sets(graph1_size_type a, 
                                 graph2_size_type b,
                                 boost::mpl::int_<subgraph_iso>) const {
        return a <= b;
      }

      // - graph isomorphism
      inline bool comp_term_sets(graph1_size_type a, 
                                 graph2_size_type b,
                                 boost::mpl::int_<isomorphism>) const {
        return a == b;
      }
      
      // Forbidden 
      state(const state&);
      state& operator=(const state&);

    public:

      state(const Graph1& graph1, const Graph2& graph2, 
            IndexMap1 index_map1, IndexMap2 index_map2, 
            EdgeEquivalencePredicate edge_comp,
            VertexEquivalencePredicate vertex_comp)
        : graph1_(graph1), graph2_(graph2), 
          index_map1_(index_map1), 
          edge_comp_(edge_comp), vertex_comp_(vertex_comp),
          state1_(graph1, graph2, index_map1, index_map2), 
          state2_(graph2, graph1, index_map2, index_map1) {}
      
            // Add vertex pair to the state
      void push(const vertex1_type& v, const vertex2_type& w) {
        state1_.push(v, w);
        state2_.push(w, v);
      }
      
      // Remove vertex pair from state
      void pop(const vertex1_type& v, const vertex2_type&) {
        vertex2_type w = state1_.core(v);
        state1_.pop(v, w);
        state2_.pop(w, v);
      }
           
      // Checks the feasibility of a new vertex pair
      bool feasible(const vertex1_type& v_new, const vertex2_type& w_new) {
        
        if (!vertex_comp_(v_new, w_new)) return false;
        
        // graph1
        graph1_size_type term_in1_count = 0, term_out1_count = 0, rest1_count = 0;
        
        {
          equivalent_edge_exists<Graph2> edge2_exists;
          
          BGL_FORALL_INEDGES_T(v_new, e1, graph1_, Graph1) {
            vertex1_type v = source(e1, graph1_);
            
            if (state1_.in_core(v) || (v == v_new)) {
              vertex2_type w = w_new;
              if (v != v_new)
                w = state1_.core(v);
              if (!edge2_exists(w, w_new,
                                edge2_predicate<Graph1, Graph2, EdgeEquivalencePredicate>(edge_comp_, e1), 
                                graph2_))
                return false;
              
            } else {
              if (0 < state1_.in_depth(v))
                ++term_in1_count;
              if (0 < state1_.out_depth(v))
                ++term_out1_count;
              if ((state1_.in_depth(v) == 0) && (state1_.out_depth(v) == 0))
                ++rest1_count;
            }
          }
        }
        
        {
          equivalent_edge_exists<Graph2> edge2_exists;
          
          BGL_FORALL_OUTEDGES_T(v_new, e1, graph1_, Graph1) {
            vertex1_type v = target(e1, graph1_);
            if (state1_.in_core(v) || (v == v_new)) {
              vertex2_type w = w_new;
              if (v != v_new)
                w = state1_.core(v);
              
              if (!edge2_exists(w_new, w,
                                edge2_predicate<Graph1, Graph2, EdgeEquivalencePredicate>(edge_comp_, e1), 
                                graph2_))
                return false;
              
            } else {
              if (0 < state1_.in_depth(v))
                ++term_in1_count;
              if (0 < state1_.out_depth(v))
                ++term_out1_count;
              if ((state1_.in_depth(v) == 0) && (state1_.out_depth(v) == 0))
                ++rest1_count;
            }
          }
        }
        
        // graph2
        graph2_size_type term_out2_count = 0, term_in2_count = 0, rest2_count = 0;
        
        {
          equivalent_edge_exists<Graph1> edge1_exists;
          
          BGL_FORALL_INEDGES_T(w_new, e2, graph2_, Graph2) {
            vertex2_type w = source(e2, graph2_);
            if (state2_.in_core(w) || (w == w_new)) {
              if (problem_selection != subgraph_mono) {
                vertex1_type v = v_new;
                if (w != w_new)
                  v = state2_.core(w);
              
                if (!edge1_exists(v, v_new,
                                  edge1_predicate<Graph1, Graph2, EdgeEquivalencePredicate>(edge_comp_, e2), 
                                  graph1_))
                  return false;
              }
            } else {
              if (0 < state2_.in_depth(w))
                ++term_in2_count;
              if (0 < state2_.out_depth(w))
                ++term_out2_count;
              if ((state2_.in_depth(w) == 0) && (state2_.out_depth(w) == 0))
                ++rest2_count;
            }
          }
        }

        {
          equivalent_edge_exists<Graph1> edge1_exists;
          
          BGL_FORALL_OUTEDGES_T(w_new, e2, graph2_, Graph2) {
            vertex2_type w = target(e2, graph2_);
            if (state2_.in_core(w) || (w == w_new)) {
              if (problem_selection != subgraph_mono) {
                vertex1_type v = v_new;
                if (w != w_new)
                  v = state2_.core(w);
              
                if (!edge1_exists(v_new, v,
                                  edge1_predicate<Graph1, Graph2, EdgeEquivalencePredicate>(edge_comp_, e2), 
                                  graph1_))
                  return false;
              }
            } else {
              if (0 < state2_.in_depth(w))
                ++term_in2_count;
              if (0 < state2_.out_depth(w))
                ++term_out2_count;
              if ((state2_.in_depth(w) == 0) && (state2_.out_depth(w) == 0))
                ++rest2_count;
            }
          }
        }

        if (problem_selection != subgraph_mono) { // subgraph_iso and isomorphism
          return comp_term_sets(term_in1_count, term_in2_count,
                                boost::mpl::int_<problem_selection>()) &&
                 comp_term_sets(term_out1_count, term_out2_count, 
                                boost::mpl::int_<problem_selection>()) &&
                 comp_term_sets(rest1_count, rest2_count, 
                                boost::mpl::int_<problem_selection>());
        } else { // subgraph_mono
          return comp_term_sets(term_in1_count, term_in2_count,
                                boost::mpl::int_<problem_selection>()) &&
                 comp_term_sets(term_out1_count, term_out2_count, 
                                boost::mpl::int_<problem_selection>()) &&
                 comp_term_sets(term_in1_count + term_out1_count + rest1_count,
                                term_in2_count + term_out2_count + rest2_count, 
                                boost::mpl::int_<problem_selection>());
        }
      }
      
      // Returns true if vertex v in graph1 is a possible candidate to
      // be added to the current state
      bool possible_candidate1(const vertex1_type& v) const {
        if (state1_.term_both() && state2_.term_both()) 
          return state1_.term_both(v);
        else if (state1_.term_out() && state2_.term_out())
          return state1_.term_out(v);
        else if (state1_.term_in() && state2_.term_in())
          return state1_.term_in(v);
        else
          return !state1_.in_core(v);
      }

      // Returns true if vertex w in graph2 is a possible candidate to
      // be added to the current state
      bool possible_candidate2(const vertex2_type& w) const {
        if (state1_.term_both() && state2_.term_both()) 
          return state2_.term_both(w);
        else if (state1_.term_out() && state2_.term_out())
          return state2_.term_out(w);
        else if (state1_.term_in() && state2_.term_in())
          return state2_.term_in(w);
        else
          return !state2_.in_core(w);
      }

      // Returns true if a mapping was found
      bool success() const {
        return state1_.count() == num_vertices(graph1_);
      }
 
      // Returns true if a state is valid
      bool valid() const {
        boost::tuple<graph1_size_type, graph1_size_type, graph1_size_type> term1;
        boost::tuple<graph2_size_type, graph2_size_type, graph2_size_type> term2;
        
        term1 = state1_.term_set();
        term2 = state2_.term_set();
        
        return comp_term_sets(boost::get<0>(term1), boost::get<0>(term2),
                              boost::mpl::int_<problem_selection>()) &&
               comp_term_sets(boost::get<1>(term1), boost::get<1>(term2),
                              boost::mpl::int_<problem_selection>()) &&
               comp_term_sets(boost::get<2>(term1), boost::get<2>(term2),
                              boost::mpl::int_<problem_selection>()); 
      }
      
      // Calls the user_callback with a graph (sub)graph mapping 
      bool call_back(SubGraphIsoMapCallback user_callback) const {
        return user_callback(state1_.get_map(), state2_.get_map());
      }
      
    };

    
    // Data structure to keep info used for back tracking during
    // matching process
    template<typename Graph1,
             typename Graph2,
             typename VertexOrder1>
    struct vf2_match_continuation {
      typename VertexOrder1::const_iterator graph1_verts_iter;
      typename graph_traits<Graph2>::vertex_iterator graph2_verts_iter;
    };

    // Non-recursive method that explores state space using a depth-first
    // search strategy.  At each depth possible pairs candidate are compute
    // and tested for feasibility to extend the mapping. If a complete
    // mapping is found, the mapping is output to user_callback in the form
    // of a correspondence map (graph1 to graph2). Returning false from the
    // user_callback will terminate the search. Function match will return
    // true if the entire search space was explored.
    template<typename Graph1,
             typename Graph2,
             typename IndexMap1,
             typename IndexMap2,
             typename VertexOrder1,
             typename EdgeEquivalencePredicate,
             typename VertexEquivalencePredicate, 
             typename SubGraphIsoMapCallback,
             problem_selector problem_selection>
    bool match(const Graph1& graph1, const Graph2& graph2, 
               SubGraphIsoMapCallback user_callback, const VertexOrder1& vertex_order1, 
               state<Graph1, Graph2, IndexMap1, IndexMap2,
               EdgeEquivalencePredicate, VertexEquivalencePredicate,
               SubGraphIsoMapCallback, problem_selection>& s) {
      
      typename VertexOrder1::const_iterator graph1_verts_iter;

      typedef typename graph_traits<Graph2>::vertex_iterator vertex2_iterator_type;
      vertex2_iterator_type graph2_verts_iter, graph2_verts_iter_end;
    
      typedef vf2_match_continuation<Graph1, Graph2, VertexOrder1> match_continuation_type;
      std::vector<match_continuation_type> k;
      bool found_match = false;
  
      recur:
      if (s.success()) {
        if (!s.call_back(user_callback)) 
          return true;
        found_match = true;

        goto back_track;
      }
 
      if (!s.valid())
        goto back_track;

      graph1_verts_iter = vertex_order1.begin();
      while (graph1_verts_iter != vertex_order1.end() && 
             !s.possible_candidate1(*graph1_verts_iter)) {
        ++graph1_verts_iter;
      }

      boost::tie(graph2_verts_iter, graph2_verts_iter_end) = vertices(graph2);
      while (graph2_verts_iter != graph2_verts_iter_end) {
        if (s.possible_candidate2(*graph2_verts_iter)) {
          if (s.feasible(*graph1_verts_iter, *graph2_verts_iter)) {
            match_continuation_type kk;
            kk.graph1_verts_iter = graph1_verts_iter;
            kk.graph2_verts_iter = graph2_verts_iter;
            k.push_back(kk);
            
            s.push(*graph1_verts_iter, *graph2_verts_iter);
            goto recur;
          }
        }
        graph2_loop: ++graph2_verts_iter;
      }

      back_track:
      if (k.empty()) 
        return found_match;    
      
      const match_continuation_type kk = k.back();
      graph1_verts_iter = kk.graph1_verts_iter;
      graph2_verts_iter = kk.graph2_verts_iter;
      k.pop_back();
      
      s.pop(*graph1_verts_iter, *graph2_verts_iter);
      
      goto graph2_loop;
    }


    // Used to sort nodes by in/out degrees
    template<typename Graph>
    struct vertex_in_out_degree_cmp {
      typedef typename graph_traits<Graph>::vertex_descriptor vertex_type;

      vertex_in_out_degree_cmp(const Graph& graph)
        : graph_(graph) {}

      bool operator()(const vertex_type& v, const vertex_type& w) const {
        // lexicographical comparison
        return std::make_pair(in_degree(v, graph_), out_degree(v, graph_)) <
               std::make_pair(in_degree(w, graph_), out_degree(w, graph_));
      }

      const Graph& graph_;
    };


    // Used to sort nodes by multiplicity of in/out degrees
    template<typename Graph,
             typename FrequencyMap>
    struct vertex_frequency_degree_cmp {
      typedef typename graph_traits<Graph>::vertex_descriptor vertex_type;
      
      vertex_frequency_degree_cmp(const Graph& graph, FrequencyMap freq)
        : graph_(graph), freq_(freq) {}
      
      bool operator()(const vertex_type& v, const vertex_type& w) const {
        // lexicographical comparison
        return std::make_pair(freq_[v], in_degree(v, graph_)+out_degree(v, graph_)) <
               std::make_pair(freq_[w], in_degree(w, graph_)+out_degree(w, graph_));
      }

      const Graph& graph_;
      FrequencyMap freq_;
    };

    
    // Sorts vertices of a graph by multiplicity of in/out degrees 
    template<typename Graph,
             typename IndexMap,
             typename VertexOrder>
    void sort_vertices(const Graph& graph, IndexMap index_map, VertexOrder& order) {
      typedef typename graph_traits<Graph>::vertices_size_type size_type;

      boost::range::sort(order, vertex_in_out_degree_cmp<Graph>(graph));

      std::vector<size_type> freq_vec(num_vertices(graph), 0);
      typedef iterator_property_map<typename std::vector<size_type>::iterator,
                                    IndexMap, size_type, size_type&> frequency_map_type;
                
      frequency_map_type freq = make_iterator_property_map(freq_vec.begin(), index_map);

      typedef typename VertexOrder::iterator order_iterator;

      for (order_iterator order_iter = order.begin(); order_iter != order.end(); ) {
        size_type count = 0;
        for (order_iterator count_iter = order_iter;
             (count_iter != order.end()) &&
             (in_degree(*order_iter, graph) == in_degree(*count_iter, graph)) &&
             (out_degree(*order_iter, graph) == out_degree(*count_iter, graph)); 
             ++count_iter)
          ++count;
      
        for (size_type i = 0; i < count; ++i) {
          freq[*order_iter] = count;
          ++order_iter;
        }
      }

      boost::range::sort(order, vertex_frequency_degree_cmp<Graph, frequency_map_type>(graph, freq));

    }

    // Enumerates all graph sub-graph mono-/iso-morphism mappings between graphs
    // graph_small and graph_large. Continues until user_callback returns true or the
    // search space has been fully explored.
    template <problem_selector problem_selection,
              typename GraphSmall,
              typename GraphLarge,
              typename IndexMapSmall,
              typename IndexMapLarge,
              typename VertexOrderSmall,
              typename EdgeEquivalencePredicate,
              typename VertexEquivalencePredicate,
              typename SubGraphIsoMapCallback>
    bool vf2_subgraph_morphism(const GraphSmall& graph_small, const GraphLarge& graph_large,
                          SubGraphIsoMapCallback user_callback,
                          IndexMapSmall index_map_small, IndexMapLarge index_map_large, 
                          const VertexOrderSmall& vertex_order_small,
                          EdgeEquivalencePredicate edge_comp,
                          VertexEquivalencePredicate vertex_comp) {

      // Graph requirements
      BOOST_CONCEPT_ASSERT(( BidirectionalGraphConcept<GraphSmall> ));
      BOOST_CONCEPT_ASSERT(( VertexListGraphConcept<GraphSmall> ));
      BOOST_CONCEPT_ASSERT(( EdgeListGraphConcept<GraphSmall> ));
      BOOST_CONCEPT_ASSERT(( AdjacencyMatrixConcept<GraphSmall> ));

      BOOST_CONCEPT_ASSERT(( BidirectionalGraphConcept<GraphLarge> ));
      BOOST_CONCEPT_ASSERT(( VertexListGraphConcept<GraphLarge> ));
      BOOST_CONCEPT_ASSERT(( EdgeListGraphConcept<GraphLarge> ));
      BOOST_CONCEPT_ASSERT(( AdjacencyMatrixConcept<GraphLarge> ));

      typedef typename graph_traits<GraphSmall>::vertex_descriptor vertex_small_type;
      typedef typename graph_traits<GraphLarge>::vertex_descriptor vertex_large_type;

      typedef typename graph_traits<GraphSmall>::vertices_size_type size_type_small;
      typedef typename graph_traits<GraphLarge>::vertices_size_type size_type_large;
        
      // Property map requirements
      BOOST_CONCEPT_ASSERT(( ReadablePropertyMapConcept<IndexMapSmall, vertex_small_type> ));
      typedef typename property_traits<IndexMapSmall>::value_type IndexMapSmallValue;
      BOOST_STATIC_ASSERT(( is_convertible<IndexMapSmallValue, size_type_small>::value ));
        
      BOOST_CONCEPT_ASSERT(( ReadablePropertyMapConcept<IndexMapLarge, vertex_large_type> ));
      typedef typename property_traits<IndexMapLarge>::value_type IndexMapLargeValue;
      BOOST_STATIC_ASSERT(( is_convertible<IndexMapLargeValue, size_type_large>::value ));

      // Edge & vertex requirements
      typedef typename graph_traits<GraphSmall>::edge_descriptor edge_small_type;
      typedef typename graph_traits<GraphLarge>::edge_descriptor edge_large_type;

      BOOST_CONCEPT_ASSERT(( BinaryPredicateConcept<EdgeEquivalencePredicate, 
                             edge_small_type, edge_large_type> ));

      BOOST_CONCEPT_ASSERT(( BinaryPredicateConcept<VertexEquivalencePredicate, 
                             vertex_small_type, vertex_large_type> ));

      // Vertex order requirements
      BOOST_CONCEPT_ASSERT(( ContainerConcept<VertexOrderSmall> )); 
      typedef typename VertexOrderSmall::value_type order_value_type;
      BOOST_STATIC_ASSERT(( is_same<vertex_small_type, order_value_type>::value ));
      BOOST_ASSERT( num_vertices(graph_small) == vertex_order_small.size() );

      if (num_vertices(graph_small) > num_vertices(graph_large))
        return false;

      typename graph_traits<GraphSmall>::edges_size_type num_edges_small = num_edges(graph_small);
      typename graph_traits<GraphLarge>::edges_size_type num_edges_large = num_edges(graph_large);

      // Double the number of edges for undirected graphs: each edge counts as
      // in-edge and out-edge
      if (is_undirected(graph_small)) num_edges_small *= 2;
      if (is_undirected(graph_large)) num_edges_large *= 2;
      if (num_edges_small > num_edges_large)
        return false;
    
      detail::state<GraphSmall, GraphLarge, IndexMapSmall, IndexMapLarge,
                    EdgeEquivalencePredicate, VertexEquivalencePredicate,
                    SubGraphIsoMapCallback, problem_selection> 
        s(graph_small, graph_large, index_map_small, index_map_large, edge_comp, vertex_comp);

      return detail::match(graph_small, graph_large, user_callback, vertex_order_small, s);
    }

  } // namespace detail


  // Returns vertex order (vertices sorted by multiplicity of in/out degrees)
  template<typename Graph>
  std::vector<typename graph_traits<Graph>::vertex_descriptor> 
    vertex_order_by_mult(const Graph& graph) {

    std::vector<typename graph_traits<Graph>::vertex_descriptor> vertex_order;
    std::copy(vertices(graph).first, vertices(graph).second, std::back_inserter(vertex_order));

    detail::sort_vertices(graph, get(vertex_index, graph), vertex_order);
    return vertex_order;
  }


  // Enumerates all graph sub-graph monomorphism mappings between graphs
  // graph_small and graph_large. Continues until user_callback returns true or the
  // search space has been fully explored.
  template <typename GraphSmall,
            typename GraphLarge,
            typename IndexMapSmall,
            typename IndexMapLarge,
            typename VertexOrderSmall,
            typename EdgeEquivalencePredicate,
            typename VertexEquivalencePredicate,
            typename SubGraphIsoMapCallback>
  bool vf2_subgraph_mono(const GraphSmall& graph_small, const GraphLarge& graph_large,
                         SubGraphIsoMapCallback user_callback,
                         IndexMapSmall index_map_small, IndexMapLarge index_map_large, 
                         const VertexOrderSmall& vertex_order_small,
                         EdgeEquivalencePredicate edge_comp,
                         VertexEquivalencePredicate vertex_comp) {
    return detail::vf2_subgraph_morphism<detail::subgraph_mono>
                                        (graph_small, graph_large,
                                         user_callback,
                                         index_map_small, index_map_large,
                                         vertex_order_small,
                                         edge_comp,
                                         vertex_comp);
  }


  // All default interface for vf2_subgraph_iso
  template <typename GraphSmall,
            typename GraphLarge,
            typename SubGraphIsoMapCallback>
  bool vf2_subgraph_mono(const GraphSmall& graph_small, const GraphLarge& graph_large, 
                         SubGraphIsoMapCallback user_callback) {
    return vf2_subgraph_mono(graph_small, graph_large, user_callback, 
                             get(vertex_index, graph_small), get(vertex_index, graph_large),
                             vertex_order_by_mult(graph_small),
                             always_equivalent(), always_equivalent());
  }


  // Named parameter interface of vf2_subgraph_iso
  template <typename GraphSmall,
            typename GraphLarge,
            typename VertexOrderSmall,
            typename SubGraphIsoMapCallback,
            typename Param,
            typename Tag,
            typename Rest>
  bool vf2_subgraph_mono(const GraphSmall& graph_small, const GraphLarge& graph_large,
                         SubGraphIsoMapCallback user_callback,
                         const VertexOrderSmall& vertex_order_small,
                         const bgl_named_params<Param, Tag, Rest>& params) {
    return vf2_subgraph_mono(graph_small, graph_large, user_callback,
                             choose_const_pmap(get_param(params, vertex_index1),
                                               graph_small, vertex_index),
                             choose_const_pmap(get_param(params, vertex_index2),
                                               graph_large, vertex_index),
                             vertex_order_small,
                             choose_param(get_param(params, edges_equivalent_t()),
                                          always_equivalent()),
                             choose_param(get_param(params, vertices_equivalent_t()),
                                          always_equivalent())
                             );
  }
  
  
  // Enumerates all graph sub-graph isomorphism mappings between graphs
  // graph_small and graph_large. Continues until user_callback returns true or the
  // search space has been fully explored.
  template <typename GraphSmall,
            typename GraphLarge,
            typename IndexMapSmall,
            typename IndexMapLarge,
            typename VertexOrderSmall,
            typename EdgeEquivalencePredicate,
            typename VertexEquivalencePredicate,
            typename SubGraphIsoMapCallback>
  bool vf2_subgraph_iso(const GraphSmall& graph_small, const GraphLarge& graph_large,
                        SubGraphIsoMapCallback user_callback,
                        IndexMapSmall index_map_small, IndexMapLarge index_map_large, 
                        const VertexOrderSmall& vertex_order_small,
                        EdgeEquivalencePredicate edge_comp,
                        VertexEquivalencePredicate vertex_comp) {
    return detail::vf2_subgraph_morphism<detail::subgraph_iso>
                                        (graph_small, graph_large,
                                         user_callback,
                                         index_map_small, index_map_large,
                                         vertex_order_small,
                                         edge_comp,
                                         vertex_comp);
  }


  // All default interface for vf2_subgraph_iso
  template <typename GraphSmall,
            typename GraphLarge,
            typename SubGraphIsoMapCallback>
  bool vf2_subgraph_iso(const GraphSmall& graph_small, const GraphLarge& graph_large, 
                        SubGraphIsoMapCallback user_callback) {

    return vf2_subgraph_iso(graph_small, graph_large, user_callback, 
                            get(vertex_index, graph_small), get(vertex_index, graph_large),
                            vertex_order_by_mult(graph_small),
                            always_equivalent(), always_equivalent());
  }


  // Named parameter interface of vf2_subgraph_iso
  template <typename GraphSmall,
            typename GraphLarge,
            typename VertexOrderSmall,
            typename SubGraphIsoMapCallback,
            typename Param,
            typename Tag,
            typename Rest>
  bool vf2_subgraph_iso(const GraphSmall& graph_small, const GraphLarge& graph_large,
                        SubGraphIsoMapCallback user_callback,
                        const VertexOrderSmall& vertex_order_small,
                        const bgl_named_params<Param, Tag, Rest>& params) {
    
    return vf2_subgraph_iso(graph_small, graph_large, user_callback,
                            choose_const_pmap(get_param(params, vertex_index1),
                                              graph_small, vertex_index),
                            choose_const_pmap(get_param(params, vertex_index2),
                                              graph_large, vertex_index),
                            vertex_order_small,
                            choose_param(get_param(params, edges_equivalent_t()),
                                         always_equivalent()),
                            choose_param(get_param(params, vertices_equivalent_t()),
                                         always_equivalent())
                            );

  }


  // Enumerates all isomorphism mappings between graphs graph1_ and graph2_.
  // Continues until user_callback returns true or the search space has been
  // fully explored.
  template <typename Graph1,
            typename Graph2,
            typename IndexMap1,
            typename IndexMap2,
            typename VertexOrder1,
            typename EdgeEquivalencePredicate,
            typename VertexEquivalencePredicate,
            typename GraphIsoMapCallback>
  bool vf2_graph_iso(const Graph1& graph1, const Graph2& graph2,
                     GraphIsoMapCallback user_callback,
                     IndexMap1 index_map1, IndexMap2 index_map2, 
                     const VertexOrder1& vertex_order1,
                     EdgeEquivalencePredicate edge_comp,
                     VertexEquivalencePredicate vertex_comp) {

    // Graph requirements
    BOOST_CONCEPT_ASSERT(( BidirectionalGraphConcept<Graph1> ));
    BOOST_CONCEPT_ASSERT(( VertexListGraphConcept<Graph1> ));
    BOOST_CONCEPT_ASSERT(( EdgeListGraphConcept<Graph1> ));
    BOOST_CONCEPT_ASSERT(( AdjacencyMatrixConcept<Graph1> ));

    BOOST_CONCEPT_ASSERT(( BidirectionalGraphConcept<Graph2> ));
    BOOST_CONCEPT_ASSERT(( VertexListGraphConcept<Graph2> ));
    BOOST_CONCEPT_ASSERT(( EdgeListGraphConcept<Graph2> ));
    BOOST_CONCEPT_ASSERT(( AdjacencyMatrixConcept<Graph2> ));
 
        
    typedef typename graph_traits<Graph1>::vertex_descriptor vertex1_type;
    typedef typename graph_traits<Graph2>::vertex_descriptor vertex2_type;
    
    typedef typename graph_traits<Graph1>::vertices_size_type size_type1;
    typedef typename graph_traits<Graph2>::vertices_size_type size_type2;
        
    // Property map requirements
    BOOST_CONCEPT_ASSERT(( ReadablePropertyMapConcept<IndexMap1, vertex1_type> ));
    typedef typename property_traits<IndexMap1>::value_type IndexMap1Value;
    BOOST_STATIC_ASSERT(( is_convertible<IndexMap1Value, size_type1>::value ));
        
    BOOST_CONCEPT_ASSERT(( ReadablePropertyMapConcept<IndexMap2, vertex2_type> ));
    typedef typename property_traits<IndexMap2>::value_type IndexMap2Value;
    BOOST_STATIC_ASSERT(( is_convertible<IndexMap2Value, size_type2>::value ));

    // Edge & vertex requirements
    typedef typename graph_traits<Graph1>::edge_descriptor edge1_type;
    typedef typename graph_traits<Graph2>::edge_descriptor edge2_type;

    BOOST_CONCEPT_ASSERT(( BinaryPredicateConcept<EdgeEquivalencePredicate, 
                           edge1_type, edge2_type> ));

    BOOST_CONCEPT_ASSERT(( BinaryPredicateConcept<VertexEquivalencePredicate, 
                           vertex1_type, vertex2_type> ));
    
    // Vertex order requirements
    BOOST_CONCEPT_ASSERT(( ContainerConcept<VertexOrder1> )); 
    typedef typename VertexOrder1::value_type order_value_type;
    BOOST_STATIC_ASSERT(( is_same<vertex1_type, order_value_type>::value ));
    BOOST_ASSERT( num_vertices(graph1) == vertex_order1.size() );

    if (num_vertices(graph1) != num_vertices(graph2))
      return false;

    typename graph_traits<Graph1>::edges_size_type num_edges1 = num_edges(graph1);
    typename graph_traits<Graph2>::edges_size_type num_edges2 = num_edges(graph2);

    // Double the number of edges for undirected graphs: each edge counts as
    // in-edge and out-edge
    if (is_undirected(graph1)) num_edges1 *= 2;
    if (is_undirected(graph2)) num_edges2 *= 2;
    if (num_edges1 != num_edges2)
      return false;

    detail::state<Graph1, Graph2, IndexMap1, IndexMap2,
                  EdgeEquivalencePredicate, VertexEquivalencePredicate,
                  GraphIsoMapCallback, detail::isomorphism> 
      s(graph1, graph2, index_map1, index_map2, edge_comp, vertex_comp);

    return detail::match(graph1, graph2, user_callback, vertex_order1, s);
  }


  // All default interface for vf2_graph_iso
  template <typename Graph1,
            typename Graph2,
            typename GraphIsoMapCallback>
  bool vf2_graph_iso(const Graph1& graph1, const Graph2& graph2, 
                     GraphIsoMapCallback user_callback) {
    
    return vf2_graph_iso(graph1, graph2, user_callback, 
                         get(vertex_index, graph1), get(vertex_index, graph2),
                         vertex_order_by_mult(graph1),
                         always_equivalent(), always_equivalent());
  }


  // Named parameter interface of vf2_graph_iso
  template <typename Graph1,
            typename Graph2,
            typename VertexOrder1,
            typename GraphIsoMapCallback,
            typename Param,
            typename Tag,
            typename Rest>
  bool vf2_graph_iso(const Graph1& graph1, const Graph2& graph2,
                     GraphIsoMapCallback user_callback,
                     const VertexOrder1& vertex_order1,
                     const bgl_named_params<Param, Tag, Rest>& params) {
    
    return vf2_graph_iso(graph1, graph2, user_callback,
                         choose_const_pmap(get_param(params, vertex_index1),
                                           graph1, vertex_index),
                         choose_const_pmap(get_param(params, vertex_index2),
                                           graph2, vertex_index),
                         vertex_order1,
                         choose_param(get_param(params, edges_equivalent_t()),
                                always_equivalent()),
                         choose_param(get_param(params, vertices_equivalent_t()),
                                      always_equivalent())
                         );

  }


  // Verifies a graph (sub)graph isomorphism map 
  template<typename Graph1,
           typename Graph2,
           typename CorresponenceMap1To2,
           typename EdgeEquivalencePredicate,
           typename VertexEquivalencePredicate>
  inline bool verify_vf2_subgraph_iso(const Graph1& graph1, const Graph2& graph2, 
                                      const CorresponenceMap1To2 f,
                                      EdgeEquivalencePredicate edge_comp, 
                                      VertexEquivalencePredicate vertex_comp) {
        
    BOOST_CONCEPT_ASSERT(( EdgeListGraphConcept<Graph1> ));
    BOOST_CONCEPT_ASSERT(( AdjacencyMatrixConcept<Graph2> ));

    detail::equivalent_edge_exists<Graph2> edge2_exists;

    BGL_FORALL_EDGES_T(e1, graph1, Graph1) {
      typename graph_traits<Graph1>::vertex_descriptor s1, t1;
      typename graph_traits<Graph2>::vertex_descriptor s2, t2;
      
      s1 = source(e1, graph1); t1 = target(e1, graph1);
      s2 = get(f, s1); t2 = get(f, t1);
      
      if (!vertex_comp(s1, s2) || !vertex_comp(t1, t2))
        return false;

      typename graph_traits<Graph2>::edge_descriptor e2;
      
      if (!edge2_exists(s2, t2,
                        detail::edge2_predicate<Graph1, Graph2, EdgeEquivalencePredicate>(edge_comp, e1), 
                        graph2))
        return false;
      
    }
  
    return true;
  }

  // Variant of verify_subgraph_iso with all default parameters
  template<typename Graph1,
           typename Graph2,
           typename CorresponenceMap1To2>
  inline bool verify_vf2_subgraph_iso(const Graph1& graph1, const Graph2& graph2, 
                                      const CorresponenceMap1To2 f) {
    return verify_vf2_subgraph_iso(graph1, graph2, f, 
                                   always_equivalent(), always_equivalent());
  }



} // namespace boost

#ifdef BOOST_ISO_INCLUDED_ITER_MACROS
#undef BOOST_ISO_INCLUDED_ITER_MACROS
#include <boost/graph/iteration_macros_undef.hpp>
#endif

#endif // BOOST_VF2_SUB_GRAPH_ISO_HPP