minimization_hopcroft.hxx

// minimization_hopcroft.hxx: this file is part of the Vaucanson project.
//
// Vaucanson, a generic library for finite state machines.
//
// Copyright (C) 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008 The Vaucanson Group.
//
// This program is free software; you can redistribute it and/or
// modify it under the terms of the GNU General Public License
// as published by the Free Software Foundation; either version 2
// of the License, or (at your option) any later version.
//
// The complete GNU General Public Licence Notice can be found as the
// `COPYING' file in the root directory.
//
// The Vaucanson Group consists of people listed in the `AUTHORS' file.
//
#ifndef VCSN_ALGORITHMS_MINIMIZATION_HOPCROFT_HXX
# define VCSN_ALGORITHMS_MINIMIZATION_HOPCROFT_HXX

# include <algorithm>
# include <list>
# include <queue>
# include <set>
# include <vector>

# include <vaucanson/algebra/implementation/semiring/numerical_semiring.hh>
# include <vaucanson/algorithms/minimization_hopcroft.hh>
# include <vaucanson/algorithms/is_deterministic.hh>
# include <vaucanson/algorithms/complete.hh>
# include <vaucanson/automata/concept/automata_base.hh>
# include <vaucanson/misc/usual_macros.hh>
# include <vaucanson/misc/bitset.hh>

# ifndef VCSN_NDEBUG
#  include <vaucanson/algorithms/is_deterministic.hh>
# endif // ! VCSN_NDEBUG

namespace vcsn
{

  namespace internal
  {
    namespace hopcroft_minimization_det
    {

# define HOPCROFT_TYPES()                                               \
      typedef std::set<hstate_t> hstates_t;                             \
      typedef std::vector<hstates_t> partition_t;                       \
      typedef std::vector<unsigned> class_of_t;                         \
      typedef std::queue<std::pair<hstates_t*, unsigned> > to_treat_t;

      template <typename input_t>
      struct splitter_functor
      {
        AUTOMATON_TYPES (input_t);
        AUTOMATON_FREEMONOID_TYPES (input_t);
        HOPCROFT_TYPES ();

        const input_t& input_;
        hstates_t going_in_;
        class_of_t& class_of_;
        std::list<unsigned> maybe_splittable_;
        std::vector<unsigned> count_for_;

        splitter_functor (const input_t& input, unsigned int max_state,
                          class_of_t& class_of)
          : input_ (input), going_in_ (), class_of_(class_of),
            count_for_ (max_state)
        {}

        bool compute_states_going_in (const hstates_t& ss, letter_t l)
        {
          going_in_.clear ();
          maybe_splittable_.clear ();
          for_all_const_ (hstates_t, i, ss)
            input_.letter_rdeltaf (*this, *i, l, delta_kind::states ());
          return not going_in_.empty ();
        }

        void operator () (hstate_t state)
        {
          unsigned class_of_state = class_of_[state];

          if (count_for_[class_of_state] == 0)
            maybe_splittable_.push_back (class_of_state);
          count_for_[class_of_state]++;
          going_in_.insert (state);
        }

        void execute (partition_t& partition, to_treat_t& to_treat,
                      unsigned& n_partition)
        {
          for_all (std::list<unsigned>, inpartition, maybe_splittable_)
          {
            hstates_t& states = partition[*inpartition];
            if (states.size () == count_for_[*inpartition])
            { // All elements in states are predecessors, no split.
              count_for_[*inpartition] = 0;
              continue;
            }
            count_for_[*inpartition] = 0;
            hstates_t states_inter_going_in;
            hstates_t& states_minus_going_in = partition[n_partition];
            // Compute @a states \ @a going_in_.
            set_difference
              (states.begin (), states.end (),
               going_in_.begin (), going_in_.end (),
               std::insert_iterator<hstates_t> (states_minus_going_in,
                                                states_minus_going_in.begin ()));
            // Compute @a states Inter @a going_in_.
            set_intersection
              (states.begin(), states.end (),
               going_in_.begin (), going_in_.end (),
               std::insert_iterator<hstates_t> (states_inter_going_in,
                                                states_inter_going_in.begin ()));
            // A split MUST occur.
            assertion (not (states_inter_going_in.empty ()
                            or states_minus_going_in.empty ()));
            // @a states must be the bigger one.
            if (states_minus_going_in.size () > states_inter_going_in.size ())
            {
              states.swap (states_minus_going_in);
              states_minus_going_in.swap (states_inter_going_in);
            }
            else
              states.swap (states_inter_going_in);
            for_all_const_ (hstates_t, istate, states_minus_going_in)
              class_of_[*istate] = n_partition;
            to_treat.push (std::make_pair (&states_minus_going_in,
                                           n_partition++));
          }
        }
      };

      template <typename input_t, typename output_t>
      struct transition_adder_functor
      {
        AUTOMATON_TYPES (input_t);
        HOPCROFT_TYPES ();

        const input_t& input_;
        output_t& output_;
        const class_of_t& class_of_;

        unsigned src_;

        transition_adder_functor (const input_t& input, output_t& output,
                                  const class_of_t& class_of)
          : input_ (input), output_ (output), class_of_ (class_of)
        {}

        void execute (hstate_t representative)
        {
          src_ = class_of_[representative];
          input_.deltaf (*this, representative, delta_kind::transitions ());
        }

        void operator () (htransition_t t)
        {
          output_.add_series_transition (src_, class_of_[input_.dst_of (t)],
                                         input_.series_of (t));
        }
      };
    }
  }


  template <typename A, typename AI1, typename AI2>
  void
  do_hopcroft_minimization_det(const AutomataBase<A>&,
                               Element<A, AI2>&         output,
                               const Element<A, AI1>&   input)
  {
    typedef Element<A, AI1> input_t;
    typedef Element<A, AI2> output_t;
    AUTOMATON_TYPES (input_t);
    AUTOMATON_FREEMONOID_TYPES (input_t);
    HOPCROFT_TYPES ();

    using namespace internal::hopcroft_minimization_det;

    precondition(is_deterministic(input));

    unsigned max_state = input.states ().back () + 1;
    partition_t partition (max_state);
    class_of_t class_of (max_state);
    to_treat_t to_treat;
    unsigned n_partition = 0;
    const alphabet_t& alphabet =
      input.structure ().series ().monoid ().alphabet ();

    {
      // Initialize Partition = {Q \ F , F }
      hstates_t* finals = 0, * others = 0;
      int n_finals = -1, n_others = -1,
        count_finals = 0, count_others = 0;

# define add_to_class(Name)                     \
      do {                                      \
        if (not Name)                           \
        {                                       \
          Name = &(partition[n_partition]);     \
          n_ ## Name = n_partition++;           \
        }                                       \
        count_ ## Name ++;                      \
        (*Name).insert (*state);                \
        class_of[*state] = n_ ## Name;          \
      } while (0)

      for_all_const_states (state, input)
        if (input.is_final (*state))
          add_to_class (finals);
        else
          add_to_class (others);
# undef add_to_class

      if (n_partition == 0)
        return;
      if (n_partition == 1)
      {
        output = input;
        return;
      }
      // Put F or Q \ F in the "To treat" list T.
      if (count_finals > count_others)
        to_treat.push (std::make_pair (others, n_others));
      else
        to_treat.push (std::make_pair (finals, n_finals));
    }

    {
      splitter_functor<input_t> splitter (input, max_state, class_of);

      // While T is not empty,
      while (not to_treat.empty () && n_partition < max_state)
      {
        // Remove a set S of T ,
        hstates_t& states = *(to_treat.front ().first);
        to_treat.pop ();

        // For each letter l in Alphabet,
        for_all_const_letters (letter, alphabet)
          {
            if (not splitter.compute_states_going_in (states, *letter))
              continue;
            splitter.execute (partition, to_treat, n_partition);
            if (n_partition == max_state)
              break;
          }
      }
    }

    // Build the automaton.
    // Assume that states are numbers starting from 0.
    for (unsigned i = 0; i < n_partition; ++i)
      output.add_state ();

    transition_adder_functor<input_t, output_t>
      transition_adder (input, output, class_of);

    typename partition_t::iterator istates = partition.begin ();
    for (unsigned i = 0; i < n_partition; ++i, ++istates)
    {
      hstate_t representative = *(*istates).begin();

      if (input.is_final (representative))
        output.set_final (class_of[representative]);
      transition_adder.execute (representative);
    }

    for_all_const_initial_states (state, input)
      output.set_initial (class_of[*state]);
  }

# undef HOPCROFT_TYPES

  template<typename A, typename AI>
  Element<A, AI>
  minimization_hopcroft(const Element<A, AI>& a)
  {
    TIMER_SCOPED ("minimization_hopcroft");
    precondition(is_deterministic(a));
    precondition(is_complete(a));
    Element<A, AI> output(a.structure());
    do_hopcroft_minimization_det(a.structure(), output, a);
    return output;
  }


  /*-------------------------------------.
  | Quotient with Boolean multiplicities |
  `-------------------------------------*/
  namespace internal
  {
    namespace hopcroft_minimization_undet
    {

# define QUOTIENT_TYPES()                                               \
      typedef std::list<hstate_t> partition_t;                          \
      typedef std::vector<partition_t> partition_set_t;                 \
      typedef typename partition_t::iterator partition_iterator;        \
      typedef std::vector<unsigned> class_of_t;                         \
      typedef std::set<hstate_t> delta_ret_t;                           \
      typedef std::pair<unsigned, letter_t> pair_t;                     \
      typedef std::list<pair_t> to_treat_t;

      template <typename input_t>
      class quotient_splitter
      {
      public:
        AUTOMATON_TYPES(input_t);
        AUTOMATON_FREEMONOID_TYPES(input_t);
        QUOTIENT_TYPES();

        typedef std::vector<bool> going_in_t;

        quotient_splitter (const automaton_t& input, class_of_t& class_of,
                           unsigned max_states)
          : input_(input),
            alphabet_(input.series().monoid().alphabet()),
            class_(class_of),
            count_for_(max_states, 0),
            twin_(max_states, 0),
            going_in_(max_states, false)
        { }

        bool compute_going_in_states (partition_t& p, letter_t a)
        {
          for_all_(going_in_t, s, going_in_)
            *s = false;

          for_all_(partition_t, s, p)
            input_.letter_rdeltaf(*this, *s, a, delta_kind::states());
          return !met_class_.empty();
        }

        void operator() (hstate_t s)
        {
          if (!going_in_[s])
          {
            going_in_[s] = true;
            const unsigned i = class_[s];
            if (count_for_[i] == 0)
              met_class_.push_back(i);
            count_for_[i]++;
          }
        }

        void split (partition_set_t& parts, unsigned& max_partitions)
        {
          for_all_(std::list<unsigned>, klass, met_class_)
          {
            // if all states are predecessors there is no needed split
            if (count_for_[*klass] == parts[*klass].size())
              continue;

            if (twin_[*klass] == 0)
              twin_[*klass] = max_partitions++;
            unsigned new_klass = twin_[*klass];

            typename partition_t::iterator q;
            for (typename partition_t::iterator next = parts[*klass].begin();
                 next != parts[*klass].end();)
            {
              q = next;
              ++next;
              if (going_in_[*q])
              {
                parts[new_klass].insert(parts[new_klass].end(), *q);
                class_[*q] = new_klass;
                parts[*klass].erase(q);
              }
            }
          }
        }

        void add_new_partitions(to_treat_t&             to_treat,
                                const partition_set_t&  part)
        {
          for_all_(std::list<unsigned>, klass, met_class_)
          {
            if (twin_[*klass] != 0)
            {
              for_all_const_letters(e, alphabet_)
              {
                if (find(to_treat.begin(), to_treat.end(), pair_t(*klass, *e)) !=
                    to_treat.end())
                  to_treat.push_back(pair_t(twin_[*klass], *e));
                else
                  if (part[*klass].size() < part[twin_[*klass]].size())
                    to_treat.push_back(pair_t(*klass, *e));
                  else
                    to_treat.push_back(pair_t(twin_[*klass], *e));
              }
            }
          }

          for_all_(std::list<unsigned>, klass, met_class_)
          {
            count_for_[*klass] = 0;
            twin_[*klass] = 0;
          }
          met_class_.clear();
        }

      private:
        const automaton_t& input_;
        const alphabet_t& alphabet_;
        class_of_t& class_;
        std::vector<unsigned> count_for_;
        std::vector<unsigned> twin_;
        going_in_t going_in_;
        std::list<unsigned> met_class_;
      };
    }
  }

  template <typename A, typename AI1, typename AI2>
  void
  do_quotient(const AutomataBase<A>&,
              const algebra::NumericalSemiring&,
              SELECTOR(bool),
              Element<A, AI2>&          output,
              const Element<A, AI1>&    input)
  {
    typedef Element<A, AI1> input_t;
    typedef Element<A, AI2> output_t;
    AUTOMATON_TYPES(input_t);
    AUTOMATON_FREEMONOID_TYPES(input_t);
    QUOTIENT_TYPES();

    using namespace internal::hopcroft_minimization_undet;

    const alphabet_t& alphabet_(input.series().monoid().alphabet());
    unsigned max_states = 0;

    for_all_const_states(i, input)
      max_states = std::max(unsigned(*i), max_states);
    ++max_states;

    /*--------------------------.
    | To label the subsets of Q |
    `--------------------------*/
    unsigned max_partitions = 0;

    /*-----------------------------------------.
    | To manage efficiently the partition of Q |
    `-----------------------------------------*/
    class_of_t          class_(max_states);
    partition_set_t     part(max_states);

    /*-------------------------.
    | To have a list of (P, a) |
    `-------------------------*/
    to_treat_t          to_treat;

    /*-------------------------.
    | Initialize the partition |
    `-------------------------*/

    // In the general case, we have two sets, part[0] and part[1] One
    // holds the final states, the other the non-final states.  In
    // some cases we may have only one set, for instance if we have no
    // final states, or if we have only final states.  Because we do
    // not want to have an empty part[0], we will fill it with the
    // first kind of state (final / non-final) we encounter.
    unsigned final = 1;

    for_all_const_states (p, input)
    {
      unsigned c;
      if (max_partitions == 0)
        {
          c = 0;
          final = !input.is_final(*p);
        }
      else
        {
          c = input.is_final(*p) ? final : (1 - final);
        }
      class_[*p] = c;
      part[c].insert(part[c].end(), *p);
      max_partitions = std::max(max_partitions, c + 1);
    }

    /*------------------------------.
    | Initialize the list of (P, a) |
    `------------------------------*/

    if (max_partitions > 0)
      for_all_const_letters (e, alphabet_)
        to_treat.push_back(pair_t(0, *e));

    if (max_partitions > 1)
      for_all_const_letters (e, alphabet_)
        to_treat.push_back(pair_t(1, *e));

    /*----------.
    | Main loop |
    `----------*/
    {
      quotient_splitter<input_t> splitter(input, class_, max_states);
      while (!to_treat.empty())
      {
        pair_t c = to_treat.front();
        to_treat.pop_front();
        unsigned p = c.first;
        letter_t a = c.second;

        if (!splitter.compute_going_in_states(part[p], a))
          continue;
        splitter.split(part, max_partitions);

        splitter.add_new_partitions(to_treat, part);
      }
    }

    /*------------------------------------.
    | Construction of the ouput automaton |
    `------------------------------------*/
    // Create the states
    for (unsigned i = 0; i < max_partitions; ++i)
      output.add_state();

    delta_ret_t delta_ret;
    std::set<unsigned> already_linked;
    for (unsigned i = 0; i < max_partitions; ++i)
    {
      // Get the first state of the partition => each state has the
      // same behaviour
      hstate_t s = part[i].front();

      if (input.is_final(s))
        output.set_final(i);

      // Create the transitions
      for_all_const_letters (e, alphabet_)
      {
        delta_ret.clear();
        already_linked.clear();

        input.letter_deltac(delta_ret, s, *e, delta_kind::states());
        for_all_(delta_ret_t, out, delta_ret)
        {
          unsigned c = class_[*out];
          if (already_linked.find(c) == already_linked.end())
          {
            already_linked.insert(c);
            output.add_letter_transition(i, c, *e);
          }
        }
      }
    }

    // Set initial states.
    for_all_const_initial_states(i, input)
      output.set_initial(class_[*i]);
  }

# undef QUOTIENT_TYPES


  /*----------------------------------------.
  | Quotient with multiplicities (covering) |
  `----------------------------------------*/

  template <class S, class T,
            typename A, typename input_t, typename output_t>
  void
  do_quotient(const AutomataBase<A>&    ,
              const S&                  ,
              const T&                  ,
              output_t&                 output,
              const input_t&            input)
  {
    AUTOMATON_TYPES(input_t);
    AUTOMATON_FREEMONOID_TYPES(input_t);
    using namespace std;

    /*----------------------------------------.
    | Declare data structures and variables.  |
    `----------------------------------------*/

    typedef set<htransition_t>                       set_transitions_t;
    typedef set<hstate_t>                      set_states_t;
    typedef set<semiring_elt_t>                set_semiring_elt_t;
    typedef vector<semiring_elt_t>             vector_semiring_elt_t;
    typedef pair<unsigned, letter_t>           pair_class_letter_t;
    typedef pair<hstate_t, semiring_elt_t>     pair_state_semiring_elt_t;
    typedef set<pair_state_semiring_elt_t>     set_pair_state_semiring_elt_t;
    typedef map<semiring_elt_t, unsigned>      map_semiring_elt_t;

    series_set_elt_t    null_series     = input.series().zero_;
    semiring_elt_t      weight_zero     = input.series().semiring().wzero_;
    monoid_elt_t        monoid_identity = input.series().monoid().VCSN_EMPTY_;
    const alphabet_t&   alphabet (input.series().monoid().alphabet());

    queue<pair_class_letter_t>                          the_queue;

    set<unsigned>       met_classes;
    set_transitions_t           transitions_leaving;

    unsigned    max_partition   = 0;
    //     unsigned     max_letters     = alphabet.size();
    unsigned    max_states      = 0;

    for_all_const_states(q, input)
      max_states = std::max(unsigned (*q), max_states);
    ++max_states;
    // Avoid special case problem (one initial and final state...)
    max_states = std::max(max_states, 2u);

    vector< vector<set_pair_state_semiring_elt_t> > inverse (max_states);

    map<letter_t, unsigned> pos_of_letter;

    unsigned max_pos = 0;

    for_all_const_letters(a, alphabet)
      pos_of_letter[*a] = max_pos++;

    set_states_t                states_visited;
    set_semiring_elt_t          semiring_had_class;
    vector<set_states_t>        classes (max_states);
    vector<unsigned>            class_of_state (max_states);
    vector_semiring_elt_t       old_weight (max_states);
    map_semiring_elt_t          class_of_weight;

    for (unsigned i = 0; i < max_states; ++i)
      inverse[i].resize(max_pos);

    for_all_const_states(q, input)
      for_all_const_letters(a, alphabet)
      {

        for_all_const_(set_states_t, r, states_visited)
          old_weight[*r] = weight_zero;
        states_visited.clear();

        set_transitions_t transitions_comming;
        input.letter_rdeltac(transitions_comming, *q, *a,
                             delta_kind::transitions());

        for_all_const_(set_transitions_t, e, transitions_comming)
          {
            hstate_t            p = input.src_of(*e);
            if (states_visited.find(p) != states_visited.end())
              inverse[*q][pos_of_letter[*a]].
                erase(pair_state_semiring_elt_t (p, old_weight[p]));
            else
              states_visited.insert(p);
            series_set_elt_t    sd = input.series_of(*e);
            monoid_elt_t        md (input.structure().series().monoid(), *a);
            semiring_elt_t      wsd = sd.get(md);
            old_weight[p] += wsd;
            inverse[*q][pos_of_letter[*a]].
              insert(pair_state_semiring_elt_t (p, old_weight[p]));
          }
      }

    /*-----------------------------------------------------------.
    | Initialize the partition with as many classes as there are |
    | final values.                                              |
    `-----------------------------------------------------------*/

    bool         empty = true;
    unsigned     class_non_final (0);

    for_all_const_states(q, input)
      {
        if (not input.is_final(*q))
        {
          if (empty == true)
          {
            empty = false;
            class_non_final = max_partition;
            max_partition++;
          }
          classes[class_non_final].insert(*q);
          class_of_state[*q] = class_non_final;
        }
        else
        {
          semiring_elt_t w = input.get_final(*q).get(monoid_identity);
          if (semiring_had_class.find(w) == semiring_had_class.end())
          {
            semiring_had_class.insert(w);
            classes[max_partition].insert(*q);
            class_of_weight[w] = max_partition;
            class_of_state[*q] = max_partition;
            max_partition++;
          }
          else
          {
            classes[class_of_weight[w]].insert(*q);
            class_of_state[*q] = class_of_weight[w];
          }
        }
      }

    /*-----------------------------------------------------.
    | Initialize the queue with pairs <class_id, letter>.  |
    `-----------------------------------------------------*/

    for (unsigned i = 0; i < max_partition; i++)
      for_all_const_letters(a, alphabet)
        the_queue.push(pair_class_letter_t (i, *a));

    /*----------------.
    | The main loop.  |
    `----------------*/

    unsigned old_max_partition = max_partition;

    while(not the_queue.empty())
    {
      pair_class_letter_t pair = the_queue.front();
      the_queue.pop();
      met_classes.clear();
      vector_semiring_elt_t val (max_states);

      // FIXME: Isn't there an easier way to do this?
      for_all_const_states(q, input)
        val[*q] = 0;

      // First, calculate val[state] and not met_classes.
      for_all_const_(set_states_t, q, classes[pair.first])
        for_all_const_(set_pair_state_semiring_elt_t, pair_,
                       inverse[*q][pos_of_letter[pair.second]])
        {
          unsigned  state = pair_->first;
          if (met_classes.find(class_of_state[state]) ==
              met_classes.end())
            met_classes.insert(class_of_state[state]);
          val[state] += pair_->second;
        }

      // Next, for each met class, do the partition.
      for_all_const_(set<unsigned>, class_id, met_classes)
      {
        if (classes[*class_id].size() == 1)
          continue ;

        queue<hstate_t> to_erase;
        semiring_elt_t  next_val;
        semiring_elt_t  first_val = val[*(classes[*class_id].begin())];
        class_of_weight.clear();
        semiring_had_class.clear();

        for_all_const_(set_states_t, p, classes[*class_id])
        {
          next_val = val[*p];
          // This state must be moved to another class!
          if (next_val != first_val)
          {
            if (semiring_had_class.find(next_val) ==
                semiring_had_class.end()) // Must create a new class
            {
              classes[max_partition].insert(*p);
              class_of_state[*p] = max_partition;
              semiring_had_class.insert(next_val);
              class_of_weight[next_val] = max_partition;
              max_partition++;
            }
            else
            {
              classes[class_of_weight[next_val]].insert(*p);
              class_of_state[*p] = class_of_weight[next_val];
            }
            to_erase.push(*p);
          }
        }

        while(not to_erase.empty())
        {
          hstate_t s = to_erase.front();
          to_erase.pop();
          classes[*class_id].erase(s);
        }

        // Push pairs <new_class_id, letter> into the queue.
        for (unsigned i = old_max_partition; i < max_partition; i++)
          for_all_const_letters(b, alphabet)
            the_queue.push(pair_class_letter_t(i, *b));
        old_max_partition = max_partition;
      }
    }

    /*------------------.
    | Form the output.  |
    `------------------*/

    typedef vector<series_set_elt_t> vector_series_set_elt_t;

    std::vector<hstate_t>       out_states (max_partition);

    // typedef map<unsigned, series_set_elt_t> map_class_series_elt_t;
    // map_class_series_elt_t   seriesof;

    // Add states.
    for(unsigned i = 0; i < max_partition; i++)
    {
      out_states[i]  = output.add_state();
      hstate_t a_state = *classes[i].begin();
      series_set_elt_t a_serie = null_series;

      for_all_const_(set_states_t, state, classes[i])
        if(input.is_initial(*state))
          a_serie += input.get_initial(*state);

      output.set_initial(out_states[i] , a_serie);

      if (input.is_final(a_state))
        output.set_final(out_states[i] , input.get_final(a_state));
    }

    // Add transitions.
    vector_series_set_elt_t seriesof (max_partition, null_series);

    for(unsigned i = 0; i < max_partition; i++)
    {
      met_classes.clear();

      transitions_leaving.clear();
      input.deltac(transitions_leaving, *classes[i].begin(),
                   delta_kind::transitions());

      for_all_const_(set_transitions_t, e, transitions_leaving)
        {
          series_set_elt_t      se = input.series_of(*e);
          unsigned              cs = class_of_state[input.dst_of(*e)];

          if (met_classes.find(cs) == met_classes.end())
          {
            met_classes.insert(cs);
            seriesof[cs] = se;
          }
          else
            seriesof[cs] += se;
        }

      for_all_const_(set<unsigned>, cs, met_classes)
        output.add_series_transition(out_states[i],
                                     out_states[*cs],
                                     seriesof[*cs]);
    }
  }

  template<typename A, typename AI>
  Element<A, AI>
  quotient(const Element<A, AI>& a)
  {
    TIMER_SCOPED ("quotient");
    precondition(is_realtime(a));
    typedef Element<A, AI> automaton_t;
    AUTOMATON_TYPES(automaton_t);
    automaton_t output(a.structure());
    do_quotient(a.structure(), a.structure().series().semiring(),
                SELECT(semiring_elt_value_t), output, a);
    return output;
  }

} // vcsn

#endif // ! VCSN_ALGORITHMS_MINIMIZATION_HOPCROFT_HXX

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